\(\int x^5 (a+b \text {csch}(c+d x^2)) \, dx\) [1]

Optimal result

Integrand size = 16, antiderivative size = 104 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {14, 5545, 4267, 2611, 2320, 6724} \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}+\frac {b \operatorname {PolyLog}\left (3,-e^{d x^2+c}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (3,e^{d x^2+c}\right )}{d^3}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{d x^2+c}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{d x^2+c}\right )}{d^2} \]

[In]

Int[x^5*(a + b*Csch[c + d*x^2]),x]

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 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}& = \int \left (a x^5+b x^5 \text {csch}\left (c+d x^2\right )\right ) \, dx \\ & = \frac {a x^6}{6}+b \int x^5 \text {csch}\left (c+d x^2\right ) \, dx \\ & = \frac {a x^6}{6}+\frac {1}{2} b \text {Subst}\left (\int x^2 \text {csch}(c+d x) \, dx,x,x^2\right ) \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b \text {Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {b \text {Subst}\left (\int x \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{d} \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {b \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2} \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {b \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3} \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.28 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}+\frac {b x^4 \log \left (1-e^{c+d x^2}\right )}{2 d}-\frac {b x^4 \log \left (1+e^{c+d x^2}\right )}{2 d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \]

[In]

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

[Out]

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

Maple [F]

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

[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. 209 vs. \(2 (95) = 190\).

Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.01 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a d^{3} x^{6} - 3 \, b d^{2} x^{4} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) + 6 \, b d x^{2} {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 6 \, b d x^{2} {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) + 3 \, b c^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 3 \, {\left (b d^{2} x^{4} - b c^{2}\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right ) - 6 \, b {\rm polylog}\left (3, \cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) + 6 \, b {\rm polylog}\left (3, -\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right )}{6 \, d^{3}} \]

[In]

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

[Out]

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

Sympy [F]

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

[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]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^5*(a + b/sinh(c + d*x^2)),x)

[Out]

int(x^5*(a + b/sinh(c + d*x^2)), x)