3.1.67 \(\int \frac {x^{3/2}}{(a+b \text {sech}(c+d \sqrt {x}))^2} \, dx\) [67]

Optimal. Leaf size=1755 \[ \frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {24 b^2 x \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {8 b^3 x^{3/2} \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {16 b x^{3/2} \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {24 b^2 x \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {8 b^3 x^{3/2} \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {16 b x^{3/2} \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {48 b^2 \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {24 b^3 x \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {48 b x \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {48 b^2 \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac {24 b^3 x \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {48 b x \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {48 b^2 \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac {48 b^3 \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac {96 b \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}-\frac {48 b^2 \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac {48 b^3 \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac {96 b \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}-\frac {48 b^3 \text {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac {96 b \text {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}+\frac {48 b^3 \text {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac {96 b \text {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )} \]

[Out]

-8*b^2*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^2+2*b^3*x^2*ln(1+a*exp(c+d*x^(1/2
))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d-8*b^2*x^(3/2)*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^
2/(a^2-b^2)/d^2-2*b^3*x^2*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d-24*b^2*x*polylo
g(2,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^3+8*b^3*x^(3/2)*polylog(2,-a*exp(c+d*x^(1/2))/(b
-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-24*b^2*x*polylog(2,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/
(a^2-b^2)/d^3-8*b^3*x^(3/2)*polylog(2,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-24*b^
3*x*polylog(3,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3+24*b^3*x*polylog(3,-a*exp(c+d
*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3-4*b*x^2*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))
/a^2/d/(-a^2+b^2)^(1/2)+4*b*x^2*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/d/(-a^2+b^2)^(1/2)-16*b*x^(3
/2)*polylog(2,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/(-a^2+b^2)^(1/2)+16*b*x^(3/2)*polylog(2,-a*exp
(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/d^2/(-a^2+b^2)^(1/2)+48*b*x*polylog(3,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2
)^(1/2)))/a^2/d^3/(-a^2+b^2)^(1/2)-48*b*x*polylog(3,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b^
2)^(1/2)+48*b^2*polylog(3,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2-b^2)/d^4+48*b^2*polylog(3
,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2-b^2)/d^4+48*b^3*polylog(4,-a*exp(c+d*x^(1/2))/(b-(
-a^2+b^2)^(1/2)))*x^(1/2)/a^2/(-a^2+b^2)^(3/2)/d^4-48*b^3*polylog(4,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))*
x^(1/2)/a^2/(-a^2+b^2)^(3/2)/d^4-96*b*polylog(4,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^4/(-a^
2+b^2)^(1/2)+96*b*polylog(4,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^4/(-a^2+b^2)^(1/2)+2/5*x^(
5/2)/a^2+2*b^2*x^2*sinh(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*cosh(c+d*x^(1/2)))-48*b^2*polylog(4,-a*exp(c+d*x^(1/2)
)/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^5-48*b^2*polylog(4,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(a^2-
b^2)/d^5-48*b^3*polylog(5,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^5+48*b^3*polylog(5,
-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^5+96*b*polylog(5,-a*exp(c+d*x^(1/2))/(b-(-a^2
+b^2)^(1/2)))/a^2/d^5/(-a^2+b^2)^(1/2)-96*b*polylog(5,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a^2/d^5/(-a^2+
b^2)^(1/2)+2*b^2*x^2/a^2/(a^2-b^2)/d

________________________________________________________________________________________

Rubi [A]
time = 1.90, antiderivative size = 1755, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5544, 4276, 3405, 3401, 2296, 2221, 2611, 6744, 2320, 6724, 5681} \begin {gather*} \frac {2 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {2 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {8 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {8 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {24 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {24 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {48 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {48 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {48 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^5}+\frac {48 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^5}+\frac {2 x^2 b^2}{a^2 \left (a^2-b^2\right ) d}-\frac {8 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {8 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {24 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac {24 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {48 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {48 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}-\frac {48 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}-\frac {48 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^5}+\frac {2 x^2 \sinh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {4 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {4 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {16 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {16 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {48 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {48 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {96 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {96 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {96 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}-\frac {96 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^5}+\frac {2 x^{5/2}}{5 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(a + b*Sech[c + d*Sqrt[x]])^2,x]

