\(\int \frac {x^3}{(a+b \text {csch}(c+d x^2))^2} \, dx\) [30]

Optimal result

Integrand size = 18, antiderivative size = 519 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )} \] Output:

1/4*x^4/a^2+1/2*b^3*x^2*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*x^2*ln(1+a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^ 
(1/2)/d-1/2*b^3*x^2*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*x^2*ln(1+a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(1/2 
)/d+1/2*b^2*ln(b+a*sinh(d*x^2+c))/a^2/(a^2+b^2)/d^2+1/2*b^3*polylog(2,-a*e 
xp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2-b*polylog(2,-a*ex 
p(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(1/2)/d^2-1/2*b^3*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*polylog(2,- 
a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(1/2)/d^2-1/2*b^2*x^2*co 
sh(d*x^2+c)/a/(a^2+b^2)/d/(b+a*sinh(d*x^2+c))
 

Mathematica [A] (warning: unable to verify)

Time = 3.49 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.42 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\frac {\text {csch}^2\left (c+d x^2\right ) \left (b+a \sinh \left (c+d x^2\right )\right ) \left (-\frac {2 a b^2 d x^2 \cosh \left (c+d x^2\right )}{a^2+b^2}+\left (-c+d x^2\right ) \left (c+d x^2\right ) \left (b+a \sinh \left (c+d x^2\right )\right )-\frac {2 b \left (a^2+b^2\right ) \left (-b \sqrt {-\left (a^2+b^2\right )^2} \left (c+d x^2\right )+2 b^2 \sqrt {a^2+b^2} \arctan \left (\frac {b+a e^{c+d x^2}}{\sqrt {-a^2-b^2}}\right )+2 b^2 \sqrt {-a^2-b^2} \text {arctanh}\left (\frac {b+a e^{c+d x^2}}{\sqrt {a^2+b^2}}\right )-4 a^2 \sqrt {-a^2-b^2} c \text {arctanh}\left (\frac {b+a e^{c+d x^2}}{\sqrt {a^2+b^2}}\right )-2 b^2 \sqrt {-a^2-b^2} c \text {arctanh}\left (\frac {b+a e^{c+d x^2}}{\sqrt {a^2+b^2}}\right )-2 a^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )-b^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )+2 a^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )+b^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )+b \sqrt {-\left (a^2+b^2\right )^2} \log \left (2 b e^{c+d x^2}+a \left (-1+e^{2 \left (c+d x^2\right )}\right )\right )-\sqrt {-a^2-b^2} \left (2 a^2+b^2\right ) \operatorname {PolyLog}\left (2,\frac {a e^{c+d x^2}}{-b+\sqrt {a^2+b^2}}\right )+\sqrt {-a^2-b^2} \left (2 a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )\right ) \left (b+a \sinh \left (c+d x^2\right )\right )}{\left (-\left (a^2+b^2\right )^2\right )^{3/2}}\right )}{4 a^2 d^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \] Input:

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

Output:

(Csch[c + d*x^2]^2*(b + a*Sinh[c + d*x^2])*((-2*a*b^2*d*x^2*Cosh[c + d*x^2 
])/(a^2 + b^2) + (-c + d*x^2)*(c + d*x^2)*(b + a*Sinh[c + d*x^2]) - (2*b*( 
a^2 + b^2)*(-(b*Sqrt[-(a^2 + b^2)^2]*(c + d*x^2)) + 2*b^2*Sqrt[a^2 + b^2]* 
ArcTan[(b + a*E^(c + d*x^2))/Sqrt[-a^2 - b^2]] + 2*b^2*Sqrt[-a^2 - b^2]*Ar 
cTanh[(b + a*E^(c + d*x^2))/Sqrt[a^2 + b^2]] - 4*a^2*Sqrt[-a^2 - b^2]*c*Ar 
cTanh[(b + a*E^(c + d*x^2))/Sqrt[a^2 + b^2]] - 2*b^2*Sqrt[-a^2 - b^2]*c*Ar 
cTanh[(b + a*E^(c + d*x^2))/Sqrt[a^2 + b^2]] - 2*a^2*Sqrt[-a^2 - b^2]*(c + 
 d*x^2)*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])] - b^2*Sqrt[-a^2 - 
 b^2]*(c + d*x^2)*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])] + 2*a^2 
*Sqrt[-a^2 - b^2]*(c + d*x^2)*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^ 
2])] + b^2*Sqrt[-a^2 - b^2]*(c + d*x^2)*Log[1 + (a*E^(c + d*x^2))/(b + Sqr 
t[a^2 + b^2])] + b*Sqrt[-(a^2 + b^2)^2]*Log[2*b*E^(c + d*x^2) + a*(-1 + E^ 
(2*(c + d*x^2)))] - Sqrt[-a^2 - b^2]*(2*a^2 + b^2)*PolyLog[2, (a*E^(c + d* 
x^2))/(-b + Sqrt[a^2 + b^2])] + Sqrt[-a^2 - b^2]*(2*a^2 + b^2)*PolyLog[2, 
-((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])*(b + a*Sinh[c + d*x^2]))/(-(a 
^2 + b^2)^2)^(3/2)))/(4*a^2*d^2*(a + b*Csch[c + d*x^2])^2)
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 510, normalized size of antiderivative = 0.98, 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.

