Integrand size = 15, antiderivative size = 67 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {\text {csch}(2 \log (c x))} \sqrt {i \sinh (2 \log (c x))}} \] Output:
-cosh(2*ln(c*x))*csch(2*ln(c*x))^(1/2)+I*EllipticE(cos(1/4*Pi+I*ln(c*x)),2 ^(1/2))/csch(2*ln(c*x))^(1/2)/(I*sinh(2*ln(c*x)))^(1/2)
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\sqrt {\text {csch}(2 \log (c x))} \left (-\cosh (2 \log (c x))+E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right ) \sqrt {i \sinh (2 \log (c x))}\right ) \] Input:
Integrate[Csch[2*Log[c*x]]^(3/2)/x,x]
Output:
Sqrt[Csch[2*Log[c*x]]]*(-Cosh[2*Log[c*x]] + EllipticE[Pi/4 - I*Log[c*x], 2 ]*Sqrt[I*Sinh[2*Log[c*x]]])
Time = 0.43 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3039, 3042, 4255, 3042, 4258, 3042, 3119}
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.
\(\displaystyle \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \int \text {csch}^{\frac {3}{2}}(2 \log (c x))d\log (c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (i \csc (2 i \log (c x)))^{3/2}d\log (c x)\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \int \frac {1}{\sqrt {\text {csch}(2 \log (c x))}}d\log (c x)-\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\int \frac {1}{\sqrt {i \csc (2 i \log (c x))}}d\log (c x)\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {\int \sqrt {i \sinh (2 \log (c x))}d\log (c x)}{\sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {\int \sqrt {\sin (2 i \log (c x))}d\log (c x)}{\sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))}}\) |
Input:
Int[Csch[2*Log[c*x]]^(3/2)/x,x]
Output:
-(Cosh[2*Log[c*x]]*Sqrt[Csch[2*Log[c*x]]]) + (I*EllipticE[Pi/4 - I*Log[c*x ], 2])/(Sqrt[Csch[2*Log[c*x]]]*Sqrt[I*Sinh[2*Log[c*x]]])
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (59 ) = 118\).
Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.43
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (2 \ln \left (x c \right )\right )^{2}}{2 \cosh \left (2 \ln \left (x c \right )\right ) \sqrt {\sinh \left (2 \ln \left (x c \right )\right )}}\) | \(163\) |
default | \(\frac {2 \sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (x c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (2 \ln \left (x c \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (2 \ln \left (x c \right )\right )^{2}}{2 \cosh \left (2 \ln \left (x c \right )\right ) \sqrt {\sinh \left (2 \ln \left (x c \right )\right )}}\) | \(163\) |
Input:
int(csch(2*ln(x*c))^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
1/2*(2*(1-I*sinh(2*ln(x*c)))^(1/2)*2^(1/2)*(I*sinh(2*ln(x*c))+1)^(1/2)*(I* sinh(2*ln(x*c)))^(1/2)*EllipticE((1-I*sinh(2*ln(x*c)))^(1/2),1/2*2^(1/2))- (1-I*sinh(2*ln(x*c)))^(1/2)*2^(1/2)*(I*sinh(2*ln(x*c))+1)^(1/2)*(I*sinh(2* ln(x*c)))^(1/2)*EllipticF((1-I*sinh(2*ln(x*c)))^(1/2),1/2*2^(1/2))-2*cosh( 2*ln(x*c))^2)/cosh(2*ln(x*c))/sinh(2*ln(x*c))^(1/2)
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} x^{2} - 2 \, \sqrt {-\frac {1}{2}} c^{2} E(\arcsin \left (c x\right )\,|\,-1) + 2 \, \sqrt {-\frac {1}{2}} c^{2} F(\arcsin \left (c x\right )\,|\,-1) \] Input:
integrate(csch(2*log(c*x))^(3/2)/x,x, algorithm="fricas")
Output:
-sqrt(2)*sqrt(c^2*x^2/(c^4*x^4 - 1))*c^2*x^2 - 2*sqrt(-1/2)*c^2*elliptic_e (arcsin(c*x), -1) + 2*sqrt(-1/2)*c^2*elliptic_f(arcsin(c*x), -1)
\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \] Input:
integrate(csch(2*ln(c*x))**(3/2)/x,x)
Output:
Integral(csch(2*log(c*x))**(3/2)/x, x)
\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int { \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x} \,d x } \] Input:
integrate(csch(2*log(c*x))^(3/2)/x,x, algorithm="maxima")
Output:
integrate(csch(2*log(c*x))^(3/2)/x, x)
Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\text {Timed out} \] Input:
integrate(csch(2*log(c*x))^(3/2)/x,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \] Input:
int((1/sinh(2*log(c*x)))^(3/2)/x,x)
Output:
int((1/sinh(2*log(c*x)))^(3/2)/x, x)
\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {\sqrt {\mathrm {csch}\left (2 \,\mathrm {log}\left (c x \right )\right )}\, \mathrm {csch}\left (2 \,\mathrm {log}\left (c x \right )\right )}{x}d x \] Input:
int(csch(2*log(c*x))^(3/2)/x,x)
Output:
int((sqrt(csch(2*log(c*x)))*csch(2*log(c*x)))/x,x)