Integrand size = 20, antiderivative size = 2663 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]
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^...
Time = 8.52 (sec) , antiderivative size = 2841, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \]
(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)]...
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 )\) |
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
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]
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]
\[\int \frac {x^{3}}{\left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}}d x\]
\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{3}}{\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
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)
\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]