3.1.23 \(\int \frac {x^5}{(a+b \text {csch}(c+d x^2))^2} \, dx\) [23]

3.1.23.1 Optimal result

Integrand size = 18, antiderivative size = 922 \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=-\frac {b^2 x^4}{2 a^2 \left (a^2+b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {b^3 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b^3 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {b^2 x^4 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )} \]

output

-1/2*b^2*x^4/a^2/(a^2+b^2)/d+1/6*x^6/a^2+b^2*x^2*ln(1+a*exp(d*x^2+c)/(b-(a 
^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2+1/2*b^3*x^4*ln(1+a*exp(d*x^2+c)/(b-(a^2+ 
b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d+b^2*x^2*ln(1+a*exp(d*x^2+c)/(b+(a^2+b^2 
)^(1/2)))/a^2/(a^2+b^2)/d^2-1/2*b^3*x^4*ln(1+a*exp(d*x^2+c)/(b+(a^2+b^2)^( 
1/2)))/a^2/(a^2+b^2)^(3/2)/d+b^2*polylog(2,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1 
/2)))/a^2/(a^2+b^2)/d^3+b^3*x^2*polylog(2,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/ 
2)))/a^2/(a^2+b^2)^(3/2)/d^2+b^2*polylog(2,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1 
/2)))/a^2/(a^2+b^2)/d^3-b^3*x^2*polylog(2,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/ 
2)))/a^2/(a^2+b^2)^(3/2)/d^2-b^3*polylog(3,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1 
/2)))/a^2/(a^2+b^2)^(3/2)/d^3+b^3*polylog(3,-a*exp(d*x^2+c)/(b+(a^2+b^2)^( 
1/2)))/a^2/(a^2+b^2)^(3/2)/d^3-1/2*b^2*x^4*cosh(d*x^2+c)/a/(a^2+b^2)/d/(b+ 
a*sinh(d*x^2+c))-b*x^4*ln(1+a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/d/(a^2 
+b^2)^(1/2)+b*x^4*ln(1+a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2) 
^(1/2)-2*b*x^2*polylog(2,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/d^2/(a^2 
+b^2)^(1/2)+2*b*x^2*polylog(2,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a^2/d^2 
/(a^2+b^2)^(1/2)+2*b*polylog(3,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/d^ 
3/(a^2+b^2)^(1/2)-2*b*polylog(3,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a^2/d 
^3/(a^2+b^2)^(1/2)
 

3.1.23.2 Mathematica [A] (verified)

Time = 4.08 (sec) , antiderivative size = 1502, normalized size of antiderivative = 1.63 \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx =\text {Too large to display} \]

input

Integrate[x^5/(a + b*Csch[c + d*x^2])^2,x]
 

output

(Csch[c + d*x^2]^2*(b + a*Sinh[c + d*x^2])*((6*b^2*x^4*Csch[c]*(b*Cosh[c] 
+ a*Sinh[d*x^2]))/((a^2 + b^2)*d) + 2*x^6*(b + a*Sinh[c + d*x^2]) - (6*b*E 
^(2*c)*(2*b*d^2*E^(2*c)*Sqrt[(a^2 + b^2)*E^(2*c)]*x^4 + 2*b*d*Sqrt[(a^2 + 
b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^ 
(2*c)])] - 2*b*d*E^(2*c)*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + 
 d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*a^2*d^2*E^c*x^4*Log[1 + 
(a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - b^2*d^2*E^c*x^4 
*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*a^2* 
d^2*E^(3*c)*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2 
*c)])] + b^2*d^2*E^(3*c)*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^ 
2 + b^2)*E^(2*c)])] + 2*b*d*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2* 
c + d*x^2))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*b*d*E^(2*c)*Sqrt[(a^2 
 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c + Sqrt[(a^2 + b^2) 
*E^(2*c)])] + 2*a^2*d^2*E^c*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c + Sqrt[ 
(a^2 + b^2)*E^(2*c)])] + b^2*d^2*E^c*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^ 
c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*a^2*d^2*E^(3*c)*x^4*Log[1 + (a*E^(2*c 
+ d*x^2))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - b^2*d^2*E^(3*c)*x^4*Log[1 
 + (a*E^(2*c + d*x^2))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*(-1 + E^(2 
*c))*(-(b*Sqrt[(a^2 + b^2)*E^(2*c)]) + 2*a^2*d*E^c*x^2 + b^2*d*E^c*x^2)*Po 
lyLog[2, -((a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - ...
 

