Integrand size = 15, antiderivative size = 118 \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {4}{77 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 x^4}{77 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^8}{11 \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {4 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{77 c^{11} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]
4/77/c^4/(c^4-1/x^4)/csch(2*ln(c*x))^(3/2)-6/77*x^4/(c^4-1/x^4)/csch(2*ln( c*x))^(3/2)+1/11*x^8/csch(2*ln(c*x))^(3/2)-4/77*EllipticF(1/c/x,I)/c^11/(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 = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {x^2 \left (\left (1-c^4 x^4\right )^{5/2}-\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},c^4 x^4\right )\right )}{22 c^6 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
(x^2*((1 - c^4*x^4)^(5/2) - Hypergeometric2F1[-3/2, 1/4, 5/4, c^4*x^4]))/( 22*c^6*Sqrt[2 - 2*c^4*x^4]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6086, 6084, 858, 809, 809, 847, 762}
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.
\(\displaystyle \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\) |
\(\Big \downarrow \) 6086 |
\(\displaystyle \frac {\int \frac {c^7 x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))}d(c x)}{c^8}\) |
\(\Big \downarrow \) 6084 |
\(\displaystyle \frac {\int c^{10} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^{10}d(c x)}{c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {\int \frac {\left (1-c^4 x^4\right )^{3/2}}{c^{12} x^{12}}d\frac {1}{c x}}{c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {-\frac {6}{11} \int \frac {\sqrt {1-c^4 x^4}}{c^8 x^8}d\frac {1}{c x}-\frac {\left (1-c^4 x^4\right )^{3/2}}{11 c^{11} x^{11}}}{c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {-\frac {6}{11} \left (-\frac {2}{7} \int \frac {1}{c^4 x^4 \sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{7 c^7 x^7}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{11 c^{11} x^{11}}}{c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {-\frac {6}{11} \left (-\frac {2}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{3 c^3 x^3}\right )-\frac {\sqrt {1-c^4 x^4}}{7 c^7 x^7}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{11 c^{11} x^{11}}}{c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {-\frac {6}{11} \left (-\frac {2}{7} \left (\frac {1}{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{3 c^3 x^3}\right )-\frac {\sqrt {1-c^4 x^4}}{7 c^7 x^7}\right )-\frac {\left (1-c^4 x^4\right )^{3/2}}{11 c^{11} x^{11}}}{c^{11} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}\) |
-((-1/11*(1 - c^4*x^4)^(3/2)/(c^11*x^11) - (6*(-1/7*Sqrt[1 - c^4*x^4]/(c^7 *x^7) - (2*(-1/3*Sqrt[1 - c^4*x^4]/(c^3*x^3) + EllipticF[ArcSin[1/(c*x)], -1]/3))/7))/11)/(c^11*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)))
3.2.44.3.1 Defintions of rubi rules used
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[((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[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 = 0.57 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {x^{2} \left (7 c^{8} x^{8}-13 c^{4} x^{4}+4\right ) \sqrt {2}}{308 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right ) \sqrt {2}\, x}{77 c^{6} \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(133\) |
1/308*x^2*(7*c^8*x^8-13*c^4*x^4+4)/c^6*2^(1/2)/(c^2*x^2/(c^4*x^4-1))^(1/2) +1/77/c^6/(-c^2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)*El lipticF(x*(-c^2)^(1/2),I)*2^(1/2)*x/(c^2*x^2/(c^4*x^4-1))^(1/2)
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (7 \, c^{14} x^{12} - 20 \, c^{10} x^{8} + 17 \, c^{6} x^{4} - 4 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 4 \, \sqrt {2} \sqrt {c^{4}} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)}{308 \, c^{10}} \]
1/308*(sqrt(2)*(7*c^14*x^12 - 20*c^10*x^8 + 17*c^6*x^4 - 4*c^2)*sqrt(c^2*x ^2/(c^4*x^4 - 1)) - 4*sqrt(2)*sqrt(c^4)*elliptic_f(arcsin(1/(c*x)), -1))/c ^10
\[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{7}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
\[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{7}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^7}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]