Optimal. Leaf size=25 \[ \frac {1}{2} x^3 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) \]
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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 = {5551, 5549, 261} \[ \frac {1}{2} x^3 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) \]
Antiderivative was successfully verified.
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Rule 261
Rule 5549
Rule 5551
Rubi steps
\begin {align*} \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx &=c \operatorname {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 ) \operatorname {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 = 0.03, size = 32, normalized size = 1.28 \[ \sqrt {2} c^2 x \sqrt {\frac {c^2 x^2}{c^4 x^4+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 28, normalized size = 1.12 \[ \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{2} x \]
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 [F] time = 0.17, 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 = 0.40, size = 39, normalized size = 1.56 \[ c {\left (\frac {\sqrt {2}}{{\left (\frac {1}{c^{4} x^{4}} + 1\right )}^{\frac {3}{2}}} + \frac {\sqrt {2}}{c^{4} x^{4} {\left (\frac {1}{c^{4} x^{4}} + 1\right )}^{\frac {3}{2}}}\right )} \]
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 \[ c^2\,x\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}} \]
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 \[ \int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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