Optimal. Leaf size=57 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a-i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 \cosh (x)}{\sqrt {a-i a \sinh (x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2830, 2728,
212} \begin {gather*} \frac {2 \cosh (x)}{\sqrt {a-i a \sinh (x)}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a-i a \sinh (x)}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2830
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx &=\frac {2 \cosh (x)}{\sqrt {a-i a \sinh (x)}}-i \int \frac {1}{\sqrt {a-i a \sinh (x)}} \, dx\\ &=\frac {2 \cosh (x)}{\sqrt {a-i a \sinh (x)}}+2 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \cosh (x)}{\sqrt {a-i a \sinh (x)}}\right )\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a-i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 \cosh (x)}{\sqrt {a-i a \sinh (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 76, normalized size = 1.33 \begin {gather*} \frac {2 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right ) \left (\cosh \left (\frac {x}{2}\right )+i \left ((1+i) (-1)^{3/4} \text {ArcTan}\left (\frac {-i+\tanh \left (\frac {x}{4}\right )}{\sqrt {2}}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )}{\sqrt {a-i a \sinh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \frac {\sinh \left (x \right )}{\sqrt {a -i a \sinh \left (x \right )}}\, dx\]
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 [A]
time = 0.40, size = 76, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {2} \sqrt {a} \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {a} + \sqrt {-\frac {1}{2} i \, a e^{\left (-x\right )}}\right ) - \sqrt {2} \sqrt {a} \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {a} + \sqrt {-\frac {1}{2} i \, a e^{\left (-x\right )}}\right ) + 2 \, \sqrt {-\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (-i \, e^{x} - 1\right )}}{a} \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 {\sinh {\left (x \right )}}{\sqrt {- i a \left (\sinh {\left (x \right )} + i\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 = -0.02 \begin {gather*} \int \frac {\mathrm {sinh}\left (x\right )}{\sqrt {a-a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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