Optimal. Leaf size=33 \[ -\frac {i \sqrt {\sinh ^2(x)-1} E(i x|-1)}{\sqrt {1-\sinh ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3178, 3177} \[ -\frac {i \sqrt {\sinh ^2(x)-1} E(i x|-1)}{\sqrt {1-\sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3177
Rule 3178
Rubi steps
\begin {align*} \int \sqrt {-1+\sinh ^2(x)} \, dx &=\frac {\sqrt {-1+\sinh ^2(x)} \int \sqrt {1-\sinh ^2(x)} \, dx}{\sqrt {1-\sinh ^2(x)}}\\ &=-\frac {i E(i x|-1) \sqrt {-1+\sinh ^2(x)}}{\sqrt {1-\sinh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 1.00 \[ \frac {i \sqrt {3-\cosh (2 x)} E(i x|-1)}{\sqrt {\cosh (2 x)-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 3.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\sinh \relax (x)^{2} - 1}, x\right ) \]
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 \[ \int \sqrt {\sinh \relax (x)^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 61, normalized size = 1.85 \[ \frac {i \sqrt {\left (-1+\sinh ^{2}\relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {1-\left (\sinh ^{2}\relax (x )\right )}\, \EllipticE \left (i \sinh \relax (x ), i\right )}{\sqrt {\sinh ^{4}\relax (x )-1}\, \cosh \relax (x ) \sqrt {-1+\sinh ^{2}\relax (x )}} \]
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 \[ \int \sqrt {\sinh \relax (x)^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {{\mathrm {sinh}\relax (x)}^2-1} \,d x \]
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 \[ \int \sqrt {\sinh ^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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