Integrand size = 15, antiderivative size = 81 \[ \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=-\frac {2 x^2}{21 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{21 c^7 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \]
-2/21*x^2/c^4/csch(2*ln(c*x))^(1/2)+1/7*x^6/csch(2*ln(c*x))^(1/2)+2/21*Ell ipticF(1/c/x,I)/c^7/x/(1-1/c^4/x^4)^(1/2)/csch(2*ln(c*x))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^2 \left (-\left (1-c^4 x^4\right )^{3/2}+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},c^4 x^4\right )\right )}{7 c^4 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
(x^2*(-(1 - c^4*x^4)^(3/2) + Hypergeometric2F1[-1/2, 1/4, 5/4, c^4*x^4]))/ (7*c^4*Sqrt[2 - 2*c^4*x^4]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6086, 6084, 858, 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.
\(\displaystyle \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx\) |
\(\Big \downarrow \) 6086 |
\(\displaystyle \frac {\int \frac {c^5 x^5}{\sqrt {\text {csch}(2 \log (c x))}}d(c x)}{c^6}\) |
\(\Big \downarrow \) 6084 |
\(\displaystyle \frac {\int c^6 \sqrt {1-\frac {1}{c^4 x^4}} x^6d(c x)}{c^7 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {\int \frac {\sqrt {1-c^4 x^4}}{c^8 x^8}d\frac {1}{c x}}{c^7 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {-\frac {2}{7} \int \frac {1}{c^4 x^4 \sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{7 c^7 x^7}}{c^7 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {-\frac {2}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{3 c^3 x^3}\right )-\frac {\sqrt {1-c^4 x^4}}{7 c^7 x^7}}{c^7 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {-\frac {2}{7} \left (\frac {1}{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{3 c^3 x^3}\right )-\frac {\sqrt {1-c^4 x^4}}{7 c^7 x^7}}{c^7 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
-((-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)/(c^7*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt [Csch[2*Log[c*x]]]))
3.2.32.3.1 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 = 0.82 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.54
method | result | size |
risch | \(\frac {x^{2} \left (3 c^{4} x^{4}-2\right ) \sqrt {2}}{42 c^{4} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right ) \sqrt {2}\, x}{21 c^{4} \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(125\) |
1/42*x^2*(3*c^4*x^4-2)/c^4*2^(1/2)/(c^2*x^2/(c^4*x^4-1))^(1/2)-1/21/c^4/(- c^2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)*EllipticF(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 = 72, normalized size of antiderivative = 0.89 \[ \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {\sqrt {2} {\left (3 \, c^{10} x^{8} - 5 \, c^{6} x^{4} + 2 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + 2 \, \sqrt {2} \sqrt {c^{4}} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)}{42 \, c^{8}} \]
1/42*(sqrt(2)*(3*c^10*x^8 - 5*c^6*x^4 + 2*c^2)*sqrt(c^2*x^2/(c^4*x^4 - 1)) + 2*sqrt(2)*sqrt(c^4)*elliptic_f(arcsin(1/(c*x)), -1))/c^8
\[ \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int \frac {x^{5}}{\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
\[ \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int { \frac {x^{5}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
\[ \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int { \frac {x^{5}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
Timed out. \[ \int \frac {x^5}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int \frac {x^5}{\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]