Integrand size = 20, antiderivative size = 597 \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 a b \operatorname {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \operatorname {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )}{d^8} \]
-840*a*b*x^2*polylog(4,-exp(c+d*x^(1/2)))/d^4+840*a*b*x^2*polylog(4,exp(c+ d*x^(1/2)))/d^4+3360*a*b*x^(3/2)*polylog(5,-exp(c+d*x^(1/2)))/d^5-3360*a*b *x^(3/2)*polylog(5,exp(c+d*x^(1/2)))/d^5-8*a*b*x^(7/2)*arctanh(exp(c+d*x^( 1/2)))/d-28*a*b*x^3*polylog(2,-exp(c+d*x^(1/2)))/d^2+28*a*b*x^3*polylog(2, exp(c+d*x^(1/2)))/d^2-20160*a*b*polylog(7,exp(c+d*x^(1/2)))*x^(1/2)/d^7-10 080*a*b*x*polylog(6,-exp(c+d*x^(1/2)))/d^6+10080*a*b*x*polylog(6,exp(c+d*x ^(1/2)))/d^6+20160*a*b*polylog(7,-exp(c+d*x^(1/2)))*x^(1/2)/d^7+168*a*b*x^ (5/2)*polylog(3,-exp(c+d*x^(1/2)))/d^3-168*a*b*x^(5/2)*polylog(3,exp(c+d*x ^(1/2)))/d^3+1/4*a^2*x^4-315/2*b^2*polylog(7,exp(2*c+2*d*x^(1/2)))/d^8-2*b ^2*x^(7/2)/d+210*b^2*x^(3/2)*polylog(4,exp(2*c+2*d*x^(1/2)))/d^5-315*b^2*x *polylog(5,exp(2*c+2*d*x^(1/2)))/d^6-20160*a*b*polylog(8,-exp(c+d*x^(1/2)) )/d^8+20160*a*b*polylog(8,exp(c+d*x^(1/2)))/d^8+315*b^2*polylog(6,exp(2*c+ 2*d*x^(1/2)))*x^(1/2)/d^7-2*b^2*x^(7/2)*coth(c+d*x^(1/2))/d+14*b^2*x^3*ln( 1-exp(2*c+2*d*x^(1/2)))/d^2+42*b^2*x^(5/2)*polylog(2,exp(2*c+2*d*x^(1/2))) /d^3-105*b^2*x^2*polylog(3,exp(2*c+2*d*x^(1/2)))/d^4
Leaf count is larger than twice the leaf count of optimal. \(1289\) vs. \(2(597)=1194\).
Time = 7.40 (sec) , antiderivative size = 1289, normalized size of antiderivative = 2.16 \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx =\text {Too large to display} \]
(a^2*x^4*(a + b*Csch[c + d*Sqrt[x]])^2*Sinh[c + d*Sqrt[x]]^2)/(4*(b + a*Si nh[c + d*Sqrt[x]])^2) - (2*b*(a + b*Csch[c + d*Sqrt[x]])^2*(2*b*d^7*x^(7/2 ) - 7*b*d^6*(-1 + E^(2*c))*x^3*Log[1 - E^(-c - d*Sqrt[x])] - 2*a*d^7*(-1 + E^(2*c))*x^(7/2)*Log[1 - E^(-c - d*Sqrt[x])] - 7*b*d^6*(-1 + E^(2*c))*x^3 *Log[1 + E^(-c - d*Sqrt[x])] + 2*a*d^7*(-1 + E^(2*c))*x^(7/2)*Log[1 + E^(- c - d*Sqrt[x])] + 42*b*d^5*(-1 + E^(2*c))*x^(5/2)*PolyLog[2, -E^(-c - d*Sq rt[x])] - 14*a*d^6*(-1 + E^(2*c))*x^3*PolyLog[2, -E^(-c - d*Sqrt[x])] + 42 *b*d^5*(-1 + E^(2*c))*x^(5/2)*PolyLog[2, E^(-c - d*Sqrt[x])] + 14*a*d^6*(- 1 + E^(2*c))*x^3*PolyLog[2, E^(-c - d*Sqrt[x])] + 210*b*d^4*(-1 + E^(2*c)) *x^2*PolyLog[3, -E^(-c - d*Sqrt[x])] - 84*a*d^5*(-1 + E^(2*c))*x^(5/2)*Pol yLog[3, -E^(-c - d*Sqrt[x])] + 210*b*d^4*(-1 + E^(2*c))*x^2*PolyLog[3, E^( -c - d*Sqrt[x])] + 84*a*d^5*(-1 + E^(2*c))*x^(5/2)*PolyLog[3, E^(-c - d*Sq rt[x])] + 840*b*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[4, -E^(-c - d*Sqrt[x])] - 420*a*d^4*(-1 + E^(2*c))*x^2*PolyLog[4, -E^(-c - d*Sqrt[x])] + 840*b*d^ 3*(-1 + E^(2*c))*x^(3/2)*PolyLog[4, E^(-c - d*Sqrt[x])] + 420*a*d^4*(-1 + E^(2*c))*x^2*PolyLog[4, E^(-c - d*Sqrt[x])] + 2520*b*d^2*(-1 + E^(2*c))*x* PolyLog[5, -E^(-c - d*Sqrt[x])] - 1680*a*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLo g[5, -E^(-c - d*Sqrt[x])] + 2520*b*d^2*(-1 + E^(2*c))*x*PolyLog[5, E^(-c - d*Sqrt[x])] + 1680*a*d^3*(-1 + E^(2*c))*x^(3/2)*PolyLog[5, E^(-c - d*Sqrt [x])] + 5040*b*d*(-1 + E^(2*c))*Sqrt[x]*PolyLog[6, -E^(-c - d*Sqrt[x])]...
