Optimal. Leaf size=17 \[ -i F\left (\left .i x+\frac {\pi }{2}\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3182} \[ -i F\left (\left .i x+\frac {\pi }{2}\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 3182
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\cosh ^2(x)}} \, dx &=-i F\left (\left .\frac {\pi }{2}+i x\right |-1\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 18, normalized size = 1.06 \[ -\frac {i F\left (i x\left |\frac {1}{2}\right .\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {\cosh \relax (x)^{2} + 1}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {\cosh \relax (x)^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 45, normalized size = 2.65 \[ -\frac {i \sqrt {\left (1+\cosh ^{2}\relax (x )\right ) \left (\sinh ^{2}\relax (x )\right )}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \EllipticF \left (i \cosh \relax (x ), i\right )}{\sqrt {\cosh ^{4}\relax (x )-1}\, \sinh \relax (x )} \]
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 \[ \int \frac {1}{\sqrt {\cosh \relax (x)^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {{\mathrm {cosh}\relax (x)}^2+1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\cosh ^{2}{\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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