Optimal. Leaf size=203 \[ \frac {2}{5 c^4 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2}{5 c^4 \left (c^2+\frac {1}{x^2}\right ) x^2 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {sech}(2 \log (c x))}}+\frac {2 \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 c^3 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}}-\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 c^3 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}} \]
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Rubi [A]
time = 0.09, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {5670, 5668,
342, 283, 331, 311, 226, 1210} \begin {gather*} \frac {2}{5 c^4 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2}{5 c^4 x^2 \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))}}-\frac {\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 c^3 x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}+\frac {2 \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 c^3 x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {sech}(2 \log (c x))}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 283
Rule 311
Rule 331
Rule 342
Rule 1210
Rule 5668
Rule 5670
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c^4}\\ &=\frac {\text {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x^4 \, dx,x,c x\right )}{c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^6} \, dx,x,\frac {1}{c x}\right )}{c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {x^4}{5 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {2}{5 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {2}{5 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}+\frac {2 \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{5 c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {2}{5 c^4 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2}{5 c^4 \left (c^2+\frac {1}{x^2}\right ) x^2 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {sech}(2 \log (c x))}}+\frac {2 \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 c^3 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}}-\frac {\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 c^3 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(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 = 0.32 \begin {gather*} \frac {\left (\frac {c^2 x^2}{1+c^4 x^4}\right )^{3/2} \left (1+c^4 x^4\right )^{3/2} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-c^4 x^4\right )}{3 \sqrt {2} c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.27, size = 134, normalized size = 0.66
method | result | size |
risch | \(\frac {x^{4} \sqrt {2}}{10 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {i \sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \left (\EllipticF \left (x \sqrt {i c^{2}}, i\right )-\EllipticE \left (x \sqrt {i c^{2}}, i\right )\right ) \sqrt {2}\, x}{5 \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(134\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^{3}}{\sqrt {\operatorname {sech}{\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3}{\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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