[Out]

(2*b^2*x^2)/(a^2*(a^2 - b^2)*d) + (2*x^(5/2))/(5*a^2) - (8*b^2*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt
[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2
*(-a^2 + b^2)^(3/2)*d) - (4*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]
*d) - (8*b^2*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x^2
*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x^2*Log[1 + (a*E^(c
+ d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (24*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))
/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt
[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (16*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^
2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (24*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))
])/(a^2*(a^2 - b^2)*d^3) - (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-
a^2 + b^2)^(3/2)*d^2) + (16*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-
a^2 + b^2]*d^2) + (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2
)*d^4) - (24*b^3*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) +
(48*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (48*b^2*Sqr
t[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (24*b^3*x*PolyLog[3,
-((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (48*b*x*PolyLog[3, -((a*E^(c
+ d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (48*b^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]
))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) + (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - S
qrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (96*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[
-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (48*b^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])
)])/(a^2*(a^2 - b^2)*d^5) - (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*
(-a^2 + b^2)^(3/2)*d^4) + (96*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt
[-a^2 + b^2]*d^4) - (48*b^3*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/
2)*d^5) + (96*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (48*
b^3*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - (96*b*PolyLog[
5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (2*b^2*x^2*Sinh[c + d*Sqrt[x
]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3401

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*(
(-I)*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 5544

