Optimal. Leaf size=35 \[ \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x))+\frac{1}{3} (a \sinh (x)+b \cosh (x))^3 \]
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Rubi [A] time = 0.0248492, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3072} \[ \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x))+\frac{1}{3} (a \sinh (x)+b \cosh (x))^3 \]
Antiderivative was successfully verified.
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Rule 3072
Rubi steps
\begin{align*} \int (a \cosh (x)+b \sinh (x))^3 \, dx &=i \operatorname{Subst}\left (\int \left (a^2-b^2-x^2\right ) \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )\\ &=\left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x))+\frac{1}{3} (b \cosh (x)+a \sinh (x))^3\\ \end{align*}
Mathematica [A] time = 0.126669, size = 63, normalized size = 1.8 \[ \frac{1}{12} \left (9 a \left (a^2-b^2\right ) \sinh (x)+a \left (a^2+3 b^2\right ) \sinh (3 x)+9 b \left (a^2-b^2\right ) \cosh (x)+b \left (3 a^2+b^2\right ) \cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 68, normalized size = 1.9 \begin{align*}{b}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( x \right ) +3\,a{b}^{2} \left ( 1/3\,\sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}-1/3\,\sinh \left ( x \right ) \right ) +3\,{a}^{2}b \left ( 1/3\,\cosh \left ( x \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}+1/3\,\cosh \left ( x \right ) \right ) +{a}^{3} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01723, size = 93, normalized size = 2.66 \begin{align*} a^{2} b \cosh \left (x\right )^{3} + a b^{2} \sinh \left (x\right )^{3} + \frac{1}{24} \, b^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac{1}{24} \, a^{3}{\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.38148, size = 261, normalized size = 7.46 \begin{align*} \frac{1}{12} \,{\left (3 \, a^{2} b + b^{3}\right )} \cosh \left (x\right )^{3} + \frac{1}{4} \,{\left (3 \, a^{2} b + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \frac{1}{12} \,{\left (a^{3} + 3 \, a b^{2}\right )} \sinh \left (x\right )^{3} + \frac{3}{4} \,{\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) + \frac{1}{4} \,{\left (3 \, a^{3} - 3 \, a b^{2} +{\left (a^{3} + 3 \, a b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.43813, size = 66, normalized size = 1.89 \begin{align*} - \frac{2 a^{3} \sinh ^{3}{\left (x \right )}}{3} + a^{3} \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )} + a^{2} b \cosh ^{3}{\left (x \right )} + a b^{2} \sinh ^{3}{\left (x \right )} + b^{3} \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )} - \frac{2 b^{3} \cosh ^{3}{\left (x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14039, size = 181, normalized size = 5.17 \begin{align*} \frac{1}{24} \, a^{3} e^{\left (3 \, x\right )} + \frac{1}{8} \, a^{2} b e^{\left (3 \, x\right )} + \frac{1}{8} \, a b^{2} e^{\left (3 \, x\right )} + \frac{1}{24} \, b^{3} e^{\left (3 \, x\right )} + \frac{3}{8} \, a^{3} e^{x} + \frac{3}{8} \, a^{2} b e^{x} - \frac{3}{8} \, a b^{2} e^{x} - \frac{3}{8} \, b^{3} e^{x} - \frac{1}{24} \,{\left (9 \, a^{3} e^{\left (2 \, x\right )} - 9 \, a^{2} b e^{\left (2 \, x\right )} - 9 \, a b^{2} e^{\left (2 \, x\right )} + 9 \, b^{3} e^{\left (2 \, x\right )} + a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} e^{\left (-3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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