Optimal. Leaf size=214 \[ -\frac {12}{5 \left (c^4+\frac {1}{x^4}\right ) \left (c^2+\frac {1}{x^2}\right ) x^4 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {6}{5 \left (c^4+\frac {1}{x^4}\right ) x^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 c \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 \left (c^4+\frac {1}{x^4}\right )^2 x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 c \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 \left (c^4+\frac {1}{x^4}\right )^2 x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \]
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Rubi [A]
time = 0.09, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5670, 5668,
342, 283, 311, 226, 1210} \begin {gather*} \frac {6}{5 x^2 \left (c^4+\frac {1}{x^4}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {12}{5 x^4 \left (c^4+\frac {1}{x^4}\right ) \left (c^2+\frac {1}{x^2}\right ) \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 c \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 ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 x^3 \left (c^4+\frac {1}{x^4}\right )^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 c \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 ) E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 x^3 \left (c^4+\frac {1}{x^4}\right )^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 283
Rule 311
Rule 342
Rule 1210
Rule 5668
Rule 5670
Rubi steps
\begin {align*} \int \frac {x}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\text {Subst}\left (\int \frac {x}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^2}\\ &=\frac {\text {Subst}\left (\int \left (1+\frac {1}{x^4}\right )^{3/2} x^4 \, dx,x,c x\right )}{c^5 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {\text {Subst}\left (\int \frac {\left (1+x^4\right )^{3/2}}{x^6} \, dx,x,\frac {1}{c x}\right )}{c^5 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {x^2}{5 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 \text {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^2} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {6}{5 \left (c^4+\frac {1}{x^4}\right ) x^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {6}{5 \left (c^4+\frac {1}{x^4}\right ) x^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {12 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \left (1+\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ &=-\frac {12}{5 \left (c^4+\frac {1}{x^4}\right ) \left (c^2+\frac {1}{x^2}\right ) x^4 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {6}{5 \left (c^4+\frac {1}{x^4}\right ) x^2 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^2}{5 \text {sech}^{\frac {3}{2}}(2 \log (c x))}+\frac {12 c \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 ) E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 \left (c^4+\frac {1}{x^4}\right )^2 x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}-\frac {6 c \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 ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 \left (c^4+\frac {1}{x^4}\right )^2 x^3 \text {sech}^{\frac {3}{2}}(2 \log (c x))}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.09, size = 65, normalized size = 0.30 \begin {gather*} -\frac {\, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};-c^4 x^4\right )}{2 \sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{1+c^4 x^4}} \sqrt {1+c^4 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.54, size = 159, normalized size = 0.74
method | result | size |
risch | \(\frac {\left (c^{8} x^{8}-4 c^{4} x^{4}-5\right ) \sqrt {2}}{20 \left (c^{4} x^{4}+1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {3 i \sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \left (\EllipticF \left (x \sqrt {i c^{2}}, i\right )-\EllipticE \left (x \sqrt {i c^{2}}, i\right )\right ) \sqrt {2}\, x}{5 \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(159\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {x}{\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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