\(\int x^5 (a+b \text {csch}(c+d x^2))^2 \, dx\) [8]

Optimal result

Integrand size = 18, antiderivative size = 196 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b^2 \operatorname {PolyLog}\left (2,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac {2 a b \operatorname {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {2 a b \operatorname {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5545, 4275, 4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438} \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^6}{6}-\frac {2 a b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}+\frac {2 a b \operatorname {PolyLog}\left (3,-e^{d x^2+c}\right )}{d^3}-\frac {2 a b \operatorname {PolyLog}\left (3,e^{d x^2+c}\right )}{d^3}-\frac {2 a b x^2 \operatorname {PolyLog}\left (2,-e^{d x^2+c}\right )}{d^2}+\frac {2 a b x^2 \operatorname {PolyLog}\left (2,e^{d x^2+c}\right )}{d^2}+\frac {b^2 \operatorname {PolyLog}\left (2,e^{2 \left (d x^2+c\right )}\right )}{2 d^3}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {b^2 x^4}{2 d} \]

[In]

Int[x^5*(a + b*Csch[c + d*x^2])^2,x]

[Out]

-1/2*(b^2*x^4)/d + (a^2*x^6)/6 - (2*a*b*x^4*ArcTanh[E^(c + d*x^2)])/d - (b^2*x^4*Coth[c + d*x^2])/(2*d) + (b^2
*x^2*Log[1 - E^(2*(c + d*x^2))])/d^2 - (2*a*b*x^2*PolyLog[2, -E^(c + d*x^2)])/d^2 + (2*a*b*x^2*PolyLog[2, E^(c
+ d*x^2)])/d^2 + (b^2*PolyLog[2, E^(2*(c + d*x^2))])/(2*d^3) + (2*a*b*PolyLog[3, -E^(c + d*x^2)])/d^3 - (2*a*
b*PolyLog[3, E^(c + d*x^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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4275

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]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(595\) vs. \(2(196)=392\).

Time = 2.23 (sec) , antiderivative size = 595, normalized size of antiderivative = 3.04 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=-\frac {12 b^2 d^2 x^4+2 a^2 d^3 x^6-2 a^2 d^3 e^{2 c} x^6+12 b^2 d x^2 \log \left (1-e^{-c-d x^2}\right )-12 b^2 d e^{2 c} x^2 \log \left (1-e^{-c-d x^2}\right )+12 a b d^2 x^4 \log \left (1-e^{-c-d x^2}\right )-12 a b d^2 e^{2 c} x^4 \log \left (1-e^{-c-d x^2}\right )+12 b^2 d x^2 \log \left (1+e^{-c-d x^2}\right )-12 b^2 d e^{2 c} x^2 \log \left (1+e^{-c-d x^2}\right )-12 a b d^2 x^4 \log \left (1+e^{-c-d x^2}\right )+12 a b d^2 e^{2 c} x^4 \log \left (1+e^{-c-d x^2}\right )+12 b \left (-1+e^{2 c}\right ) \left (b-2 a d x^2\right ) \operatorname {PolyLog}\left (2,-e^{-c-d x^2}\right )+12 b \left (-1+e^{2 c}\right ) \left (b+2 a d x^2\right ) \operatorname {PolyLog}\left (2,e^{-c-d x^2}\right )+24 a b \operatorname {PolyLog}\left (3,-e^{-c-d x^2}\right )-24 a b e^{2 c} \operatorname {PolyLog}\left (3,-e^{-c-d x^2}\right )-24 a b \operatorname {PolyLog}\left (3,e^{-c-d x^2}\right )+24 a b e^{2 c} \operatorname {PolyLog}\left (3,e^{-c-d x^2}\right )+3 b^2 d^2 x^4 \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )-3 b^2 d^2 e^{2 c} x^4 \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )-3 b^2 d^2 x^4 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )+3 b^2 d^2 e^{2 c} x^4 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )}{12 d^3 \left (-1+e^{2 c}\right )} \]

[In]

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

[Out]

-1/12*(12*b^2*d^2*x^4 + 2*a^2*d^3*x^6 - 2*a^2*d^3*E^(2*c)*x^6 + 12*b^2*d*x^2*Log[1 - E^(-c - d*x^2)] - 12*b^2*
d*E^(2*c)*x^2*Log[1 - E^(-c - d*x^2)] + 12*a*b*d^2*x^4*Log[1 - E^(-c - d*x^2)] - 12*a*b*d^2*E^(2*c)*x^4*Log[1
- E^(-c - d*x^2)] + 12*b^2*d*x^2*Log[1 + E^(-c - d*x^2)] - 12*b^2*d*E^(2*c)*x^2*Log[1 + E^(-c - d*x^2)] - 12*a
*b*d^2*x^4*Log[1 + E^(-c - d*x^2)] + 12*a*b*d^2*E^(2*c)*x^4*Log[1 + E^(-c - d*x^2)] + 12*b*(-1 + E^(2*c))*(b -
2*a*d*x^2)*PolyLog[2, -E^(-c - d*x^2)] + 12*b*(-1 + E^(2*c))*(b + 2*a*d*x^2)*PolyLog[2, E^(-c - d*x^2)] + 24*
a*b*PolyLog[3, -E^(-c - d*x^2)] - 24*a*b*E^(2*c)*PolyLog[3, -E^(-c - d*x^2)] - 24*a*b*PolyLog[3, E^(-c - d*x^2
)] + 24*a*b*E^(2*c)*PolyLog[3, E^(-c - d*x^2)] + 3*b^2*d^2*x^4*Csch[c/2]*Csch[(c + d*x^2)/2]*Sinh[(d*x^2)/2] -
3*b^2*d^2*E^(2*c)*x^4*Csch[c/2]*Csch[(c + d*x^2)/2]*Sinh[(d*x^2)/2] - 3*b^2*d^2*x^4*Sech[c/2]*Sech[(c + d*x^2
)/2]*Sinh[(d*x^2)/2] + 3*b^2*d^2*E^(2*c)*x^4*Sech[c/2]*Sech[(c + d*x^2)/2]*Sinh[(d*x^2)/2])/(d^3*(-1 + E^(2*c)
))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (180) = 360\).