Time = 1.13 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5960, 3042, 4678, 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.
| \(\displaystyle \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 5960 |
| \(\displaystyle 2 \int x^{7/2} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2d\sqrt {x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 \int x^{7/2} \left (a+i b \csc \left (i c+i d \sqrt {x}\right )\right )^2d\sqrt {x}\) |
| \(\Big \downarrow \) 4678 |
| \(\displaystyle 2 \int \left (a^2 x^{7/2}+b^2 \text {csch}^2\left (c+d \sqrt {x}\right ) x^{7/2}+2 a b \text {csch}\left (c+d \sqrt {x}\right ) x^{7/2}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {a^2 x^4}{8}-\frac {4 a b x^{7/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {10080 a b \operatorname {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {10080 a b \operatorname {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )}{d^8}+\frac {10080 a b \sqrt {x} \operatorname {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {10080 a b \sqrt {x} \operatorname {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {5040 a b x \operatorname {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {5040 a b x \operatorname {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {1680 a b x^{3/2} \operatorname {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {1680 a b x^{3/2} \operatorname {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {420 a b x^2 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {420 a b x^2 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {84 a b x^{5/2} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {84 a b x^{5/2} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {14 a b x^3 \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {14 a b x^3 \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}-\frac {315 b^2 \operatorname {PolyLog}\left (7,e^{2 \left (c+d \sqrt {x}\right )}\right )}{4 d^8}+\frac {315 b^2 \sqrt {x} \operatorname {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^7}-\frac {315 b^2 x \operatorname {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^6}+\frac {105 b^2 x^{3/2} \operatorname {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {105 b^2 x^2 \operatorname {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^4}+\frac {21 b^2 x^{5/2} \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {7 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {b^2 x^{7/2}}{d}\right )\) |
2*(-((b^2*x^(7/2))/d) + (a^2*x^4)/8 - (4*a*b*x^(7/2)*ArcTanh[E^(c + d*Sqrt [x])])/d - (b^2*x^(7/2)*Coth[c + d*Sqrt[x]])/d + (7*b^2*x^3*Log[1 - E^(2*( c + d*Sqrt[x]))])/d^2 - (14*a*b*x^3*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (14*a*b*x^3*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (21*b^2*x^(5/2)*PolyLog[2 , E^(2*(c + d*Sqrt[x]))])/d^3 + (84*a*b*x^(5/2)*PolyLog[3, -E^(c + d*Sqrt[ x])])/d^3 - (84*a*b*x^(5/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (105*b^2* x^2*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/(2*d^4) - (420*a*b*x^2*PolyLog[4, - E^(c + d*Sqrt[x])])/d^4 + (420*a*b*x^2*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (105*b^2*x^(3/2)*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (1680*a*b*x^(3 /2)*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (1680*a*b*x^(3/2)*PolyLog[5, E^( c + d*Sqrt[x])])/d^5 - (315*b^2*x*PolyLog[5, E^(2*(c + d*Sqrt[x]))])/(2*d^ 6) - (5040*a*b*x*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (5040*a*b*x*PolyLog [6, E^(c + d*Sqrt[x])])/d^6 + (315*b^2*Sqrt[x]*PolyLog[6, E^(2*(c + d*Sqrt [x]))])/(2*d^7) + (10080*a*b*Sqrt[x]*PolyLog[7, -E^(c + d*Sqrt[x])])/d^7 - (10080*a*b*Sqrt[x]*PolyLog[7, E^(c + d*Sqrt[x])])/d^7 - (315*b^2*PolyLog[ 7, E^(2*(c + d*Sqrt[x]))])/(4*d^8) - (10080*a*b*PolyLog[8, -E^(c + d*Sqrt[ x])])/d^8 + (10080*a*b*PolyLog[8, E^(c + d*Sqrt[x])])/d^8)
3.1.36.3.1 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, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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]
\[\int x^{3} \left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2}d x\]
\[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]
\[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{3} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
