3.1.46 \(\int \frac {x^3}{(a+b \text {csch}(c+d \sqrt {x}))^2} \, dx\) [46]

3.1.46.1 Optimal result

Integrand size = 20, antiderivative size = 2663 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]

output

168*b*x^(5/2)*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)-168*b*x^(5/2)*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)-840*b*x^2*polylog(4,-a*exp(c+d*x^(1/2))/(b-( 
a^2+b^2)^(1/2)))/a^2/d^4/(a^2+b^2)^(1/2)+840*b*x^2*polylog(4,-a*exp(c+d*x^ 
(1/2))/(b+(a^2+b^2)^(1/2)))/a^2/d^4/(a^2+b^2)^(1/2)+3360*b*x^(3/2)*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)-3360*b 
*x^(3/2)*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)-10080*b*x*polylog(6,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a^2 
/d^6/(a^2+b^2)^(1/2)+10080*b*x*polylog(6,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^ 
(1/2)))/a^2/d^6/(a^2+b^2)^(1/2)+10080*b^2*polylog(6,-a*exp(c+d*x^(1/2))/(b 
-(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)/d^7+10080*b^2*polylog(6,-a*exp(c+ 
d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/(a^2+b^2)/d^7-10080*b^3*polylo 
g(7,-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 
^7+10080*b^3*polylog(7,-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^7+20160*b*polylog(7,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^( 
1/2)))*x^(1/2)/a^2/d^7/(a^2+b^2)^(1/2)-20160*b*polylog(7,-a*exp(c+d*x^(1/2 
))/(b+(a^2+b^2)^(1/2)))*x^(1/2)/a^2/d^7/(a^2+b^2)^(1/2)+14*b^2*x^3*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^(7/2)*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+14*b^2*x^3* 
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^...
 

3.1.46.2 Mathematica [A] (verified)

Time = 8.52 (sec) , antiderivative size = 2841, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \]

input

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

output

(Csch[c + d*Sqrt[x]]^2*(b + a*Sinh[c + d*Sqrt[x]])*(x^4*(b + a*Sinh[c + d* 
Sqrt[x]]) - (8*b*E^c*(2*b*E^c*x^(7/2) + ((-1 + E^(2*c))*(-7*b*d^6*Sqrt[(a^ 
2 + b^2)*E^(2*c)]*x^3*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + 
 b^2)*E^(2*c)])] + 2*a^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/( 
b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + b^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2* 
c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 7*b*d^6*Sqrt[(a^2 + 
 b^2)*E^(2*c)]*x^3*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^ 
2)*E^(2*c)])] - 2*a^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E 
^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - b^2*d^7*E^c*x^(7/2)*Log[1 + (a*E^(2*c + 
 d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 7*d^5*(-6*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^(5/2)*PolyLog 
[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 7*d^ 
5*(6*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^(5/2)*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E 
^(2*c)]))] + 210*b*d^4*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2*PolyLog[3, -((a*E^(2* 
c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 84*a^2*d^5*E^c*x^( 
5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c 
)]))] - 42*b^2*d^5*E^c*x^(5/2)*PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c 
 - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 210*b*d^4*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2* 
PolyLog[3, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]...
 

3.1.46.3 Rubi [A] (verified)

Time = 4.22 (sec) , antiderivative size = 2664, 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 = {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.

