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 = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 2656
Rule 2666
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*}
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Mathematica [A] time = 0.27, size = 77, normalized size = 0.71 \[ \frac {9 \left (16 a^2 b-b^3\right ) \cosh (2 (c+d x))+6 a \left (4 \left (8 a^2-3 b^2\right ) (c+d x)+3 b^2 \sinh (4 (c+d x))\right )+b^3 \cosh (6 (c+d x))}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 164, normalized size = 1.50 \[ \frac {b^{3} \cosh \left (d x + c\right )^{6} + 15 \, b^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{4} + b^{3} \sinh \left (d x + c\right )^{6} + 72 \, a b^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 72 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 24 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} d x + 9 \, {\left (16 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 48 \, a^{2} b - 3 \, b^{3}\right )} \sinh \left (d x + c\right )^{2}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 138, normalized size = 1.27 \[ \frac {b^{3} e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {3 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} - \frac {3 \, a b^{2} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} + \frac {b^{3} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} + \frac {1}{8} \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} x + \frac {3 \, {\left (16 \, a^{2} b - b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} + \frac {3 \, {\left (16 \, a^{2} b - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 106, normalized size = 0.97 \[ \frac {b^{3} \left (\frac {\left (\sinh ^{2}\left (d x +c \right )\right ) \left (\cosh ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (d x +c \right )\right )}{12}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{4}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {d x}{8}-\frac {c}{8}\right )+\frac {3 a^{2} b \left (\cosh ^{2}\left (d x +c \right )\right )}{2}+a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 126, normalized size = 1.16 \[ a^{3} x - \frac {1}{384} \, b^{3} {\left (\frac {{\left (9 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {9 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} - \frac {3}{64} \, a b^{2} {\left (\frac {8 \, {\left (d x + c\right )}}{d} - \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {3 \, a^{2} b \cosh \left (d x + c\right )^{2}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.87, size = 79, normalized size = 0.72 \[ \frac {\frac {b^3\,\mathrm {cosh}\left (6\,c+6\,d\,x\right )}{8}-\frac {9\,b^3\,\mathrm {cosh}\left (2\,c+2\,d\,x\right )}{8}+18\,a^2\,b\,\mathrm {cosh}\left (2\,c+2\,d\,x\right )+\frac {9\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+24\,a^3\,d\,x-9\,a\,b^2\,d\,x}{24\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.25, size = 190, normalized size = 1.74 \[ \begin {cases} a^{3} x + \frac {3 a^{2} b \sinh ^{2}{\left (c + d x \right )}}{2 d} - \frac {3 a b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} + \frac {3 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {3 a b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {3 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {3 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{4 d} - \frac {b^{3} \cosh ^{6}{\left (c + d x \right )}}{12 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh {\relax (c )} \cosh {\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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