3.1.23.3 Rubi [A] (verified)

Time = 2.30 (sec) , antiderivative size = 924, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5960, 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.

\(\displaystyle \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx\)

\(\Big \downarrow \) 5960

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \text {csch}\left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+i b \csc \left (i d x^2+i c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 4679

\(\displaystyle \frac {1}{2} \int \left (-\frac {2 b x^4}{a^2 \left (b+a \sinh \left (d x^2+c\right )\right )}+\frac {x^4}{a^2}+\frac {b^2 x^4}{a^2 \left (b+a \sinh \left (d x^2+c\right )\right )^2}\right )dx^2\)

\(\Big \downarrow \) 2009

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

input

Int[x^5/(a + b*Csch[c + d*x^2])^2,x]
 

output

(-((b^2*x^4)/(a^2*(a^2 + b^2)*d)) + x^6/(3*a^2) + (2*b^2*x^2*Log[1 + (a*E^ 
(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (b^3*x^4*Log[ 
1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - 
(2*b*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + 
 b^2]*d) + (2*b^2*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(a 
^2*(a^2 + b^2)*d^2) - (b^3*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b 
^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (2*b*x^4*Log[1 + (a*E^(c + d*x^2))/(b + 
 Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (2*b^2*PolyLog[2, -((a*E^(c 
+ d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (2*b^3*x^2*Poly 
Log[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2) 
*d^2) - (4*b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/( 
a^2*Sqrt[a^2 + b^2]*d^2) + (2*b^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt 
[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (2*b^3*x^2*PolyLog[2, -((a*E^(c + 
d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (4*b*x^2*Po 
lyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2] 
*d^2) - (2*b^3*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^ 
2*(a^2 + b^2)^(3/2)*d^3) + (4*b*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a 
^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (2*b^3*PolyLog[3, -((a*E^(c + d* 
x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (4*b*PolyLog[ 
3, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^...
 

3.1.23.3.1 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 5960

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

3.1.23.4 Maple [F]

\[\int \frac {x^{5}}{{\left (a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )\right )}^{2}}d x\]

input

int(x^5/(a+b*csch(d*x^2+c))^2,x)
 

output

int(x^5/(a+b*csch(d*x^2+c))^2,x)
 

3.1.23.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3756 vs. \(2 (834) = 1668\).

Time = 0.32 (sec) , antiderivative size = 3756, normalized size of antiderivative = 4.07 \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \]

input

integrate(x^5/(a+b*csch(d*x^2+c))^2,x, algorithm="fricas")
 