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

\(\Big \downarrow \) 5960

\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+i b \csc \left (i c+i d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (-\frac {2 b x^{7/2}}{a^2 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {x^{7/2}}{a^2}+\frac {b^2 x^{7/2}}{a^2 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^4}{8 a^2}-\frac {2 b \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{7/2}}{a^2 \sqrt {a^2+b^2} d}+\frac {b^3 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{7/2}}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {2 b \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{7/2}}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{7/2}}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 x^{7/2}}{a^2 \left (a^2+b^2\right ) d}-\frac {b^2 \cosh \left (c+d \sqrt {x}\right ) x^{7/2}}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {7 b^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) x^3}{a^2 \left (a^2+b^2\right ) d^2}+\frac {7 b^2 \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) x^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {14 b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^3}{a^2 \sqrt {a^2+b^2} d^2}+\frac {7 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {14 b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^3}{a^2 \sqrt {a^2+b^2} d^2}-\frac {7 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {42 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^{5/2}}{a^2 \left (a^2+b^2\right ) d^3}+\frac {42 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^{5/2}}{a^2 \left (a^2+b^2\right ) d^3}+\frac {84 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^{5/2}}{a^2 \sqrt {a^2+b^2} d^3}-\frac {42 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^{5/2}}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {84 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^{5/2}}{a^2 \sqrt {a^2+b^2} d^3}+\frac {42 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^{5/2}}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {210 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^2}{a^2 \left (a^2+b^2\right ) d^4}-\frac {210 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^2}{a^2 \left (a^2+b^2\right ) d^4}-\frac {420 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^2}{a^2 \sqrt {a^2+b^2} d^4}+\frac {210 b^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^2}{a^2 \left (a^2+b^2\right )^{3/2} d^4}+\frac {420 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^2}{a^2 \sqrt {a^2+b^2} d^4}-\frac {210 b^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^2}{a^2 \left (a^2+b^2\right )^{3/2} d^4}+\frac {840 b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^{3/2}}{a^2 \left (a^2+b^2\right ) d^5}+\frac {840 b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^{3/2}}{a^2 \left (a^2+b^2\right ) d^5}+\frac {1680 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^{3/2}}{a^2 \sqrt {a^2+b^2} d^5}-\frac {840 b^3 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^{3/2}}{a^2 \left (a^2+b^2\right )^{3/2} d^5}-\frac {1680 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^{3/2}}{a^2 \sqrt {a^2+b^2} d^5}+\frac {840 b^3 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x^{3/2}}{a^2 \left (a^2+b^2\right )^{3/2} d^5}-\frac {2520 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x}{a^2 \left (a^2+b^2\right ) d^6}-\frac {2520 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x}{a^2 \left (a^2+b^2\right ) d^6}-\frac {5040 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x}{a^2 \sqrt {a^2+b^2} d^6}+\frac {2520 b^3 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x}{a^2 \left (a^2+b^2\right )^{3/2} d^6}+\frac {5040 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x}{a^2 \sqrt {a^2+b^2} d^6}-\frac {2520 b^3 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) x}{a^2 \left (a^2+b^2\right )^{3/2} d^6}+\frac {5040 b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) \sqrt {x}}{a^2 \left (a^2+b^2\right ) d^7}+\frac {5040 b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) \sqrt {x}}{a^2 \left (a^2+b^2\right ) d^7}+\frac {10080 b \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) \sqrt {x}}{a^2 \sqrt {a^2+b^2} d^7}-\frac {5040 b^3 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) \sqrt {x}}{a^2 \left (a^2+b^2\right )^{3/2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) \sqrt {x}}{a^2 \sqrt {a^2+b^2} d^7}+\frac {5040 b^3 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) \sqrt {x}}{a^2 \left (a^2+b^2\right )^{3/2} d^7}-\frac {5040 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^8}-\frac {5040 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^8}-\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^8}+\frac {5040 b^3 \operatorname {PolyLog}\left (8,-\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^8}+\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^8}-\frac {5040 b^3 \operatorname {PolyLog}\left (8,-\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^8}\right )\)

input

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

output

2*(-((b^2*x^(7/2))/(a^2*(a^2 + b^2)*d)) + x^4/(8*a^2) + (7*b^2*x^3*Log[1 + 
 (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (b^ 
3*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 
+ b^2)^(3/2)*d) - (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 
 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (7*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x 
]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (b^3*x^(7/2)*Log[1 + ( 
a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + ( 
2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqr 
t[a^2 + b^2]*d) + (42*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - 
Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (7*b^3*x^3*PolyLog[2, -((a*E^( 
c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (14 
*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sq 
rt[a^2 + b^2]*d^2) + (42*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b 
 + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (7*b^3*x^3*PolyLog[2, -((a* 
E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + 
(14*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2 
*Sqrt[a^2 + b^2]*d^2) - (210*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b 
 - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) - (42*b^3*x^(5/2)*PolyLog[3, 
-((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^ 
3) + (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b...
 

3.1.46.3.1 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]
 

3.1.46.4 Maple [F]

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

input

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

output

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

3.1.46.5 Fricas [F]

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

input

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

output

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

3.1.46.6 Sympy [F]

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

input

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

output

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

3.1.46.7 Maxima [F]

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

input

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

output

1/4*(16*a*b^2*x^(7/2) - (a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^4*e^(2*d*sqrt( 
x)) + (a^3*d + a*b^2*d)*x^4 - 2*(8*b^3*x^(7/2)*e^c + (a^2*b*d*e^c + b^3*d* 
e^c)*x^4)*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*(7*a*b^2*x^(5/2) - (7*b^3*x^(5/2)*e^c + (2*a^2*b*d*e^c + b^3* 
d*e^c)*x^3)*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))) 
, x)
 

3.1.46.8 Giac [F]

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

input

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

output

sage0*x
 

3.1.46.9 Mupad [F(-1)]

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

input

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

output

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