Optimal. Leaf size=109 \[ \frac{b \left (16 a^2-b^2\right ) \cosh (2 c+2 d x)}{24 d}+\frac{1}{8} a x \left (8 a^2-3 b^2\right )+\frac{5 a b^2 \sinh (2 c+2 d x) \cosh (2 c+2 d x)}{48 d}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^2}{48 d} \]
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Rubi [A] time = 0.0988748, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2666, 2656, 2734} \[ \frac{b \left (16 a^2-b^2\right ) \cosh (2 c+2 d x)}{24 d}+\frac{1}{8} a x \left (8 a^2-3 b^2\right )+\frac{5 a b^2 \sinh (2 c+2 d x) \cosh (2 c+2 d x)}{48 d}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^2}{48 d} \]
Antiderivative was successfully verified.
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Rule 2666
Rule 2656
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x) \sinh (c+d x))^3 \, dx &=\int \left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right )^3 \, dx\\ &=\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^2}{48 d}+\frac{1}{3} \int \left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right ) \left (\frac{1}{2} \left (6 a^2-b^2\right )+\frac{5}{2} a b \sinh (2 c+2 d x)\right ) \, dx\\ &=\frac{1}{8} a \left (8 a^2-3 b^2\right ) x+\frac{b \left (16 a^2-b^2\right ) \cosh (2 c+2 d x)}{24 d}+\frac{5 a b^2 \cosh (2 c+2 d x) \sinh (2 c+2 d x)}{48 d}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^2}{48 d}\\ \end{align*}
Mathematica [A] time = 0.259059, size = 77, normalized size = 0.71 \[ \frac{6 a \left (4 \left (8 a^2-3 b^2\right ) (c+d x)+3 b^2 \sinh (4 (c+d x))\right )+9 \left (16 a^2 b-b^3\right ) \cosh (2 (c+d x))+b^3 \cosh (6 (c+d x))}{192 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 124, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}{6}}-{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{12}}-{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{12}} \right ) +3\,a{b}^{2} \left ( 1/4\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}-1/8\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/8\,dx-c/8 \right ) +{\frac{3\,{a}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}b}{2}}+{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24572, size = 170, normalized size = 1.56 \begin{align*} a^{3} x - \frac{1}{384} \, b^{3}{\left (\frac{{\left (9 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac{9 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} - \frac{3}{64} \, a b^{2}{\left (\frac{8 \,{\left (d x + c\right )}}{d} - \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac{3 \, a^{2} b \cosh \left (d x + c\right )^{2}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01423, size = 409, normalized size = 3.75 \begin{align*} \frac{b^{3} \cosh \left (d x + c\right )^{6} + 15 \, b^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{4} + b^{3} \sinh \left (d x + c\right )^{6} + 72 \, a b^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 72 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 24 \,{\left (8 \, a^{3} - 3 \, a b^{2}\right )} d x + 9 \,{\left (16 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 48 \, a^{2} b - 3 \, b^{3}\right )} \sinh \left (d x + c\right )^{2}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.90162, size = 190, normalized size = 1.74 \begin{align*} \begin{cases} a^{3} x + \frac{3 a^{2} b \cosh ^{2}{\left (c + d x \right )}}{2 d} - \frac{3 a b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} + \frac{3 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac{3 a b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{3 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} + \frac{3 a b^{2} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} - \frac{b^{3} \sinh ^{6}{\left (c + d x \right )}}{12 d} + \frac{b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh{\left (c \right )} \cosh{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14285, size = 232, normalized size = 2.13 \begin{align*} \frac{b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 18 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 144 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 48 \,{\left (8 \, a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} -{\left (352 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 132 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 144 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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