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 = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.14, size = 63, normalized size = 1.80 \[ \frac {1}{12} \left (9 a \left (a^2-b^2\right ) \sinh (x)+a \left (a^2+3 b^2\right ) \sinh (3 x)+9 b \left (a^2-b^2\right ) \cosh (x)+b \left (3 a^2+b^2\right ) \cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 97, normalized size = 2.77 \[ \frac {1}{12} \, {\left (3 \, a^{2} b + b^{3}\right )} \cosh \relax (x)^{3} + \frac {1}{4} \, {\left (3 \, a^{2} b + b^{3}\right )} \cosh \relax (x) \sinh \relax (x)^{2} + \frac {1}{12} \, {\left (a^{3} + 3 \, a b^{2}\right )} \sinh \relax (x)^{3} + \frac {3}{4} \, {\left (a^{2} b - b^{3}\right )} \cosh \relax (x) + \frac {1}{4} \, {\left (3 \, a^{3} - 3 \, a b^{2} + {\left (a^{3} + 3 \, a b^{2}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 134, normalized size = 3.83 \[ \frac {1}{24} \, a^{3} e^{\left (3 \, x\right )} + \frac {1}{8} \, a^{2} b e^{\left (3 \, x\right )} + \frac {1}{8} \, a b^{2} e^{\left (3 \, x\right )} + \frac {1}{24} \, b^{3} e^{\left (3 \, x\right )} + \frac {3}{8} \, a^{3} e^{x} + \frac {3}{8} \, a^{2} b e^{x} - \frac {3}{8} \, a b^{2} e^{x} - \frac {3}{8} \, b^{3} e^{x} - \frac {1}{24} \, {\left (9 \, a^{3} e^{\left (2 \, x\right )} - 9 \, a^{2} b e^{\left (2 \, x\right )} - 9 \, a b^{2} e^{\left (2 \, x\right )} + 9 \, b^{3} e^{\left (2 \, x\right )} + a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 48, normalized size = 1.37 \[ b^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\relax (x )\right )}{3}\right ) \cosh \relax (x )+a \,b^{2} \left (\sinh ^{3}\relax (x )\right )+a^{2} b \left (\cosh ^{3}\relax (x )\right )+a^{3} \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\relax (x )\right )}{3}\right ) \sinh \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 69, normalized size = 1.97 \[ a^{2} b \cosh \relax (x)^{3} + a b^{2} \sinh \relax (x)^{3} + \frac {1}{24} \, b^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac {1}{24} \, a^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 53, normalized size = 1.51 \[ {\mathrm {cosh}\relax (x)}^3\,\left (a^2\,b-\frac {2\,b^3}{3}\right )+{\mathrm {sinh}\relax (x)}^3\,\left (a\,b^2-\frac {2\,a^3}{3}\right )+a^3\,{\mathrm {cosh}\relax (x)}^2\,\mathrm {sinh}\relax (x)+b^3\,\mathrm {cosh}\relax (x)\,{\mathrm {sinh}\relax (x)}^2 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.33, size = 66, normalized size = 1.89 \[ - \frac {2 a^{3} \sinh ^{3}{\relax (x )}}{3} + a^{3} \sinh {\relax (x )} \cosh ^{2}{\relax (x )} + a^{2} b \cosh ^{3}{\relax (x )} + a b^{2} \sinh ^{3}{\relax (x )} + b^{3} \sinh ^{2}{\relax (x )} \cosh {\relax (x )} - \frac {2 b^{3} \cosh ^{3}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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