Optimal. Leaf size=59 \[ \frac {x}{2 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\text {csch}^{-1}\left (c^2 x^2\right )}{2 c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {5664, 5662,
342, 281, 283, 221} \begin {gather*} \frac {x}{2 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\text {csch}^{-1}\left (c^2 x^2\right )}{2 c^2 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 283
Rule 342
Rule 5662
Rule 5664
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\text {sech}(2 \log (c x))}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c}\\ &=\frac {\text {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x \, dx,x,c x\right )}{c^2 \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^3} \, dx,x,\frac {1}{c x}\right )}{c^2 \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^2}}{x^2} \, dx,x,\frac {1}{c^2 x^2}\right )}{2 c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {x}{2 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\frac {1}{c^2 x^2}\right )}{2 c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {x}{2 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\text {csch}^{-1}\left (c^2 x^2\right )}{2 c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 77, normalized size = 1.31 \begin {gather*} \frac {x \left (2 \sqrt {1+c^4 x^4}-2 \tanh ^{-1}\left (\sqrt {1+c^4 x^4}\right )\right )}{4 \sqrt {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 [F]
time = 0.92, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {\mathrm {sech}\left (2 \ln \left (c x \right )\right )}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (49) = 98\).
time = 0.37, size = 100, normalized size = 1.69 \begin {gather*} \frac {\sqrt {2} c x \log \left (\frac {c^{5} x^{5} + 2 \, c x - 2 \, {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) + 2 \, \sqrt {2} {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{8 \, c^{2} x} \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 {1}{\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(-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 = -0.02 \begin {gather*} \int \frac {1}{\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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