Optimal. Leaf size=25 \[ \frac {1}{2} \left (c^4+\frac {1}{x^4}\right ) x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x)) \]
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Rubi [A]
time = 0.03, 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 = {5670, 5668,
267} \begin {gather*} \frac {1}{2} x^3 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5668
Rule 5670
Rubi steps
\begin {align*} \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx &=c \text {Subst}\left (\int \frac {\text {sech}^{\frac {3}{2}}(2 \log (x))}{x^2} \, dx,x,c x\right )\\ &=\left (c^4 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {1}{x^4}\right )^{3/2} x^5} \, dx,x,c x\right )\\ &=\frac {1}{2} \left (c^4+\frac {1}{x^4}\right ) x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 1.28 \begin {gather*} \sqrt {2} c^2 x \sqrt {\frac {c^2 x^2}{1+c^4 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.74, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 39, normalized size = 1.56 \begin {gather*} c {\left (\frac {\sqrt {2}}{{\left (\frac {1}{c^{4} x^{4}} + 1\right )}^{\frac {3}{2}}} + \frac {\sqrt {2}}{c^{4} x^{4} {\left (\frac {1}{c^{4} x^{4}} + 1\right )}^{\frac {3}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 28, normalized size = 1.12 \begin {gather*} \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{2} x \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 28, normalized size = 1.12 \begin {gather*} c^2\,x\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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