Optimal. Leaf size=92 \[ -\frac {3}{4 x^3 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 \tanh ^{-1}\left (\sqrt {\frac {1}{c^4 x^4}+1}\right )}{4 c^4 x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5545, 5543, 266, 47, 50, 63, 207} \[ -\frac {3}{4 x^3 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 \tanh ^{-1}\left (\sqrt {\frac {1}{c^4 x^4}+1}\right )}{4 c^4 x^3 \left (\frac {1}{c^4 x^4}+1\right )^{3/2} \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 207
Rule 266
Rule 5543
Rule 5545
Rubi steps
\begin {align*} \int \frac {1}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}\right )^{3/2} x^3 \, dx,x,c x\right )}{c^4 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(1+x)^{3/2}}{x^2} \, dx,x,\frac {1}{c^4 x^4}\right )}{4 c^4 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x}{4 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,\frac {1}{c^4 x^4}\right )}{8 c^4 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {3}{4 \left (c^4+\frac {1}{x^4}\right ) x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\frac {1}{c^4 x^4}\right )}{8 c^4 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {3}{4 \left (c^4+\frac {1}{x^4}\right ) x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\frac {1}{c^4 x^4}}\right )}{4 c^4 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {3}{4 \left (c^4+\frac {1}{x^4}\right ) x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{4 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {3 \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^4 x^4}}\right )}{4 c^4 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 64, normalized size = 0.70 \[ -\frac {\sqrt {c^4 x^4+1} \sqrt {\frac {c^2 x^2}{2 c^4 x^4+2}} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-c^4 x^4\right )}{4 c^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 106, normalized size = 1.15 \[ \frac {3 \, \sqrt {2} c^{3} x^{3} \log \left (-2 \, c^{4} x^{4} - 2 \, {\left (c^{5} x^{5} + c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} - 1\right ) + 2 \, \sqrt {2} {\left (c^{8} x^{8} - c^{4} x^{4} - 2\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{32 \, c^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 131, normalized size = 1.42 \[ \frac {\left (c^{8} x^{8}-c^{4} x^{4}-2\right ) \sqrt {2}}{16 x \left (c^{4} x^{4}+1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {3 c^{2} \ln \left (\frac {c^{4} x^{2}}{\sqrt {c^{4}}}+\sqrt {c^{4} x^{4}+1}\right ) \sqrt {2}\, x}{16 \sqrt {c^{4}}\, \sqrt {c^{4} x^{4}+1}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}} \]
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 \[ \int \frac {1}{\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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