Optimal. Leaf size=41 \[ -\frac {1}{2} c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \csc ^{-1}\left (c^2 x^2\right ) \sqrt {\text {csch}(2 \log (c x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5552, 5550, 335, 275, 216} \[ -\frac {1}{2} c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \csc ^{-1}\left (c^2 x^2\right ) \sqrt {\text {csch}(2 \log (c x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 275
Rule 335
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx &=c \operatorname {Subst}\left (\int \frac {\sqrt {\text {csch}(2 \log (x))}}{x^2} \, dx,x,c x\right )\\ &=\left (c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^4}} x^3} \, dx,x,c x\right )\\ &=-\left (\left (c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )\right )\\ &=-\left (\frac {1}{2} \left (c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c^2 x^2}\right )\right )\\ &=-\frac {1}{2} c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \csc ^{-1}\left (c^2 x^2\right ) \sqrt {\text {csch}(2 \log (c x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 54, normalized size = 1.32 \[ \frac {\sqrt {c^4 x^4-1} \sqrt {\frac {c^2 x^2}{2 c^4 x^4-2}} \tan ^{-1}\left (\sqrt {c^4 x^4-1}\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 43, normalized size = 1.05 \[ \frac {1}{2} \, \sqrt {2} c \arctan \left (\frac {{\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\mathrm {csch}\left (2 \ln \left (c x \right )\right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________