Integrand size = 15, antiderivative size = 119 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=-\frac {2}{5 c^4 \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {csch}(2 \log (c x))}}-\frac {2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \]
-2/5/c^4/csch(2*ln(c*x))^(1/2)+1/5*x^4/csch(2*ln(c*x))^(1/2)-2/5*EllipticE (1/c/x,I)/c^5/x/(1-1/c^4/x^4)^(1/2)/csch(2*ln(c*x))^(1/2)+2/5*EllipticF(1/ c/x,I)/c^5/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 = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.50 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right )}{3 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]
(x^4*Hypergeometric2F1[-1/2, 3/4, 7/4, c^4*x^4])/(3*Sqrt[2 - 2*c^4*x^4]*Sq rt[(c^2*x^2)/(-1 + c^4*x^4)])
Time = 0.36 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6086, 6084, 858, 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.
\(\displaystyle \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx\) |
\(\Big \downarrow \) 6086 |
\(\displaystyle \frac {\int \frac {c^3 x^3}{\sqrt {\text {csch}(2 \log (c x))}}d(c x)}{c^4}\) |
\(\Big \downarrow \) 6084 |
\(\displaystyle \frac {\int c^4 \sqrt {1-\frac {1}{c^4 x^4}} x^4d(c x)}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {\int \frac {\sqrt {1-c^4 x^4}}{c^6 x^6}d\frac {1}{c x}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {-\frac {2}{5} \int \frac {1}{c^2 x^2 \sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {-\frac {2}{5} \left (-\int \frac {c^2 x^2}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle -\frac {-\frac {2}{5} \left (\int \frac {1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {-\frac {2}{5} \left (-\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}+\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -\frac {-\frac {2}{5} \left (-\int \frac {\sqrt {c^2 x^2+1}}{\sqrt {1-c^2 x^2}}d\frac {1}{c x}+\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {-\frac {2}{5} \left (\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )-E\left (\left .\arcsin \left (\frac {1}{c x}\right )\right |-1\right )-\frac {\sqrt {1-c^4 x^4}}{c x}\right )-\frac {\sqrt {1-c^4 x^4}}{5 c^5 x^5}}{c^5 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}\) |
-((-1/5*Sqrt[1 - c^4*x^4]/(c^5*x^5) - (2*(-(Sqrt[1 - c^4*x^4]/(c*x)) - Ell ipticE[ArcSin[1/(c*x)], -1] + EllipticF[ArcSin[1/(c*x)], -1]))/5)/(c^5*Sqr t[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]))
3.2.34.3.1 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 = 0.65 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {\sqrt {2}\, x^{4}}{10 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, x}{5 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\) | \(127\) |
1/10*2^(1/2)*x^4/(c^2*x^2/(c^4*x^4-1))^(1/2)-1/5/(-c^2)^(1/2)*(c^2*x^2+1)^ (1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)/c^2*(EllipticF(x*(-c^2)^(1/2),I)-Elli pticE(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 = 97, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {\sqrt {2} {\left (c^{10} x^{8} - 3 \, c^{6} x^{4} + 2 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 2 \, \sqrt {c^{4}} {\left (\sqrt {2} x^{2} E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) - \sqrt {2} x^{2} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{10 \, c^{8} x^{2}} \]
1/10*(sqrt(2)*(c^10*x^8 - 3*c^6*x^4 + 2*c^2)*sqrt(c^2*x^2/(c^4*x^4 - 1)) - 2*sqrt(c^4)*(sqrt(2)*x^2*elliptic_e(arcsin(1/(c*x)), -1) - sqrt(2)*x^2*el liptic_f(arcsin(1/(c*x)), -1)))/(c^8*x^2)
\[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int \frac {x^{3}}{\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
\[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int { \frac {x^{3}}{\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}} \,d x } \]
Exception generated. \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly exception caught Unable to convert to real %%{poly1[1.0000000000000000000000000000000,0.000000000000 000000000
Timed out. \[ \int \frac {x^3}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\int \frac {x^3}{\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]