3.1.66 \(\int \frac {x^{3/2}}{(a+b \text {csch}(c+d \sqrt {x}))^2} \, dx\) [66]

Optimal. Leaf size=1639 \[ -\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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )} \]

[Out]

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+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)+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^5-2*b^2*x^2/a^2/(a^2+b^2)/d+2/5*x^(5/2)/a^2-2*b^2*x^2*cosh(c+d*x^(1/2))/a/(a
^2+b^2)/d/(b+a*sinh(c+d*x^(1/2)))+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)*polyl
og(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*ex
p(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)+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*polylog(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.83, antiderivative size = 1639, 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 = {5545, 4276, 3405, 3403, 2296, 2221, 2611, 6744, 2320, 6724, 5680} \begin {gather*} \frac {2 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {8 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {8 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {24 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {24 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {48 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {48 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {48 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^5}+\frac {48 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^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 {a^2+b^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 {a^2+b^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 {a^2+b^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 {a^2+b^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 {a^2+b^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 {a^2+b^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 {a^2+b^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 {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^5}-\frac {2 x^2 \cosh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {4 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}+\frac {4 x^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}-\frac {16 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {16 x^{3/2} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {48 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {48 x \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {96 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {96 \sqrt {x} \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {96 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^5}-\frac {96 \text {Li}_5\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^5}+\frac {2 x^{5/2}}{5 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(a + b*Csch[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 - Sqr
t[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*Sqr
t[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 - Sqr
t[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*P
olyLog[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*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*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^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*Cosh[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(b + a*Sinh[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 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && 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 5545

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[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 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), 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 {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^4}{(a+b \text {csch}(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 \sinh (c+d x))^2}-\frac {2 b x^4}{a^2 (b+a \sinh (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 \sinh (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 \sinh (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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \sinh (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 \cosh (c+d x)}{b+a \sinh (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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \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 \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 12.72, size = 1682, normalized size = 1.03 \begin {gather*} \frac {2 \text {csch}^2\left (c+d \sqrt {x}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right ) \left (x^{5/2} \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )+\frac {5 b \left (-\frac {2 b d^4 e^{2 c} x^2}{-1+e^{2 c}}+\frac {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 )}{\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}{\left (a^2+b^2\right ) d^5}+\frac {5 b^2 x^2 \text {csch}(c) \left (b \cosh (c)+a \sinh \left (d \sqrt {x}\right )\right )}{\left (a^2+b^2\right ) d}\right )}{5 a^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Csch[c + d*Sqrt[x]]^2*(b + a*Sinh[c + d*Sqrt[x]])*(x^(5/2)*(b + a*Sinh[c + d*Sqrt[x]]) + (5*b*((-2*b*d^4*E^
(2*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*S
qrt[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*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*d*Sqrt[(a^2 + b^2)*E^(2*c)]*Sqrt[x]*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + S
qrt[(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*Sqr
t[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)]))])/Sqrt[(a^2 + b^2)*E^(2*c)])*(b + a
*Sinh[c + d*Sqrt[x]]))/((a^2 + b^2)*d^5) + (5*b^2*x^2*Csch[c]*(b*Cosh[c] + a*Sinh[d*Sqrt[x]]))/((a^2 + b^2)*d)
))/(5*a^2*(a + b*Csch[c + d*Sqrt[x]])^2)

________________________________________________________________________________________

Maple [F]
time = 6.02, size = 0, normalized size = 0.00 \[\int \frac {x^{\frac {3}{2}}}{\left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(10*a*b^2*x^2 - (a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^(5/2)*e^(2*d*sqrt(x)) + (a^3*d + a*b^2*d)*x^(5/2) - 2*
(5*b^3*x^2*e^c + (a^2*b*d*e^c + b^3*d*e^c)*x^(5/2))*e^(d*sqrt(x)))/(a^5*d + a^3*b^2*d - (a^5*d*e^(2*c) + a^3*b
^2*d*e^(2*c))*e^(2*d*sqrt(x)) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*e^(d*sqrt(x))) - integrate(-2*(4*a*b^2*x^2 - (
4*b^3*x^2*e^c + (2*a^2*b*d*e^c + b^3*d*e^c)*x^(5/2))*e^(d*sqrt(x)))/((a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x*e^(
2*d*sqrt(x)) + 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*x*e^(d*sqrt(x)) - (a^5*d + a^3*b^2*d)*x), x)

________________________________________________________________________________________

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*csch(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3}{2}}}{\left (a + b \operatorname {csch}{\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*csch(c+d*x**(1/2)))**2,x)

[Out]

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

________________________________________________________________________________________

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*csch(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________