3.1.43 \(\int \frac {x^2}{a+b \text {sech}(c+d \sqrt {x})} \, dx\) [43]

3.1.43.1 Optimal result

Integrand size = 20, antiderivative size = 721 \[ \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\frac {x^3}{3 a}-\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {2 b x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {10 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {120 b x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {120 b x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {240 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {240 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6} \]

output

1/3*x^3/a-2*b*x^(5/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d/(- 
a^2+b^2)^(1/2)+2*b*x^(5/2)*ln(1+a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a 
/d/(-a^2+b^2)^(1/2)-10*b*x^2*polylog(2,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^( 
1/2)))/a/d^2/(-a^2+b^2)^(1/2)+10*b*x^2*polylog(2,-a*exp(c+d*x^(1/2))/(b+(- 
a^2+b^2)^(1/2)))/a/d^2/(-a^2+b^2)^(1/2)+40*b*x^(3/2)*polylog(3,-a*exp(c+d* 
x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d^3/(-a^2+b^2)^(1/2)-40*b*x^(3/2)*polylog 
(3,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a/d^3/(-a^2+b^2)^(1/2)-120*b* 
x*polylog(4,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d^4/(-a^2+b^2)^(1/ 
2)+120*b*x*polylog(4,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2)))/a/d^4/(-a^2 
+b^2)^(1/2)-240*b*polylog(6,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2)^(1/2)))/a/d^ 
6/(-a^2+b^2)^(1/2)+240*b*polylog(6,-a*exp(c+d*x^(1/2))/(b+(-a^2+b^2)^(1/2) 
))/a/d^6/(-a^2+b^2)^(1/2)+240*b*polylog(5,-a*exp(c+d*x^(1/2))/(b-(-a^2+b^2 
)^(1/2)))*x^(1/2)/a/d^5/(-a^2+b^2)^(1/2)-240*b*polylog(5,-a*exp(c+d*x^(1/2 
))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a/d^5/(-a^2+b^2)^(1/2)
 

3.1.43.2 Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 547, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\frac {\sqrt {-a^2+b^2} d^6 x^3-6 b d^5 x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {-a^2+b^2}}\right )+6 b d^5 x^{5/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )-30 b d^4 x^2 \operatorname {PolyLog}\left (2,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )+30 b d^4 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )+120 b d^3 x^{3/2} \operatorname {PolyLog}\left (3,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )-120 b d^3 x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )-360 b d^2 x \operatorname {PolyLog}\left (4,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )+360 b d^2 x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )+720 b d \sqrt {x} \operatorname {PolyLog}\left (5,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )-720 b d \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )-720 b \operatorname {PolyLog}\left (6,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {-a^2+b^2}}\right )+720 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {-a^2+b^2}}\right )}{3 a \sqrt {-a^2+b^2} d^6} \]

input

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

output

(Sqrt[-a^2 + b^2]*d^6*x^3 - 6*b*d^5*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/ 
(b - Sqrt[-a^2 + b^2])] + 6*b*d^5*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b 
 + Sqrt[-a^2 + b^2])] - 30*b*d^4*x^2*PolyLog[2, (a*E^(c + d*Sqrt[x]))/(-b 
+ Sqrt[-a^2 + b^2])] + 30*b*d^4*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b 
+ Sqrt[-a^2 + b^2]))] + 120*b*d^3*x^(3/2)*PolyLog[3, (a*E^(c + d*Sqrt[x])) 
/(-b + Sqrt[-a^2 + b^2])] - 120*b*d^3*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqr 
t[x]))/(b + Sqrt[-a^2 + b^2]))] - 360*b*d^2*x*PolyLog[4, (a*E^(c + d*Sqrt[ 
x]))/(-b + Sqrt[-a^2 + b^2])] + 360*b*d^2*x*PolyLog[4, -((a*E^(c + d*Sqrt[ 
x]))/(b + Sqrt[-a^2 + b^2]))] + 720*b*d*Sqrt[x]*PolyLog[5, (a*E^(c + d*Sqr 
t[x]))/(-b + Sqrt[-a^2 + b^2])] - 720*b*d*Sqrt[x]*PolyLog[5, -((a*E^(c + d 
*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))] - 720*b*PolyLog[6, (a*E^(c + d*Sqrt[x] 
))/(-b + Sqrt[-a^2 + b^2])] + 720*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b 
+ Sqrt[-a^2 + b^2]))])/(3*a*Sqrt[-a^2 + b^2]*d^6)
 

3.1.43.3 Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5959, 3042, 4679, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx\)

\(\Big \downarrow \) 5959

\(\displaystyle 2 \int \frac {x^{5/2}}{a+b \text {sech}\left (c+d \sqrt {x}\right )}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{5/2}}{a+b \csc \left (i c+i d \sqrt {x}+\frac {\pi }{2}\right )}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (\frac {x^{5/2}}{a}-\frac {b x^{5/2}}{a \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {120 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^6 \sqrt {b^2-a^2}}+\frac {120 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^6 \sqrt {b^2-a^2}}+\frac {120 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^5 \sqrt {b^2-a^2}}-\frac {120 b \sqrt {x} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^5 \sqrt {b^2-a^2}}-\frac {60 b x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {60 b x \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {20 b x^{3/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {5 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {5 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {b x^{5/2} \log \left (\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {b^2-a^2}}+1\right )}{a d \sqrt {b^2-a^2}}+\frac {b x^{5/2} \log \left (\frac {a e^{c+d \sqrt {x}}}{\sqrt {b^2-a^2}+b}+1\right )}{a d \sqrt {b^2-a^2}}+\frac {x^3}{6 a}\right )\)

input

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

output

2*(x^3/(6*a) - (b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b 
^2])])/(a*Sqrt[-a^2 + b^2]*d) + (b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/( 
b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (5*b*x^2*PolyLog[2, -((a* 
E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (5 
*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqr 
t[-a^2 + b^2]*d^2) + (20*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - 
 Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (20*b*x^(3/2)*PolyLog[3, 
-((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) 
 - (60*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a 
*Sqrt[-a^2 + b^2]*d^4) + (60*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + S 
qrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (120*b*Sqrt[x]*PolyLog[5, - 
((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) 
- (120*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]) 
)])/(a*Sqrt[-a^2 + b^2]*d^5) - (120*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/( 
b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + (120*b*PolyLog[6, -((a 
*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6))
 

3.1.43.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

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*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 5959

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(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[Simplify[(m 
 + 1)/n], 0] && IntegerQ[p]
 

3.1.43.4 Maple [F]

\[\int \frac {x^{2}}{a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )}d x\]

input

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

output

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

3.1.43.5 Fricas [F]

\[ \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{2}}{b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a} \,d x } \]

input

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

output

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

3.1.43.6 Sympy [F]

\[ \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^{2}}{a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}}\, dx \]

input

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

output

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

3.1.43.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\text {Exception raised: ValueError} \]

input

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

output

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

3.1.43.8 Giac [F]

\[ \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{2}}{b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a} \,d x } \]

input

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

output

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

3.1.43.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{a+b \text {sech}\left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^2}{a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}} \,d x \]

input

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

output

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