Optimal. Leaf size=248 \[ \frac{i \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{6 \sqrt{2} d \sqrt{2 a+b \sinh (2 c+2 d x)}}+\frac{b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{6 \sqrt{2} d}-\frac{2 i \sqrt{2} a \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{3 d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
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Rubi [A] time = 0.253067, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2666, 2656, 2752, 2663, 2661, 2655, 2653} \[ \frac{i \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{6 \sqrt{2} d \sqrt{2 a+b \sinh (2 c+2 d x)}}+\frac{b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{6 \sqrt{2} d}-\frac{2 i \sqrt{2} a \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{3 d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]
Antiderivative was successfully verified.
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Rule 2666
Rule 2656
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x) \sinh (c+d x))^{3/2} \, dx &=\int \left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right )^{3/2} \, dx\\ &=\frac{b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{6 \sqrt{2} d}+\frac{2}{3} \int \frac{\frac{1}{8} \left (12 a^2-b^2\right )+a b \sinh (2 c+2 d x)}{\sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac{b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{6 \sqrt{2} d}+\frac{1}{3} (4 a) \int \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)} \, dx+\frac{1}{12} \left (-4 a^2-b^2\right ) \int \frac{1}{\sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac{b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{6 \sqrt{2} d}+\frac{\left (4 a \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}\right ) \int \sqrt{\frac{a}{a-\frac{i b}{2}}+\frac{b \sinh (2 c+2 d x)}{2 \left (a-\frac{i b}{2}\right )}} \, dx}{3 \sqrt{\frac{a+\frac{1}{2} b \sinh (2 c+2 d x)}{a-\frac{i b}{2}}}}+\frac{\left (\left (-4 a^2-b^2\right ) \sqrt{\frac{a+\frac{1}{2} b \sinh (2 c+2 d x)}{a-\frac{i b}{2}}}\right ) \int \frac{1}{\sqrt{\frac{a}{a-\frac{i b}{2}}+\frac{b \sinh (2 c+2 d x)}{2 \left (a-\frac{i b}{2}\right )}}} \, dx}{12 \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}}\\ &=\frac{b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{6 \sqrt{2} d}-\frac{2 i \sqrt{2} a E\left (\frac{1}{2} \left (2 i c-\frac{\pi }{2}+2 i d x\right )|\frac{2 b}{2 i a+b}\right ) \sqrt{2 a+b \sinh (2 c+2 d x)}}{3 d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac{i \left (4 a^2+b^2\right ) F\left (\frac{1}{2} \left (2 i c-\frac{\pi }{2}+2 i d x\right )|\frac{2 b}{2 i a+b}\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{6 \sqrt{2} d \sqrt{2 a+b \sinh (2 c+2 d x)}}\\ \end{align*}
Mathematica [A] time = 0.803819, size = 202, normalized size = 0.81 \[ \frac{-2 i \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 (c+d x))}{2 a-i b}} F\left (\frac{1}{4} (-4 i c-4 i d x+\pi )|-\frac{2 i b}{2 a-i b}\right )+b (4 a \cosh (2 (c+d x))+b \sinh (4 (c+d x)))+16 a (b+2 i a) \sqrt{\frac{2 a+b \sinh (2 (c+d x))}{2 a-i b}} E\left (\frac{1}{4} (-4 i c-4 i d x+\pi )|-\frac{2 i b}{2 a-i b}\right )}{12 d \sqrt{4 a+2 b \sinh (2 (c+d x))}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.49, size = 935, normalized size = 3.8 \begin{align*} \text{result too large to display} \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{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{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 ({\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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