\(\int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\) [146]

Optimal result

Integrand size = 15, antiderivative size = 162 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

[Out]

4/15/c^4/(c^4-1/x^4)/x^2/csch(2*ln(c*x))^(3/2)-2/15*x^2/(c^4-1/x^4)/csch(2*ln(c*x))^(3/2)+1/9*x^6/csch(2*ln(c*
x))^(3/2)+4/15*EllipticE(1/c/x,I)/c^9/(1-1/c^4/x^4)^(3/2)/x^3/csch(2*ln(c*x))^(3/2)-4/15*EllipticF(1/c/x,I)/c^
9/(1-1/c^4/x^4)^(3/2)/x^3/csch(2*ln(c*x))^(3/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5671, 5669, 342, 283, 331, 313, 227, 1195, 435} \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4}{15 c^4 x^2 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{15 c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

[In]

Int[x^5/Csch[2*Log[c*x]]^(3/2),x]

[Out]

4/(15*c^4*(c^4 - x^(-4))*x^2*Csch[2*Log[c*x]]^(3/2)) - (2*x^2)/(15*(c^4 - x^(-4))*Csch[2*Log[c*x]]^(3/2)) + x^
6/(9*Csch[2*Log[c*x]]^(3/2)) + (4*EllipticE[ArcCsc[c*x], -1])/(15*c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c
*x]]^(3/2)) - (4*EllipticF[ArcCsc[c*x], -1])/(15*c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c], Int
[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]

Rule 5669

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[Csch[d*(a + b*Log[x])]^p*(
(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p/x^((-b)*d*p)), Int[(e*x)^m*(1/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)), x]
, x] /; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 5671

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1
)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[
{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^6} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {1}{x^4}\right )^{3/2} x^8 \, dx,x,c x\right )}{c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {\text {Subst}\left (\int \frac {\left (1-x^4\right )^{3/2}}{x^{10}} \, dx,x,\frac {1}{c x}\right )}{c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {2 \text {Subst}\left (\int \frac {\sqrt {1-x^4}}{x^6} \, dx,x,\frac {1}{c x}\right )}{3 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = -\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c x}\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.39 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {x^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right )}{6 c^2 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]

[In]

Integrate[x^5/Csch[2*Log[c*x]]^(3/2),x]

[Out]

-1/6*(x^4*Hypergeometric2F1[-3/2, 3/4, 7/4, c^4*x^4])/(c^2*Sqrt[2 - 2*c^4*x^4]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {x^{4} \left (5 c^{4} x^{4}-11\right ) \sqrt {2}}{180 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{15 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{4} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\)	\(140\)

[In]

int(x^5/csch(2*ln(c*x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/180*x^4*(5*c^4*x^4-11)*2^(1/2)/c^2/(c^2*x^2/(c^4*x^4-1))^(1/2)+1/15/(-c^2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2
+1)^(1/2)/(c^4*x^4-1)/c^4*(EllipticF(x*(-c^2)^(1/2),I)-EllipticE(x*(-c^2)^(1/2),I))*2^(1/2)*x/(c^2*x^2/(c^4*x^
4-1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.65 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (5 \, c^{14} x^{12} - 16 \, c^{10} x^{8} + 23 \, c^{6} x^{4} - 12 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + 12 \, \sqrt {c^{4}} {\left (\sqrt {2} x^{2} E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - \sqrt {2} x^{2} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{180 \, c^{10} x^{2}} \]

[In]

integrate(x^5/csch(2*log(c*x))^(3/2),x, algorithm="fricas")

[Out]

1/180*(sqrt(2)*(5*c^14*x^12 - 16*c^10*x^8 + 23*c^6*x^4 - 12*c^2)*sqrt(c^2*x^2/(c^4*x^4 - 1)) + 12*sqrt(c^4)*(s
qrt(2)*x^2*elliptic_e(arcsin(1/(c*x)), -1) - sqrt(2)*x^2*elliptic_f(arcsin(1/(c*x)), -1)))/(c^10*x^2)

Sympy [F]

\[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{5}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]

[In]

integrate(x**5/csch(2*ln(c*x))**(3/2),x)

[Out]

Integral(x**5/csch(2*log(c*x))**(3/2), x)

Maxima [F]

\[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{5}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^5/csch(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^5/csch(2*log(c*x))^(3/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]

[In]

integrate(x^5/csch(2*log(c*x))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^5}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]

[In]

int(x^5/(1/sinh(2*log(c*x)))^(3/2),x)

[Out]

int(x^5/(1/sinh(2*log(c*x)))^(3/2), x)