output

-1/6*((a^5 + 2*a^3*b^2 + a*b^4)*d^3*x^6 + 6*(a^3*b^2 + a*b^4)*c^2 - ((a^5 
+ 2*a^3*b^2 + a*b^4)*d^3*x^6 - 6*(a^3*b^2 + a*b^4)*d^2*x^4 + 6*(a^3*b^2 + 
a*b^4)*c^2)*cosh(d*x^2 + c)^2 - ((a^5 + 2*a^3*b^2 + a*b^4)*d^3*x^6 - 6*(a^ 
3*b^2 + a*b^4)*d^2*x^4 + 6*(a^3*b^2 + a*b^4)*c^2)*sinh(d*x^2 + c)^2 + 6*(2 
*a^4*b + a^2*b^3 - (2*a^4*b + a^2*b^3)*cosh(d*x^2 + c)^2 - (2*a^4*b + a^2* 
b^3)*sinh(d*x^2 + c)^2 - 2*(2*a^3*b^2 + a*b^4)*cosh(d*x^2 + c) - 2*(2*a^3* 
b^2 + a*b^4 + (2*a^4*b + a^2*b^3)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt(( 
a^2 + b^2)/a^2)*polylog(3, (b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) + (a*cos 
h(d*x^2 + c) + a*sinh(d*x^2 + c))*sqrt((a^2 + b^2)/a^2))/a) - 6*(2*a^4*b + 
 a^2*b^3 - (2*a^4*b + a^2*b^3)*cosh(d*x^2 + c)^2 - (2*a^4*b + a^2*b^3)*sin 
h(d*x^2 + c)^2 - 2*(2*a^3*b^2 + a*b^4)*cosh(d*x^2 + c) - 2*(2*a^3*b^2 + a* 
b^4 + (2*a^4*b + a^2*b^3)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt((a^2 + b^ 
2)/a^2)*polylog(3, (b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) - (a*cosh(d*x^2 
+ c) + a*sinh(d*x^2 + c))*sqrt((a^2 + b^2)/a^2))/a) - 2*((a^4*b + 2*a^2*b^ 
3 + b^5)*d^3*x^6 - 3*(a^2*b^3 + b^5)*d^2*x^4 + 6*(a^2*b^3 + b^5)*c^2)*cosh 
(d*x^2 + c) + 6*(a^3*b^2 + a*b^4 - (a^3*b^2 + a*b^4)*cosh(d*x^2 + c)^2 - ( 
a^3*b^2 + a*b^4)*sinh(d*x^2 + c)^2 - 2*(a^2*b^3 + b^5)*cosh(d*x^2 + c) - 2 
*(a^2*b^3 + b^5 + (a^3*b^2 + a*b^4)*cosh(d*x^2 + c))*sinh(d*x^2 + c) + ((2 
*a^4*b + a^2*b^3)*d*x^2*cosh(d*x^2 + c)^2 + (2*a^4*b + a^2*b^3)*d*x^2*sinh 
(d*x^2 + c)^2 + 2*(2*a^3*b^2 + a*b^4)*d*x^2*cosh(d*x^2 + c) - (2*a^4*b ...
 

3.1.23.6 Sympy [F]

\[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

input

integrate(x**5/(a+b*csch(d*x**2+c))**2,x)
 

output

Integral(x**5/(a + b*csch(c + d*x**2))**2, x)
 

3.1.23.7 Maxima [F]

\[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^5/(a+b*csch(d*x^2+c))^2,x, algorithm="maxima")
 

output

-1/6*((a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^6*e^(2*d*x^2) - 6*a*b^2*x^4 - (a 
^3*d + a*b^2*d)*x^6 + 2*(3*b^3*x^4*e^c + (a^2*b*d*e^c + b^3*d*e^c)*x^6)*e^ 
(d*x^2))/(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*x 
^2) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*e^(d*x^2)) - integrate(2*(2*a*b^2*x^ 
2 - (2*b^3*x^2*e^c + (2*a^2*b*d*e^c + b^3*d*e^c)*x^4)*e^(d*x^2))*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*x^2) - 2*(a^4*b*d 
*e^c + a^2*b^3*d*e^c)*e^(d*x^2)), x)
 

3.1.23.8 Giac [F]

\[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^5/(a+b*csch(d*x^2+c))^2,x, algorithm="giac")
 

output

integrate(x^5/(b*csch(d*x^2 + c) + a)^2, x)
 

3.1.23.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^5}{{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]

input

int(x^5/(a + b/sinh(c + d*x^2))^2,x)
 

output

int(x^5/(a + b/sinh(c + d*x^2))^2, x)