Optimal. Leaf size=11 \[ -i F(i x|-1) \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3261}
\begin {gather*} -i F(i x|-1) \end {gather*}
Antiderivative was successfully verified.
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Rule 3261
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\sinh ^2(x)}} \, dx &=-i F(i x|-1)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 11, normalized size = 1.00 \begin {gather*} -i F(i x|-1) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 41, normalized size = 3.73
method | result | size |
default | \(\frac {\sqrt {-\left (-1+\sinh ^{2}\left (x \right )\right ) \left (\cosh ^{2}\left (x \right )\right )}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticF \left (\sinh \left (x \right ), i\right )}{\sqrt {1-\left (\sinh ^{4}\left (x \right )\right )}\, \cosh \left (x \right )}\) | \(41\) |
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 = 0.09, size = 1, normalized size = 0.09 \begin {gather*} 0 \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 {1 - \sinh ^{2}{\left (x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.09 \begin {gather*} \int \frac {1}{\sqrt {1-{\mathrm {sinh}\left (x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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