Integrand size = 15, antiderivative size = 69 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=-\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))+\frac {1}{2} c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x)) \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right ) \] Output:
-1/2*(c^4-1/x^4)*x^2*csch(2*ln(c*x))^(3/2)+1/2*c^5*(1-1/c^4/x^4)^(3/2)*x^3 *csch(2*ln(c*x))^(3/2)*InverseJacobiAM(arccsc(c*x),I)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=-\sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \left (1+\sqrt {1-c^4 x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^4 x^4\right )\right ) \] Input:
Integrate[Csch[2*Log[c*x]]^(3/2)/x^3,x]
Output:
-(Sqrt[2]*c^2*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)]*(1 + Sqrt[1 - c^4*x^4]*Hyperg eometric2F1[1/4, 1/2, 5/4, c^4*x^4]))
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6086, 6084, 858, 817, 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.
\(\displaystyle \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx\) |
\(\Big \downarrow \) 6086 |
\(\displaystyle c^2 \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{c^3 x^3}d(c x)\) |
\(\Big \downarrow \) 6084 |
\(\displaystyle c^5 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x)) \int \frac {1}{c^6 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^6}d(c x)\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -c^5 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x)) \int \frac {c^4 x^4}{\left (1-c^4 x^4\right )^{3/2}}d\frac {1}{c x}\) |
\(\Big \downarrow \) 817 |
\(\displaystyle -c^5 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x)) \left (\frac {1}{2 c x \sqrt {1-c^4 x^4}}-\frac {1}{2} \int \frac {1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}\right )\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -c^5 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \left (\frac {1}{2 c x \sqrt {1-c^4 x^4}}-\frac {1}{2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))\) |
Input:
Int[Csch[2*Log[c*x]]^(3/2)/x^3,x]
Output:
-(c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)*(1/(2*c*x*Sqrt[1 - c^4*x^4]) - EllipticF[ArcSin[1/(c*x)], -1]/2))
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^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && 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])
\[\int \frac {\operatorname {csch}\left (2 \ln \left (x c \right )\right )^{\frac {3}{2}}}{x^{3}}d x\]
Input:
int(csch(2*ln(x*c))^(3/2)/x^3,x)
Output:
int(csch(2*ln(x*c))^(3/2)/x^3,x)
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.62 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=2 \, \sqrt {-\frac {1}{2}} c^{2} F(\arcsin \left (c x\right )\,|\,-1) - \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} \] Input:
integrate(csch(2*log(c*x))^(3/2)/x^3,x, algorithm="fricas")
Output:
2*sqrt(-1/2)*c^2*elliptic_f(arcsin(c*x), -1) - sqrt(2)*sqrt(c^2*x^2/(c^4*x ^4 - 1))*c^2
\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{3}}\, dx \] Input:
integrate(csch(2*ln(c*x))**(3/2)/x**3,x)
Output:
Integral(csch(2*log(c*x))**(3/2)/x**3, x)
\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\int { \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{3}} \,d x } \] Input:
integrate(csch(2*log(c*x))^(3/2)/x^3,x, algorithm="maxima")
Output:
integrate(csch(2*log(c*x))^(3/2)/x^3, x)
Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\text {Timed out} \] Input:
integrate(csch(2*log(c*x))^(3/2)/x^3,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x^3} \,d x \] Input:
int((1/sinh(2*log(c*x)))^(3/2)/x^3,x)
Output:
int((1/sinh(2*log(c*x)))^(3/2)/x^3, x)
\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\int \frac {\sqrt {\mathrm {csch}\left (2 \,\mathrm {log}\left (c x \right )\right )}\, \mathrm {csch}\left (2 \,\mathrm {log}\left (c x \right )\right )}{x^{3}}d x \] Input:
int(csch(2*log(c*x))^(3/2)/x^3,x)
Output:
int((sqrt(csch(2*log(c*x)))*csch(2*log(c*x)))/x**3,x)