Time = 0.28 (sec) , antiderivative size = 1031, normalized size of antiderivative = 5.26 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*(a^2*d^3*x^6 + 6*b^2*c^2 - (a^2*d^3*x^6 - 6*b^2*d^2*x^4 + 6*b^2*c^2)*cosh(d*x^2 + c)^2 - 2*(a^2*d^3*x^6 -
6*b^2*d^2*x^4 + 6*b^2*c^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (a^2*d^3*x^6 - 6*b^2*d^2*x^4 + 6*b^2*c^2)*sinh(d
*x^2 + c)^2 + 6*(2*a*b*d*x^2 - (2*a*b*d*x^2 + b^2)*cosh(d*x^2 + c)^2 - 2*(2*a*b*d*x^2 + b^2)*cosh(d*x^2 + c)*s
inh(d*x^2 + c) - (2*a*b*d*x^2 + b^2)*sinh(d*x^2 + c)^2 + b^2)*dilog(cosh(d*x^2 + c) + sinh(d*x^2 + c)) - 6*(2*
a*b*d*x^2 - (2*a*b*d*x^2 - b^2)*cosh(d*x^2 + c)^2 - 2*(2*a*b*d*x^2 - b^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (2
*a*b*d*x^2 - b^2)*sinh(d*x^2 + c)^2 - b^2)*dilog(-cosh(d*x^2 + c) - sinh(d*x^2 + c)) - 6*(a*b*d^2*x^4 - b^2*d*
x^2 - (a*b*d^2*x^4 - b^2*d*x^2)*cosh(d*x^2 + c)^2 - 2*(a*b*d^2*x^4 - b^2*d*x^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c
) - (a*b*d^2*x^4 - b^2*d*x^2)*sinh(d*x^2 + c)^2)*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) + 1) + 6*(a*b*c^2 - b^2
*c - (a*b*c^2 - b^2*c)*cosh(d*x^2 + c)^2 - 2*(a*b*c^2 - b^2*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (a*b*c^2 - b^
2*c)*sinh(d*x^2 + c)^2)*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) - 1) + 6*(a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b^
2*c - (a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b^2*c)*cosh(d*x^2 + c)^2 - 2*(a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b
^2*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b^2*c)*sinh(d*x^2 + c)^2)*log(-co
sh(d*x^2 + c) - sinh(d*x^2 + c) + 1) + 12*(a*b*cosh(d*x^2 + c)^2 + 2*a*b*cosh(d*x^2 + c)*sinh(d*x^2 + c) + a*b
*sinh(d*x^2 + c)^2 - a*b)*polylog(3, cosh(d*x^2 + c) + sinh(d*x^2 + c)) - 12*(a*b*cosh(d*x^2 + c)^2 + 2*a*b*co
sh(d*x^2 + c)*sinh(d*x^2 + c) + a*b*sinh(d*x^2 + c)^2 - a*b)*polylog(3, -cosh(d*x^2 + c) - sinh(d*x^2 + c)))/(
d^3*cosh(d*x^2 + c)^2 + 2*d^3*cosh(d*x^2 + c)*sinh(d*x^2 + c) + d^3*sinh(d*x^2 + c)^2 - d^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.38 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=\frac {1}{6} \, a^{2} x^{6} - \frac {b^{2} x^{4}}{d e^{\left (2 \, d x^{2} + 2 \, c\right )} - d} - \frac {{\left (d^{2} x^{4} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2} {\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac {{\left (d^{2} x^{4} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2} {\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac {{\left (d x^{2} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} + \frac {{\left (d x^{2} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} - \frac {2 \, a b d^{3} x^{6} + 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} + \frac {2 \, a b d^{3} x^{6} - 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} \]

[In]

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

[Out]

1/6*a^2*x^6 - b^2*x^4/(d*e^(2*d*x^2 + 2*c) - d) - (d^2*x^4*log(e^(d*x^2 + c) + 1) + 2*d*x^2*dilog(-e^(d*x^2 +
c)) - 2*polylog(3, -e^(d*x^2 + c)))*a*b/d^3 + (d^2*x^4*log(-e^(d*x^2 + c) + 1) + 2*d*x^2*dilog(e^(d*x^2 + c))
- 2*polylog(3, e^(d*x^2 + c)))*a*b/d^3 + (d*x^2*log(e^(d*x^2 + c) + 1) + dilog(-e^(d*x^2 + c)))*b^2/d^3 + (d*x
^2*log(-e^(d*x^2 + c) + 1) + dilog(e^(d*x^2 + c)))*b^2/d^3 - 1/6*(2*a*b*d^3*x^6 + 3*b^2*d^2*x^4)/d^3 + 1/6*(2*
a*b*d^3*x^6 - 3*b^2*d^2*x^4)/d^3

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^5*(a + b/sinh(c + d*x^2))^2,x)

[Out]

int(x^5*(a + b/sinh(c + d*x^2))^2, x)