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 5681

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {x^{3/2}}{\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^4}{(a+b \text {sech}(c+d x))^2} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {x^4}{a^2}+\frac {b^2 x^4}{a^2 (b+a \cosh (c+d x))^2}-\frac {2 b x^4}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^{5/2}}{5 a^2}-\frac {(4 b) \text {Subst}\left (\int \frac {x^4}{b+a \cosh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {x^4}{(b+a \cosh (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=\frac {2 x^{5/2}}{5 a^2}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {(8 b) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {x^4}{b+a \cosh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^3 \sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,\sqrt {x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}-\frac {(8 b) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2}}+\frac {(8 b) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2}}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^3}{b-\sqrt {-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2-b^2\right ) d}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^3}{b+\sqrt {-a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac {\left (4 b^3\right ) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac {\left (24 b^2\right ) \text {Subst}\left (\int x^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {\left (24 b^2\right ) \text {Subst}\left (\int x^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {(16 b) \text {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {(16 b) \text {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (48 b^2\right ) \text {Subst}\left (\int x \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {\left (48 b^2\right ) \text {Subst}\left (\int x \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {(48 b) \text {Subst}\left (\int x^2 \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {(48 b) \text {Subst}\left (\int x^2 \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {\left (8 b^3\right ) \text {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {\left (8 b^3\right ) \text {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (48 b^2\right ) \text {Subst}\left (\int \text {Li}_3\left (-\frac {a e^{c+d x}}{b-\sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {\left (48 b^2\right ) \text {Subst}\left (\int \text {Li}_3\left (-\frac {a e^{c+d x}}{b+\sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {(96 b) \text {Subst}\left (\int x \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {(96 b) \text {Subst}\left (\int x \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {\left (24 b^3\right ) \text {Subst}\left (\int x^2 \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {\left (24 b^3\right ) \text {Subst}\left (\int x^2 \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}+\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (48 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac {\left (48 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac {(96 b) \text {Subst}\left (\int \text {Li}_4\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^4}-\frac {(96 b) \text {Subst}\left (\int \text {Li}_4\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {-a^2+b^2} d^4}+\frac {\left (48 b^3\right ) \text {Subst}\left (\int x \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {\left (48 b^3\right ) \text {Subst}\left (\int x \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {48 b^2 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac {48 b^3 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}-\frac {48 b^2 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac {48 b^3 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}+\frac {(96 b) \text {Subst}\left (\int \frac {\text {Li}_4\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}-\frac {(96 b) \text {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}-\frac {\left (48 b^3\right ) \text {Subst}\left (\int \text {Li}_4\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac {\left (48 b^3\right ) \text {Subst}\left (\int \text {Li}_4\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {48 b^2 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac {48 b^3 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}-\frac {48 b^2 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac {48 b^3 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}+\frac {96 b \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}-\frac {96 b \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (48 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_4\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac {\left (48 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}\\ &=\frac {2 b^2 x^2}{a^2 \left (a^2-b^2\right ) d}+\frac {2 x^{5/2}}{5 a^2}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {8 b^2 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^3 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {4 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {24 b^2 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {8 b^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {16 b x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}-\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {48 b^2 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^4}+\frac {24 b^3 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {48 b x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {48 b^2 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}+\frac {48 b^3 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}-\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}-\frac {48 b^2 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^5}-\frac {48 b^3 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^4}+\frac {96 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^4}-\frac {48 b^3 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}+\frac {96 b \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}+\frac {48 b^3 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^5}-\frac {96 b \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^5}+\frac {2 b^2 x^2 \sinh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 9.99, size = 1769, normalized size = 1.01 \begin {gather*} \frac {2 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right ) \text {sech}^2\left (c+d \sqrt {x}\right ) \left (x^{5/2} \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )+\frac {5 b e^c \left (b+a \cosh \left (c+d \sqrt {x}\right )\right ) \left (2 b e^c x^2-\frac {e^{-c} \left (1+e^{2 c}\right ) \left (4 b d^3 \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^{3/2} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+2 a^2 d^4 e^c x^2 \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-b^2 d^4 e^c x^2 \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+4 b d^3 \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^{3/2} \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 a^2 d^4 e^c x^2 \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+b^2 d^4 e^c x^2 \log \left (1+\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+4 d^2 \left (3 b \sqrt {\left (-a^2+b^2\right ) e^{2 c}}+2 a^2 d e^c \sqrt {x}-b^2 d e^c \sqrt {x}\right ) x \text {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+4 d^2 \left (3 b \sqrt {\left (-a^2+b^2\right ) e^{2 c}}-2 a^2 d e^c \sqrt {x}+b^2 d e^c \sqrt {x}\right ) x \text {PolyLog}\left (2,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-24 b d \sqrt {\left (-a^2+b^2\right ) e^{2 c}} \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-24 a^2 d^2 e^c x \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+12 b^2 d^2 e^c x \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-24 b d \sqrt {\left (-a^2+b^2\right ) e^{2 c}} \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+24 a^2 d^2 e^c x \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-12 b^2 d^2 e^c x \text {PolyLog}\left (3,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+24 b \sqrt {\left (-a^2+b^2\right ) e^{2 c}} \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+48 a^2 d e^c \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-24 b^2 d e^c \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+24 b \sqrt {\left (-a^2+b^2\right ) e^{2 c}} \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-48 a^2 d e^c \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+24 b^2 d e^c \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-48 a^2 e^c \text {PolyLog}\left (5,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+24 b^2 e^c \text {PolyLog}\left (5,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+48 a^2 e^c \text {PolyLog}\left (5,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-24 b^2 e^c \text {PolyLog}\left (5,-\frac {a e^{2 c+d \sqrt {x}}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )\right )}{d^4 \sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )}{\left (a^2-b^2\right ) d \left (1+e^{2 c}\right )}+\frac {5 b^2 x^2 \text {sech}(c) \left (-b \sinh (c)+a \sinh \left (d \sqrt {x}\right )\right )}{(a-b) (a+b) d}\right )}{5 a^2 \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(a + b*Sech[c + d*Sqrt[x]])^2,x]

[Out]

