Optimal. Leaf size=128 \[ \frac{2 i \sqrt{a+b \sinh (x)} E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{b \sqrt{\frac{a+b \sinh (x)}{a-i b}}}-\frac{2 i a \sqrt{\frac{a+b \sinh (x)}{a-i b}} \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},\frac{2 b}{b+i a}\right )}{b \sqrt{a+b \sinh (x)}} \]
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Rubi [A] time = 0.114254, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2752, 2663, 2661, 2655, 2653} \[ \frac{2 i \sqrt{a+b \sinh (x)} E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{b \sqrt{\frac{a+b \sinh (x)}{a-i b}}}-\frac{2 i a \sqrt{\frac{a+b \sinh (x)}{a-i b}} F\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{b \sqrt{a+b \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{\sqrt{a+b \sinh (x)}} \, dx &=\frac{\int \sqrt{a+b \sinh (x)} \, dx}{b}-\frac{a \int \frac{1}{\sqrt{a+b \sinh (x)}} \, dx}{b}\\ &=\frac{\sqrt{a+b \sinh (x)} \int \sqrt{\frac{a}{a-i b}+\frac{b \sinh (x)}{a-i b}} \, dx}{b \sqrt{\frac{a+b \sinh (x)}{a-i b}}}-\frac{\left (a \sqrt{\frac{a+b \sinh (x)}{a-i b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a-i b}+\frac{b \sinh (x)}{a-i b}}} \, dx}{b \sqrt{a+b \sinh (x)}}\\ &=\frac{2 i E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right ) \sqrt{a+b \sinh (x)}}{b \sqrt{\frac{a+b \sinh (x)}{a-i b}}}-\frac{2 i a F\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right ) \sqrt{\frac{a+b \sinh (x)}{a-i b}}}{b \sqrt{a+b \sinh (x)}}\\ \end{align*}
Mathematica [A] time = 0.351524, size = 101, normalized size = 0.79 \[ \frac{2 \sqrt{\frac{a+b \sinh (x)}{a-i b}} \left ((b+i a) E\left (\frac{1}{4} (\pi -2 i x)|-\frac{2 i b}{a-i b}\right )-i a \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),-\frac{2 i b}{a-i b}\right )\right )}{b \sqrt{a+b \sinh (x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 218, normalized size = 1.7 \begin{align*} 2\,{\frac{ib-a}{{b}^{2}\cosh \left ( x \right ) \sqrt{a+b\sinh \left ( x \right ) }}\sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}}\sqrt{{\frac{ \left ( i-\sinh \left ( x \right ) \right ) b}{ib+a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( x \right ) \right ) b}{ib-a}}} \left ( i{\it EllipticE} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+a}}} \right ) b-i{\it EllipticF} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+a}}} \right ) b+{\it EllipticE} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+a}}} \right ) a \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (x\right )}{\sqrt{b \sinh \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sinh \left (x\right )}{\sqrt{b \sinh \left (x\right ) + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (x \right )}}{\sqrt{a + b \sinh{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (x\right )}{\sqrt{b \sinh \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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