Time = 0.37 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.09 \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {1}{4} \, a^{2} x^{4} - \frac {4 \, b^{2} x^{\frac {7}{2}}}{d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - d} - \frac {4 \, {\left (d^{7} x^{\frac {7}{2}} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 7 \, d^{6} x^{3} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 42 \, d^{5} x^{\frac {5}{2}} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 210 \, d^{4} x^{2} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 840 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )}) + 2520 \, d^{2} x {\rm Li}_{6}(-e^{\left (d \sqrt {x} + c\right )}) - 5040 \, d \sqrt {x} {\rm Li}_{7}(-e^{\left (d \sqrt {x} + c\right )}) + 5040 \, {\rm Li}_{8}(-e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{8}} + \frac {4 \, {\left (d^{7} x^{\frac {7}{2}} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 7 \, d^{6} x^{3} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 42 \, d^{5} x^{\frac {5}{2}} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 210 \, d^{4} x^{2} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 840 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )}) + 2520 \, d^{2} x {\rm Li}_{6}(e^{\left (d \sqrt {x} + c\right )}) - 5040 \, d \sqrt {x} {\rm Li}_{7}(e^{\left (d \sqrt {x} + c\right )}) + 5040 \, {\rm Li}_{8}(e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{8}} + \frac {14 \, {\left (d^{6} x^{3} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 6 \, d^{5} x^{\frac {5}{2}} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 30 \, d^{4} x^{2} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 120 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 360 \, d^{2} x {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )}) + 720 \, d \sqrt {x} {\rm Li}_{6}(-e^{\left (d \sqrt {x} + c\right )}) - 720 \, {\rm Li}_{7}(-e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{8}} + \frac {14 \, {\left (d^{6} x^{3} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 6 \, d^{5} x^{\frac {5}{2}} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 30 \, d^{4} x^{2} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 120 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 360 \, d^{2} x {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )}) + 720 \, d \sqrt {x} {\rm Li}_{6}(e^{\left (d \sqrt {x} + c\right )}) - 720 \, {\rm Li}_{7}(e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{8}} - \frac {a b d^{8} x^{4} + 4 \, b^{2} d^{7} x^{\frac {7}{2}}}{2 \, d^{8}} + \frac {a b d^{8} x^{4} - 4 \, b^{2} d^{7} x^{\frac {7}{2}}}{2 \, d^{8}} \]
1/4*a^2*x^4 - 4*b^2*x^(7/2)/(d*e^(2*d*sqrt(x) + 2*c) - d) - 4*(d^7*x^(7/2) *log(e^(d*sqrt(x) + c) + 1) + 7*d^6*x^3*dilog(-e^(d*sqrt(x) + c)) - 42*d^5 *x^(5/2)*polylog(3, -e^(d*sqrt(x) + c)) + 210*d^4*x^2*polylog(4, -e^(d*sqr t(x) + c)) - 840*d^3*x^(3/2)*polylog(5, -e^(d*sqrt(x) + c)) + 2520*d^2*x*p olylog(6, -e^(d*sqrt(x) + c)) - 5040*d*sqrt(x)*polylog(7, -e^(d*sqrt(x) + c)) + 5040*polylog(8, -e^(d*sqrt(x) + c)))*a*b/d^8 + 4*(d^7*x^(7/2)*log(-e ^(d*sqrt(x) + c) + 1) + 7*d^6*x^3*dilog(e^(d*sqrt(x) + c)) - 42*d^5*x^(5/2 )*polylog(3, e^(d*sqrt(x) + c)) + 210*d^4*x^2*polylog(4, e^(d*sqrt(x) + c) ) - 840*d^3*x^(3/2)*polylog(5, e^(d*sqrt(x) + c)) + 2520*d^2*x*polylog(6, e^(d*sqrt(x) + c)) - 5040*d*sqrt(x)*polylog(7, e^(d*sqrt(x) + c)) + 5040*p olylog(8, e^(d*sqrt(x) + c)))*a*b/d^8 + 14*(d^6*x^3*log(e^(d*sqrt(x) + c) + 1) + 6*d^5*x^(5/2)*dilog(-e^(d*sqrt(x) + c)) - 30*d^4*x^2*polylog(3, -e^ (d*sqrt(x) + c)) + 120*d^3*x^(3/2)*polylog(4, -e^(d*sqrt(x) + c)) - 360*d^ 2*x*polylog(5, -e^(d*sqrt(x) + c)) + 720*d*sqrt(x)*polylog(6, -e^(d*sqrt(x ) + c)) - 720*polylog(7, -e^(d*sqrt(x) + c)))*b^2/d^8 + 14*(d^6*x^3*log(-e ^(d*sqrt(x) + c) + 1) + 6*d^5*x^(5/2)*dilog(e^(d*sqrt(x) + c)) - 30*d^4*x^ 2*polylog(3, e^(d*sqrt(x) + c)) + 120*d^3*x^(3/2)*polylog(4, e^(d*sqrt(x) + c)) - 360*d^2*x*polylog(5, e^(d*sqrt(x) + c)) + 720*d*sqrt(x)*polylog(6, e^(d*sqrt(x) + c)) - 720*polylog(7, e^(d*sqrt(x) + c)))*b^2/d^8 - 1/2*(a* b*d^8*x^4 + 4*b^2*d^7*x^(7/2))/d^8 + 1/2*(a*b*d^8*x^4 - 4*b^2*d^7*x^(7/...
\[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^3\,{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]