Optimal. Leaf size=61 \[ \frac{2}{3} \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x))^3+\left (a^2-b^2\right )^2 (a \sinh (x)+b \cosh (x))+\frac{1}{5} (a \sinh (x)+b \cosh (x))^5 \]
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Rubi [A] time = 0.0450399, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3072, 194} \[ \frac{2}{3} \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x))^3+\left (a^2-b^2\right )^2 (a \sinh (x)+b \cosh (x))+\frac{1}{5} (a \sinh (x)+b \cosh (x))^5 \]
Antiderivative was successfully verified.
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Rule 3072
Rule 194
Rubi steps
\begin{align*} \int (a \cosh (x)+b \sinh (x))^5 \, dx &=i \operatorname{Subst}\left (\int \left (a^2-b^2-x^2\right )^2 \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )\\ &=i \operatorname{Subst}\left (\int \left (a^4 \left (1+\frac{-2 a^2 b^2+b^4}{a^4}\right )-2 a^2 \left (1-\frac{b^2}{a^2}\right ) x^2+x^4\right ) \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )\\ &=\left (a^2-b^2\right )^2 (b \cosh (x)+a \sinh (x))+\frac{2}{3} \left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x))^3+\frac{1}{5} (b \cosh (x)+a \sinh (x))^5\\ \end{align*}
Mathematica [B] time = 0.225237, size = 133, normalized size = 2.18 \[ \frac{1}{240} \left (150 a \left (a^2-b^2\right )^2 \sinh (x)+25 a \left (2 a^2 b^2+a^4-3 b^4\right ) \sinh (3 x)+3 a \left (10 a^2 b^2+a^4+5 b^4\right ) \sinh (5 x)+150 b \left (a^2-b^2\right )^2 \cosh (x)-25 b \left (2 a^2 b^2-3 a^4+b^4\right ) \cosh (3 x)+3 b \left (10 a^2 b^2+5 a^4+b^4\right ) \cosh (5 x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 160, normalized size = 2.6 \begin{align*}{b}^{5} \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( x \right ) +5\,a{b}^{4} \left ( 1/5\, \left ( \sinh \left ( x \right ) \right ) ^{3} \left ( \cosh \left ( x \right ) \right ) ^{2}-1/5\,\sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}+1/5\,\sinh \left ( x \right ) \right ) +10\,{a}^{2}{b}^{3} \left ( 1/5\, \left ( \sinh \left ( x \right ) \right ) ^{2} \left ( \cosh \left ( x \right ) \right ) ^{3}-2/15\,\cosh \left ( x \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}-2/15\,\cosh \left ( x \right ) \right ) +10\,{a}^{3}{b}^{2} \left ( 1/5\, \left ( \cosh \left ( x \right ) \right ) ^{4}\sinh \left ( x \right ) -1/5\, \left ( 2/3+1/3\, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \sinh \left ( x \right ) \right ) +5\,{a}^{4}b \left ( 1/5\, \left ( \sinh \left ( x \right ) \right ) ^{2} \left ( \cosh \left ( x \right ) \right ) ^{3}+1/5\,\cosh \left ( x \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}+1/5\,\cosh \left ( x \right ) \right ) +{a}^{5} \left ({\frac{8}{15}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{4}}{5}}+{\frac{4\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{15}} \right ) \sinh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16099, size = 258, normalized size = 4.23 \begin{align*} a^{4} b \cosh \left (x\right )^{5} + a b^{4} \sinh \left (x\right )^{5} + \frac{1}{48} \,{\left ({\left (5 \, e^{\left (-2 \, x\right )} - 30 \, e^{\left (-4 \, x\right )} + 3\right )} e^{\left (5 \, x\right )} + 30 \, e^{\left (-x\right )} - 5 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )}\right )} a^{3} b^{2} - \frac{1}{48} \,{\left ({\left (5 \, e^{\left (-2 \, x\right )} + 30 \, e^{\left (-4 \, x\right )} - 3\right )} e^{\left (5 \, x\right )} + 30 \, e^{\left (-x\right )} + 5 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )}\right )} a^{2} b^{3} + \frac{1}{480} \, a^{5}{\left (3 \, e^{\left (5 \, x\right )} + 25 \, e^{\left (3 \, x\right )} - 150 \, e^{\left (-x\right )} - 25 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )} + 150 \, e^{x}\right )} + \frac{1}{480} \, b^{5}{\left (3 \, e^{\left (5 \, x\right )} - 25 \, e^{\left (3 \, x\right )} + 150 \, e^{\left (-x\right )} - 25 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} + 150 \, e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.46245, size = 747, normalized size = 12.25 \begin{align*} \frac{1}{80} \,{\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{5} + \frac{1}{16} \,{\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + \frac{1}{80} \,{\left (a^{5} + 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sinh \left (x\right )^{5} + \frac{5}{48} \,{\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{3} + \frac{1}{48} \,{\left (5 \, a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4} + 6 \,{\left (a^{5} + 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{3} + \frac{1}{16} \,{\left (2 \,{\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{3} + 5 \,{\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac{5}{8} \,{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right ) + \frac{1}{16} \,{\left (10 \, a^{5} - 20 \, a^{3} b^{2} + 10 \, a b^{4} +{\left (a^{5} + 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cosh \left (x\right )^{4} + 5 \,{\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4}\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 = 1.65456, size = 172, normalized size = 2.82 \begin{align*} \frac{8 a^{5} \sinh ^{5}{\left (x \right )}}{15} - \frac{4 a^{5} \sinh ^{3}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{3} + a^{5} \sinh{\left (x \right )} \cosh ^{4}{\left (x \right )} + a^{4} b \cosh ^{5}{\left (x \right )} - \frac{4 a^{3} b^{2} \sinh ^{5}{\left (x \right )}}{3} + \frac{10 a^{3} b^{2} \sinh ^{3}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{3} + \frac{10 a^{2} b^{3} \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} - \frac{4 a^{2} b^{3} \cosh ^{5}{\left (x \right )}}{3} + a b^{4} \sinh ^{5}{\left (x \right )} + b^{5} \sinh ^{4}{\left (x \right )} \cosh{\left (x \right )} - \frac{4 b^{5} \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} + \frac{8 b^{5} \cosh ^{5}{\left (x \right )}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16068, size = 464, normalized size = 7.61 \begin{align*} \frac{1}{160} \, a^{5} e^{\left (5 \, x\right )} + \frac{1}{32} \, a^{4} b e^{\left (5 \, x\right )} + \frac{1}{16} \, a^{3} b^{2} e^{\left (5 \, x\right )} + \frac{1}{16} \, a^{2} b^{3} e^{\left (5 \, x\right )} + \frac{1}{32} \, a b^{4} e^{\left (5 \, x\right )} + \frac{1}{160} \, b^{5} e^{\left (5 \, x\right )} + \frac{5}{96} \, a^{5} e^{\left (3 \, x\right )} + \frac{5}{32} \, a^{4} b e^{\left (3 \, x\right )} + \frac{5}{48} \, a^{3} b^{2} e^{\left (3 \, x\right )} - \frac{5}{48} \, a^{2} b^{3} e^{\left (3 \, x\right )} - \frac{5}{32} \, a b^{4} e^{\left (3 \, x\right )} - \frac{5}{96} \, b^{5} e^{\left (3 \, x\right )} + \frac{5}{16} \, a^{5} e^{x} + \frac{5}{16} \, a^{4} b e^{x} - \frac{5}{8} \, a^{3} b^{2} e^{x} - \frac{5}{8} \, a^{2} b^{3} e^{x} + \frac{5}{16} \, a b^{4} e^{x} + \frac{5}{16} \, b^{5} e^{x} - \frac{1}{480} \,{\left (150 \, a^{5} e^{\left (4 \, x\right )} - 150 \, a^{4} b e^{\left (4 \, x\right )} - 300 \, a^{3} b^{2} e^{\left (4 \, x\right )} + 300 \, a^{2} b^{3} e^{\left (4 \, x\right )} + 150 \, a b^{4} e^{\left (4 \, x\right )} - 150 \, b^{5} e^{\left (4 \, x\right )} + 25 \, a^{5} e^{\left (2 \, x\right )} - 75 \, a^{4} b e^{\left (2 \, x\right )} + 50 \, a^{3} b^{2} e^{\left (2 \, x\right )} + 50 \, a^{2} b^{3} e^{\left (2 \, x\right )} - 75 \, a b^{4} e^{\left (2 \, x\right )} + 25 \, b^{5} e^{\left (2 \, x\right )} + 3 \, a^{5} - 15 \, a^{4} b + 30 \, a^{3} b^{2} - 30 \, a^{2} b^{3} + 15 \, a b^{4} - 3 \, b^{5}\right )} e^{\left (-5 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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