(2*(b + a*Cosh[c + d*Sqrt[x]])*Sech[c + d*Sqrt[x]]^2*(x^(5/2)*(b + a*Cosh[c + d*Sqrt[x]]) + (5*b*E^c*(b + a*Co
sh[c + d*Sqrt[x]])*(2*b*E^c*x^2 - ((1 + E^(2*c))*(4*b*d^3*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^(3/2)*Log[1 + (a*E^(2*c
+ d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)])] + 2*a^2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^
c - Sqrt[(-a^2 + b^2)*E^(2*c)])] - b^2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*
E^(2*c)])] + 4*b*d^3*Sqrt[(-a^2 + b^2)*E^(2*c)]*x^(3/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 +
b^2)*E^(2*c)])] - 2*a^2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + b^
2*d^4*E^c*x^2*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])] + 4*d^2*(3*b*Sqrt[(-a^2 +
b^2)*E^(2*c)] + 2*a^2*d*E^c*Sqrt[x] - b^2*d*E^c*Sqrt[x])*x*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[
(-a^2 + b^2)*E^(2*c)]))] + 4*d^2*(3*b*Sqrt[(-a^2 + b^2)*E^(2*c)] - 2*a^2*d*E^c*Sqrt[x] + b^2*d*E^c*Sqrt[x])*x*
PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 24*b*d*Sqrt[(-a^2 + b^2)*E^(2*c)
]*Sqrt[x]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 24*a^2*d^2*E^c*x*PolyL
og[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 12*b^2*d^2*E^c*x*PolyLog[3, -((a*E^(2
*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 24*b*d*Sqrt[(-a^2 + b^2)*E^(2*c)]*Sqrt[x]*PolyLog[3,
-((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 24*a^2*d^2*E^c*x*PolyLog[3, -((a*E^(2*c +
d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 12*b^2*d^2*E^c*x*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*
E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 24*b*Sqrt[(-a^2 + b^2)*E^(2*c)]*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*
E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 48*a^2*d*E^c*Sqrt[x]*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[
(-a^2 + b^2)*E^(2*c)]))] - 24*b^2*d*E^c*Sqrt[x]*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2
)*E^(2*c)]))] + 24*b*Sqrt[(-a^2 + b^2)*E^(2*c)]*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2
)*E^(2*c)]))] - 48*a^2*d*E^c*Sqrt[x]*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)])
)] + 24*b^2*d*E^c*Sqrt[x]*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 48*a^2
*E^c*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 24*b^2*E^c*PolyLog[5, -((a*
E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(-a^2 + b^2)*E^(2*c)]))] + 48*a^2*E^c*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/
(b*E^c + Sqrt[(-a^2 + b^2)*E^(2*c)]))] - 24*b^2*E^c*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(-a^2 +
b^2)*E^(2*c)]))]))/(d^4*E^c*Sqrt[(-a^2 + b^2)*E^(2*c)])))/((a^2 - b^2)*d*(1 + E^(2*c))) + (5*b^2*x^2*Sech[c]*
(-(b*Sinh[c]) + a*Sinh[d*Sqrt[x]]))/((a - b)*(a + b)*d)))/(5*a^2*(a + b*Sech[c + d*Sqrt[x]])^2)

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Maple [F]
time = 3.72, size = 0, normalized size = 0.00 \[\int \frac {x^{\frac {3}{2}}}{\left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x)

[Out]

int(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-b>0)', see `assume?` for mor
e details)Is

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(x^(3/2)/(b^2*sech(d*sqrt(x) + c)^2 + 2*a*b*sech(d*sqrt(x) + c) + a^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3}{2}}}{\left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)/(a+b*sech(c+d*x**(1/2)))**2,x)

[Out]

Integral(x**(3/2)/(a + b*sech(c + d*sqrt(x)))**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b*sech(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate(x^(3/2)/(b*sech(d*sqrt(x) + c) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{3/2}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(a + b/cosh(c + d*x^(1/2)))^2,x)

[Out]

int(x^(3/2)/(a + b/cosh(c + d*x^(1/2)))^2, x)

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