Optimal. Leaf size=72 \[ \frac{3}{8} x \left (a^2-b^2\right )^2+\frac{3}{8} \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))+\frac{1}{4} (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))^3 \]
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Rubi [A] time = 0.037078, antiderivative size = 72, 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 = {3073, 8} \[ \frac{3}{8} x \left (a^2-b^2\right )^2+\frac{3}{8} \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))+\frac{1}{4} (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))^3 \]
Antiderivative was successfully verified.
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Rule 3073
Rule 8
Rubi steps
\begin{align*} \int (a \cosh (x)+b \sinh (x))^4 \, dx &=\frac{1}{4} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^3+\frac{1}{4} \left (3 \left (a^2-b^2\right )\right ) \int (a \cosh (x)+b \sinh (x))^2 \, dx\\ &=\frac{3}{8} \left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))+\frac{1}{4} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^3+\frac{1}{8} \left (3 \left (a^2-b^2\right )^2\right ) \int 1 \, dx\\ &=\frac{3}{8} \left (a^2-b^2\right )^2 x+\frac{3}{8} \left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))+\frac{1}{4} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^3\\ \end{align*}
Mathematica [A] time = 0.14543, size = 87, normalized size = 1.21 \[ \frac{1}{32} \left (8 \left (a^4-b^4\right ) \sinh (2 x)+\left (6 a^2 b^2+a^4+b^4\right ) \sinh (4 x)+16 a b \left (a^2-b^2\right ) \cosh (2 x)+4 a b \left (a^2+b^2\right ) \cosh (4 x)+12 x (a-b)^2 (a+b)^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 118, normalized size = 1.6 \begin{align*}{b}^{4} \left ( \left ({\frac{ \left ( \sinh \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( x \right ) }{8}} \right ) \cosh \left ( x \right ) +{\frac{3\,x}{8}} \right ) +4\,a{b}^{3} \left ( 1/4\, \left ( \sinh \left ( x \right ) \right ) ^{2} \left ( \cosh \left ( x \right ) \right ) ^{2}-1/4\, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) +6\,{a}^{2}{b}^{2} \left ( 1/4\,\sinh \left ( x \right ) \left ( \cosh \left ( x \right ) \right ) ^{3}-1/8\,\cosh \left ( x \right ) \sinh \left ( x \right ) -x/8 \right ) +4\,{a}^{3}b \left ( 1/4\, \left ( \sinh \left ( x \right ) \right ) ^{2} \left ( \cosh \left ( x \right ) \right ) ^{2}+1/4\, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) +{a}^{4} \left ( \left ({\frac{ \left ( \cosh \left ( x \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( x \right ) }{8}} \right ) \sinh \left ( x \right ) +{\frac{3\,x}{8}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07815, size = 139, normalized size = 1.93 \begin{align*} a^{3} b \cosh \left (x\right )^{4} + a b^{3} \sinh \left (x\right )^{4} + \frac{1}{64} \, a^{4}{\left (24 \, x + e^{\left (4 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} + \frac{1}{64} \, b^{4}{\left (24 \, x + e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} - \frac{3}{32} \, a^{2} b^{2}{\left (8 \, x - e^{\left (4 \, x\right )} + e^{\left (-4 \, x\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35033, size = 425, normalized size = 5.9 \begin{align*} \frac{1}{8} \,{\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{4} + \frac{1}{8} \,{\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \frac{1}{8} \,{\left (a^{3} b + a b^{3}\right )} \sinh \left (x\right )^{4} + \frac{1}{2} \,{\left (a^{3} b - a b^{3}\right )} \cosh \left (x\right )^{2} + \frac{1}{4} \,{\left (2 \, a^{3} b - 2 \, a b^{3} + 3 \,{\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + \frac{3}{8} \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x + \frac{1}{8} \,{\left ({\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{3} + 4 \,{\left (a^{4} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.946163, size = 265, normalized size = 3.68 \begin{align*} \frac{3 a^{4} x \sinh ^{4}{\left (x \right )}}{8} - \frac{3 a^{4} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{4} + \frac{3 a^{4} x \cosh ^{4}{\left (x \right )}}{8} - \frac{3 a^{4} \sinh ^{3}{\left (x \right )} \cosh{\left (x \right )}}{8} + \frac{5 a^{4} \sinh{\left (x \right )} \cosh ^{3}{\left (x \right )}}{8} + a^{3} b \cosh ^{4}{\left (x \right )} - \frac{3 a^{2} b^{2} x \sinh ^{4}{\left (x \right )}}{4} + \frac{3 a^{2} b^{2} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{2} - \frac{3 a^{2} b^{2} x \cosh ^{4}{\left (x \right )}}{4} + \frac{3 a^{2} b^{2} \sinh ^{3}{\left (x \right )} \cosh{\left (x \right )}}{4} + \frac{3 a^{2} b^{2} \sinh{\left (x \right )} \cosh ^{3}{\left (x \right )}}{4} + a b^{3} \sinh ^{4}{\left (x \right )} + \frac{3 b^{4} x \sinh ^{4}{\left (x \right )}}{8} - \frac{3 b^{4} x \sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{4} + \frac{3 b^{4} x \cosh ^{4}{\left (x \right )}}{8} + \frac{5 b^{4} \sinh ^{3}{\left (x \right )} \cosh{\left (x \right )}}{8} - \frac{3 b^{4} \sinh{\left (x \right )} \cosh ^{3}{\left (x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14752, size = 281, normalized size = 3.9 \begin{align*} \frac{1}{64} \, a^{4} e^{\left (4 \, x\right )} + \frac{1}{16} \, a^{3} b e^{\left (4 \, x\right )} + \frac{3}{32} \, a^{2} b^{2} e^{\left (4 \, x\right )} + \frac{1}{16} \, a b^{3} e^{\left (4 \, x\right )} + \frac{1}{64} \, b^{4} e^{\left (4 \, x\right )} + \frac{1}{8} \, a^{4} e^{\left (2 \, x\right )} + \frac{1}{4} \, a^{3} b e^{\left (2 \, x\right )} - \frac{1}{4} \, a b^{3} e^{\left (2 \, x\right )} - \frac{1}{8} \, b^{4} e^{\left (2 \, x\right )} + \frac{3}{8} \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x - \frac{1}{64} \,{\left (18 \, a^{4} e^{\left (4 \, x\right )} - 36 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 18 \, b^{4} e^{\left (4 \, x\right )} + 8 \, a^{4} e^{\left (2 \, x\right )} - 16 \, a^{3} b e^{\left (2 \, x\right )} + 16 \, a b^{3} e^{\left (2 \, x\right )} - 8 \, b^{4} e^{\left (2 \, x\right )} + a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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