Optimal. Leaf size=92 \[ \frac {1}{2} x^2 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {x^3 \left (c^4+\frac {1}{x^4}\right ) \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{4 c} \]
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Rubi [A] time = 0.08, antiderivative size = 92, 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 = {5551, 5549, 335, 288, 220} \[ \frac {1}{2} x^2 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {x^3 \left (c^4+\frac {1}{x^4}\right ) \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x)) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 288
Rule 335
Rule 5549
Rule 5551
Rubi steps
\begin {align*} \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx &=c^2 \operatorname {Subst}\left (\int \frac {\text {sech}^{\frac {3}{2}}(2 \log (x))}{x^3} \, dx,x,c x\right )\\ &=\left (c^5 \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^6} \, dx,x,c x\right )\\ &=-\left (\left (c^5 \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 {x^4}{\left (1+x^4\right )^{3/2}} \, dx,x,\frac {1}{c x}\right )\right )\\ &=\frac {1}{2} \left (c^4+\frac {1}{x^4}\right ) x^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} \left (c^5 \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}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )\\ &=\frac {1}{2} \left (c^4+\frac {1}{x^4}\right ) x^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))-\frac {\left (c^4+\frac {1}{x^4}\right ) \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) x^3 F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}{4 c}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 65, normalized size = 0.71 \[ \sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{c^4 x^4+1}} \left (\sqrt {c^4 x^4+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-c^4 x^4\right )+1\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{3}}, x\right ) \]
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 = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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