\(\int \frac {x^5}{a+b \text {csch}(c+d x^2)} \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 325 \[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {x^6}{6 a}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3} \]

[Out]

1/6*x^6/a-1/2*b*x^4*ln(1+a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a/d/(a^2+b^2)^(1/2)+1/2*b*x^4*ln(1+a*exp(d*x^2+c)
/(b+(a^2+b^2)^(1/2)))/a/d/(a^2+b^2)^(1/2)-b*x^2*polylog(2,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a/d^2/(a^2+b^2)
^(1/2)+b*x^2*polylog(2,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a/d^2/(a^2+b^2)^(1/2)+b*polylog(3,-a*exp(d*x^2+c)/
(b-(a^2+b^2)^(1/2)))/a/d^3/(a^2+b^2)^(1/2)-b*polylog(3,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a/d^3/(a^2+b^2)^(1
/2)

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5545, 4276, 3403, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}+\frac {b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}-\frac {b x^4 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {b x^4 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {x^6}{6 a} \]

[In]

Int[x^5/(a + b*Csch[c + d*x^2]),x]

[Out]

x^6/(6*a) - (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) + (b*x^4*Log[1 +
(a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) - (b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b -
Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/
(a*Sqrt[a^2 + b^2]*d^2) + (b*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) -
(b*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3)

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]

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.79 \[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {\sqrt {a^2+b^2} d^3 x^6-3 b d^2 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )+3 b d^2 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )-6 b d x^2 \operatorname {PolyLog}\left (2,\frac {a e^{c+d x^2}}{-b+\sqrt {a^2+b^2}}\right )+6 b d x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )+6 b \operatorname {PolyLog}\left (3,\frac {a e^{c+d x^2}}{-b+\sqrt {a^2+b^2}}\right )-6 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{6 a \sqrt {a^2+b^2} d^3} \]

[In]

Integrate[x^5/(a + b*Csch[c + d*x^2]),x]

[Out]

(Sqrt[a^2 + b^2]*d^3*x^6 - 3*b*d^2*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])] + 3*b*d^2*x^4*Log[1 +
(a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])] - 6*b*d*x^2*PolyLog[2, (a*E^(c + d*x^2))/(-b + Sqrt[a^2 + b^2])] + 6*
b*d*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))] + 6*b*PolyLog[3, (a*E^(c + d*x^2))/(-b + Sqrt[a
^2 + b^2])] - 6*b*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(6*a*Sqrt[a^2 + b^2]*d^3)

Maple [F]

\[\int \frac {x^{5}}{a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )}d x\]

[In]

int(x^5/(a+b*csch(d*x^2+c)),x)

[Out]

int(x^5/(a+b*csch(d*x^2+c)),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (287) = 574\).

Time = 0.27 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.11 \[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {{\left (a^{2} + b^{2}\right )} d^{3} x^{6} - 6 \, a b d x^{2} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) + 6 \, a b d x^{2} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) + 3 \, a b c^{2} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) + 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) - 3 \, a b c^{2} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) - 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) + 6 \, a b \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm polylog}\left (3, \frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}}}{a}\right ) - 6 \, a b \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm polylog}\left (3, \frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}}}{a}\right ) - 3 \, {\left (a b d^{2} x^{4} - a b c^{2}\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right ) + 3 \, {\left (a b d^{2} x^{4} - a b c^{2}\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right )}{6 \, {\left (a^{3} + a b^{2}\right )} d^{3}} \]

[In]

integrate(x^5/(a+b*csch(d*x^2+c)),x, algorithm="fricas")

[Out]

1/6*((a^2 + b^2)*d^3*x^6 - 6*a*b*d*x^2*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) + 6*a*b*d*x^2*sqrt((a^2 + b^2)/a^2)*di
log((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) + 3*a*b*c^2*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x^2 + c) + 2*a*sinh(d*x^2 + c) + 2*a*sqrt((a^2 + b^2
)/a^2) + 2*b) - 3*a*b*c^2*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x^2 + c) + 2*a*sinh(d*x^2 + c) - 2*a*sqrt((a^2
+ b^2)/a^2) + 2*b) + 6*a*b*sqrt((a^2 + b^2)/a^2)*polylog(3, (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) - 6*a*b*sqrt((a^2 + b^2)/a^2)*polylog(3, (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) - 3*(a*b*d^2*x^
4 - a*b*c^2)*sqrt((a^2 + b^2)/a^2)*log(-(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) + 3*(a*b*d^2*x^4 - a*b*c^2)*sqrt((a^2 + b^2)/a^2)*log(-(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))/((a^3 + a*b^
2)*d^3)

Sympy [F]

\[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^{5}}{a + b \operatorname {csch}{\left (c + d x^{2} \right )}}\, dx \]

[In]

integrate(x**5/(a+b*csch(d*x**2+c)),x)

[Out]

Integral(x**5/(a + b*csch(c + d*x**2)), x)

Maxima [F]

\[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int { \frac {x^{5}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a} \,d x } \]

[In]

integrate(x^5/(a+b*csch(d*x^2+c)),x, algorithm="maxima")

[Out]

1/6*x^6/a - 2*b*integrate(x^5*e^(d*x^2 + c)/(a^2*e^(2*d*x^2 + 2*c) + 2*a*b*e^(d*x^2 + c) - a^2), x)

Giac [F]

\[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int { \frac {x^{5}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a} \,d x } \]

[In]

integrate(x^5/(a+b*csch(d*x^2+c)),x, algorithm="giac")

[Out]

integrate(x^5/(b*csch(d*x^2 + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^5}{a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}} \,d x \]

[In]

int(x^5/(a + b/sinh(c + d*x^2)),x)

[Out]

int(x^5/(a + b/sinh(c + d*x^2)), x)