Optimal. Leaf size=103 \[ \frac{2}{3} (a \sinh (x)+b \cosh (x)) \sqrt{a \cosh (x)+b \sinh (x)}-\frac{2 i \left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}} \text{EllipticF}\left (\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right )}{3 \sqrt{a \cosh (x)+b \sinh (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0532752, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3073, 3078, 2641} \[ \frac{2}{3} (a \sinh (x)+b \cosh (x)) \sqrt{a \cosh (x)+b \sinh (x)}-\frac{2 i \left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}} F\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{3 \sqrt{a \cosh (x)+b \sinh (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3073
Rule 3078
Rule 2641
Rubi steps
\begin{align*} \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx &=\frac{2}{3} (b \cosh (x)+a \sinh (x)) \sqrt{a \cosh (x)+b \sinh (x)}+\frac{1}{3} \left (a^2-b^2\right ) \int \frac{1}{\sqrt{a \cosh (x)+b \sinh (x)}} \, dx\\ &=\frac{2}{3} (b \cosh (x)+a \sinh (x)) \sqrt{a \cosh (x)+b \sinh (x)}+\frac{\left (\left (a^2-b^2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}\right ) \int \frac{1}{\sqrt{\cosh \left (x+i \tan ^{-1}(a,-i b)\right )}} \, dx}{3 \sqrt{a \cosh (x)+b \sinh (x)}}\\ &=\frac{2}{3} (b \cosh (x)+a \sinh (x)) \sqrt{a \cosh (x)+b \sinh (x)}-\frac{2 i \left (a^2-b^2\right ) F\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}{3 \sqrt{a \cosh (x)+b \sinh (x)}}\\ \end{align*}
Mathematica [C] time = 0.573857, size = 92, normalized size = 0.89 \[ \frac{2}{3} \sqrt{a \cosh (x)+b \sinh (x)} \left (-b \sqrt{1-\frac{a^2}{b^2}} \sqrt{\cosh ^2\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right )} \text{sech}\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},-\sinh ^2\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right )\right )+a \sinh (x)+b \cosh (x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.238, size = 171, normalized size = 1.7 \begin{align*} -{\frac{1}{2\, \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{-\sqrt{{a}^{2}-{b}^{2}} \left ( \sinh \left ( x \right ) \right ) ^{3}} \left ( \cosh \left ( x \right ) \sqrt{\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}\sqrt{-\sqrt{{a}^{2}-{b}^{2}} \left ( \sinh \left ( x \right ) \right ) ^{3}} \left ({a}^{2}-{b}^{2} \right ) +\sinh \left ( x \right ) \arctan \left ({\cosh \left ( x \right ) \sqrt{\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}{\frac{1}{\sqrt{-\sqrt{{a}^{2}-{b}^{2}} \left ( \sinh \left ( x \right ) \right ) ^{3}}}}} \right ) \left ({a}^{2}-{b}^{2} \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}{\frac{1}{\sqrt{\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}}}{\frac{1}{\sqrt{-\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]