Optimal. Leaf size=25 \[ \frac {1}{2} x \left (c^4-\frac {1}{x^4}\right ) \sqrt {\text {csch}(2 \log (c x))} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5552, 5550, 261} \[ \frac {1}{2} x \left (c^4-\frac {1}{x^4}\right ) \sqrt {\text {csch}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5550
Rule 5552
Rubi steps
\begin {align*} \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^4} \, dx &=c^3 \operatorname {Subst}\left (\int \frac {\sqrt {\text {csch}(2 \log (x))}}{x^4} \, dx,x,c x\right )\\ &=\left (c^4 \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^5} \, dx,x,c x\right )\\ &=\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x \sqrt {\text {csch}(2 \log (c x))}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 33, normalized size = 1.32 \[ \frac {c^2}{2 x \sqrt {\frac {c^2 x^2}{2 c^4 x^4-2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 37, normalized size = 1.48 \[ \frac {\sqrt {2} {\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{2 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 0.14, size = 38, normalized size = 1.52 \[ \frac {\sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}\, \left (c^{4} x^{4}-1\right )}{2 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 89, normalized size = 3.56 \[ \frac {1}{2} \, c^{3} {\left (\frac {\sqrt {2}}{\sqrt {\frac {1}{c x} + 1} \sqrt {-\frac {1}{c x} + 1} \sqrt {\frac {1}{c^{2} x^{2}} + 1}} - \frac {\sqrt {2}}{c^{4} x^{4} \sqrt {\frac {1}{c x} + 1} \sqrt {-\frac {1}{c x} + 1} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 58, normalized size = 2.32 \[ \frac {c^4\,x\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4-1}}}{2}-\frac {\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4-1}}}{2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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