Integrand size = 15, antiderivative size = 162 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {4}{15 c^4 \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {2 x^2}{15 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^6}{9 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{15 c^9 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
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)
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 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=-\frac {x^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right )}{6 c^2 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
-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)])
Time = 0.39 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6086, 6084, 858, 809, 809, 847, 836, 762, 1388, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\) |
\(\Big \downarrow \) 6086 |
\(\displaystyle \frac {\int \frac {c^5 x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))}d(c x)}{c^6}\) |
\(\Big \downarrow \) 6084 |
\(\displaystyle \frac {\int c^8 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^8d(c x)}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {\int \frac {\left (1-c^4 x^4\right )^{3/2}}{c^{10} x^{10}}d\frac {1}{c x}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {-\frac {2}{3} \int \frac {\sqrt {1-c^4 x^4}}{c^6 x^6}d\frac {1}{c x}-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \int \frac {1}{c^2 x^2 \sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (-\int \frac {c^2 x^2}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (\int \frac {1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (-\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}+\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (-\int \frac {\sqrt {c^2 x^2+1}}{\sqrt {1-c^2 x^2}}d\frac {1}{c x}+\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {-\frac {2}{3} \left (-\frac {2}{5} \left (\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-E\left (\left .\arcsin \left (\frac {1}{c x}\right )\right |-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{9 c^9 x^9}}{c^9 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
-((-1/9*(1 - c^4*x^4)^(3/2)/(c^9*x^9) - (2*(-1/5*Sqrt[1 - c^4*x^4]/(c^5*x^ 5) - (2*(-(Sqrt[1 - c^4*x^4]/(c*x)) - EllipticE[ArcSin[1/(c*x)], -1] + Ell ipticF[ArcSin[1/(c*x)], -1]))/5))/3)/(c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch [2*Log[c*x]]^(3/2)))
3.2.46.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
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] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(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] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
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] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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] && Int egerQ[m]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[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]
Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m _.), x_Symbol] :> Simp[(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])
Time = 0.66 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {x^{4} \left (5 c^{4} x^{4}-11\right ) \sqrt {2}}{180 c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{15 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{4} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(140\) |
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)
Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.65 \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (5 \, c^{14} x^{12} - 16 \, c^{10} x^{8} + 23 \, c^{6} x^{4} - 12 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + 12 \, \sqrt {c^{4}} {\left (\sqrt {2} x^{2} E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - \sqrt {2} x^{2} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{180 \, c^{10} x^{2}} \]
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)*(sqrt(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)
\[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{5}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
\[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{5}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^5}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^5}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]