\(\displaystyle \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx\)

\(\Big \downarrow \) 5960

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(x^4/(2*a^2) + (b^3*x^2*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^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a 
^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^3*x^2*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^2*Log[1 + (a*E^ 
(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (b^2*Log[b 
+ a*Sinh[c + d*x^2]])/(a^2*(a^2 + b^2)*d^2) + (b^3*PolyLog[2, -((a*E^(c + 
d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (2*b*PolyLo 
g[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2 
) - (b^3*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 
 + b^2)^(3/2)*d^2) + (2*b*PolyLog[2, -((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*Cosh[c + d*x^2])/(a*(a^2 + b^ 
2)*d*(b + a*Sinh[c + d*x^2])))/2
 

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]
 

Maple [F]

\[\int \frac {x^{3}}{{\left (a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )\right )}^{2}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2383 vs. \(2 (461) = 922\).

Time = 0.12 (sec) , antiderivative size = 2383, normalized size of antiderivative = 4.59 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/4*((a^5 + 2*a^3*b^2 + a*b^4)*d^2*x^4 - ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*x 
^4 - 4*(a^3*b^2 + a*b^4)*d*x^2 - 4*(a^3*b^2 + a*b^4)*c)*cosh(d*x^2 + c)^2 
- ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*x^4 - 4*(a^3*b^2 + a*b^4)*d*x^2 - 4*(a^3* 
b^2 + a*b^4)*c)*sinh(d*x^2 + c)^2 - 2*(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)*dilog((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)/a + 1) + 2*(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)*dilog((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)/a + 1) - 2*((2*a^4*b + a^2*b^3)*d*x^2 - ((2*a^4*b + a^2*b 
^3)*d*x^2 + (2*a^4*b + a^2*b^3)*c)*cosh(d*x^2 + c)^2 - ((2*a^4*b + a^2*b^3 
)*d*x^2 + (2*a^4*b + a^2*b^3)*c)*sinh(d*x^2 + c)^2 + (2*a^4*b + a^2*b^3)*c 
 - 2*((2*a^3*b^2 + a*b^4)*d*x^2 + (2*a^3*b^2 + a*b^4)*c)*cosh(d*x^2 + c) - 
 2*((2*a^3*b^2 + a*b^4)*d*x^2 + (2*a^3*b^2 + a*b^4)*c + ((2*a^4*b + a^2*b^ 
3)*d*x^2 + (2*a^4*b + a^2*b^3)*c)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt(( 
a^2 + b^2)/a^2)*log(-(b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) + (a*cosh(d...
 

Sympy [F]

\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{3}}{\left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:

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

Output:

-4*a^2*b*d*integrate(x^3*e^(d*x^2 + c)/(a^5*d*e^(2*d*x^2 + 2*c) + a^3*b^2* 
d*e^(2*d*x^2 + 2*c) + 2*a^4*b*d*e^(d*x^2 + c) + 2*a^2*b^3*d*e^(d*x^2 + c) 
- a^5*d - a^3*b^2*d), x) - 2*b^3*d*integrate(x^3*e^(d*x^2 + c)/(a^5*d*e^(2 
*d*x^2 + 2*c) + a^3*b^2*d*e^(2*d*x^2 + 2*c) + 2*a^4*b*d*e^(d*x^2 + c) + 2* 
a^2*b^3*d*e^(d*x^2 + c) - a^5*d - a^3*b^2*d), x) + 1/2*a*b^2*(b*log((a*e^( 
d*x^2 + c) + b - sqrt(a^2 + b^2))/(a*e^(d*x^2 + c) + b + sqrt(a^2 + b^2))) 
/((a^5 + a^3*b^2)*sqrt(a^2 + b^2)*d^2) - 2*(d*x^2 + c)/((a^5 + a^3*b^2)*d^ 
2) + log(a*e^(2*d*x^2 + 2*c) + 2*b*e^(d*x^2 + c) - a)/((a^5 + a^3*b^2)*d^2 
)) - 1/2*b^3*log((a*e^(d*x^2 + c) + b - sqrt(a^2 + b^2))/(a*e^(d*x^2 + c) 
+ b + sqrt(a^2 + b^2)))/((a^4 + a^2*b^2)*sqrt(a^2 + b^2)*d^2) - 1/4*((a^3* 
d*e^(2*c) + a*b^2*d*e^(2*c))*x^4*e^(2*d*x^2) - 4*a*b^2*x^2 - (a^3*d + a*b^ 
2*d)*x^4 + 2*(2*b^3*x^2*e^c + (a^2*b*d*e^c + b^3*d*e^c)*x^4)*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))
 

Giac [F]

\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{3}}{\mathrm {csch}\left (d \,x^{2}+c \right )^{2} b^{2}+2 \,\mathrm {csch}\left (d \,x^{2}+c \right ) a b +a^{2}}d x \] Input:

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

Output:

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