\(\int \frac {x^3}{a+b \text {csch}(c+d \sqrt {x})} \, dx\) [41]

Optimal result

Integrand size = 20, antiderivative size = 897 \[ \int \frac {x^3}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^8} \]

[Out]

1/4*x^4/a-2*b*x^(7/2)*ln(1+a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a/d/(a^2+b^2)^(1/2)+2*b*x^(7/2)*ln(1+a*exp(
c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a/d/(a^2+b^2)^(1/2)-14*b*x^3*polylog(2,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2
)))/a/d^2/(a^2+b^2)^(1/2)+14*b*x^3*polylog(2,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a/d^2/(a^2+b^2)^(1/2)+84
*b*x^(5/2)*polylog(3,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a/d^3/(a^2+b^2)^(1/2)-84*b*x^(5/2)*polylog(3,-a*
exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a/d^3/(a^2+b^2)^(1/2)-420*b*x^2*polylog(4,-a*exp(c+d*x^(1/2))/(b-(a^2+b^
2)^(1/2)))/a/d^4/(a^2+b^2)^(1/2)+420*b*x^2*polylog(4,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a/d^4/(a^2+b^2)^
(1/2)+1680*b*x^(3/2)*polylog(5,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a/d^5/(a^2+b^2)^(1/2)-1680*b*x^(3/2)*p
olylog(5,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a/d^5/(a^2+b^2)^(1/2)-5040*b*x*polylog(6,-a*exp(c+d*x^(1/2))
/(b-(a^2+b^2)^(1/2)))/a/d^6/(a^2+b^2)^(1/2)+5040*b*x*polylog(6,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a/d^6/
(a^2+b^2)^(1/2)-10080*b*polylog(8,-a*exp(c+d*x^(1/2))/(b-(a^2+b^2)^(1/2)))/a/d^8/(a^2+b^2)^(1/2)+10080*b*polyl
og(8,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))/a/d^8/(a^2+b^2)^(1/2)+10080*b*polylog(7,-a*exp(c+d*x^(1/2))/(b-(
a^2+b^2)^(1/2)))*x^(1/2)/a/d^7/(a^2+b^2)^(1/2)-10080*b*polylog(7,-a*exp(c+d*x^(1/2))/(b+(a^2+b^2)^(1/2)))*x^(1
/2)/a/d^7/(a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 897, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5545, 4276, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {x^3}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\frac {x^4}{4 a}-\frac {2 b \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{7/2}}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d}-\frac {14 b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^3}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^2}+\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 \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^3}-\frac {420 b \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x^2}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^4}+\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 \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^5}-\frac {5040 b \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) x}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^6}+\frac {10080 b \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) \sqrt {x}}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^8} \]

[In]

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

[Out]

x^4/(4*a) - (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x^
(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (14*b*x^3*PolyLog[2, -((a*
E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (14*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[
x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b -
Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 +
b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*
Sqrt[a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^
2]*d^4) + (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5)
- (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (5040*
b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6) + (5040*b*x*PolyLog[6,
-((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6) + (10080*b*Sqrt[x]*PolyLog[7, -((a*E
^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^7) - (10080*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d
*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^7) - (10080*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b -
Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^8) + (10080*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2
]))])/(a*Sqrt[a^2 + b^2]*d^8)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

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*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 5545

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(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[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^7}{a+b \text {csch}(c+d x)} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^7}{a}-\frac {b x^7}{a (b+a \sinh (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {x^4}{4 a}-\frac {(2 b) \text {Subst}\left (\int \frac {x^7}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {x^4}{4 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^7}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {x^4}{4 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^7}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+b^2}}+\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^7}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+b^2}} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {(14 b) \text {Subst}\left (\int x^6 \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d}-\frac {(14 b) \text {Subst}\left (\int x^6 \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {(84 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^2}-\frac {(84 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^2} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {(420 b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^3}+\frac {(420 b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^3} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {(1680 b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^4}-\frac {(1680 b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (4,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^4} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {(5040 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^5}+\frac {(5040 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (5,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^5} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {(10080 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^6}-\frac {(10080 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (6,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^6} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {(10080 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^7}+\frac {(10080 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (7,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^7} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {(10080 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (7,\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a \sqrt {a^2+b^2} d^8}+\frac {(10080 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (7,-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a \sqrt {a^2+b^2} d^8} \\ & = \frac {x^4}{4 a}-\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {14 b x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {84 b x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {420 b x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {1680 b x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {5040 b x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^6}+\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {10080 b \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^7}-\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^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 \sqrt {a^2+b^2} d^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 685, normalized size of antiderivative = 0.76 \[ \int \frac {x^3}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\frac {\sqrt {a^2+b^2} d^8 x^4-8 b d^7 x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )+8 b d^7 x^{7/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )-56 b d^6 x^3 \operatorname {PolyLog}\left (2,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )+56 b d^6 x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )+336 b d^5 x^{5/2} \operatorname {PolyLog}\left (3,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )-336 b d^5 x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )-1680 b d^4 x^2 \operatorname {PolyLog}\left (4,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )+1680 b d^4 x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )+6720 b d^3 x^{3/2} \operatorname {PolyLog}\left (5,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )-6720 b d^3 x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )-20160 b d^2 x \operatorname {PolyLog}\left (6,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )+20160 b d^2 x \operatorname {PolyLog}\left (6,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )+40320 b d \sqrt {x} \operatorname {PolyLog}\left (7,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )-40320 b d \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )-40320 b \operatorname {PolyLog}\left (8,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )+40320 b \operatorname {PolyLog}\left (8,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{4 a \sqrt {a^2+b^2} d^8} \]

[In]

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

[Out]

(Sqrt[a^2 + b^2]*d^8*x^4 - 8*b*d^7*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])] + 8*b*d^7*x^(7
/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])] - 56*b*d^6*x^3*PolyLog[2, (a*E^(c + d*Sqrt[x]))/(-b +
Sqrt[a^2 + b^2])] + 56*b*d^6*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))] + 336*b*d^5*x^(5/
2)*PolyLog[3, (a*E^(c + d*Sqrt[x]))/(-b + Sqrt[a^2 + b^2])] - 336*b*d^5*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[
x]))/(b + Sqrt[a^2 + b^2]))] - 1680*b*d^4*x^2*PolyLog[4, (a*E^(c + d*Sqrt[x]))/(-b + Sqrt[a^2 + b^2])] + 1680*
b*d^4*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))] + 6720*b*d^3*x^(3/2)*PolyLog[5, (a*E^(c +
d*Sqrt[x]))/(-b + Sqrt[a^2 + b^2])] - 6720*b*d^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b
^2]))] - 20160*b*d^2*x*PolyLog[6, (a*E^(c + d*Sqrt[x]))/(-b + Sqrt[a^2 + b^2])] + 20160*b*d^2*x*PolyLog[6, -((
a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))] + 40320*b*d*Sqrt[x]*PolyLog[7, (a*E^(c + d*Sqrt[x]))/(-b + Sqrt[a
^2 + b^2])] - 40320*b*d*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))] - 40320*b*PolyLog[8
, (a*E^(c + d*Sqrt[x]))/(-b + Sqrt[a^2 + b^2])] + 40320*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b
^2]))])/(4*a*Sqrt[a^2 + b^2]*d^8)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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