Optimal. Leaf size=128 \[ \frac {b \left (64 a^2-54 a b+15 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{48 d}+\frac {1}{16} x (2 a-b) \left (8 a^2-8 a b+5 b^2\right )+\frac {5 b^2 (2 a-b) \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \]
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Rubi [A] time = 0.10, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3180, 3169} \[ \frac {b \left (64 a^2-54 a b+15 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{48 d}+\frac {1}{16} x (2 a-b) \left (8 a^2-8 a b+5 b^2\right )+\frac {5 b^2 (2 a-b) \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 3169
Rule 3180
Rubi steps
\begin {align*} \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d}+\frac {1}{6} \int \left (a+b \sinh ^2(c+d x)\right ) \left (a (6 a-b)+5 (2 a-b) b \sinh ^2(c+d x)\right ) \, dx\\ &=\frac {1}{16} (2 a-b) \left (8 a^2-8 a b+5 b^2\right ) x+\frac {b \left (64 a^2-54 a b+15 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{48 d}+\frac {5 (2 a-b) b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 95, normalized size = 0.74 \[ \frac {12 (2 a-b) \left (8 a^2-8 a b+5 b^2\right ) (c+d x)+9 b \left (16 a^2-16 a b+5 b^2\right ) \sinh (2 (c+d x))+9 b^2 (2 a-b) \sinh (4 (c+d x))+b^3 \sinh (6 (c+d x))}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 165, normalized size = 1.29 \[ \frac {3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} + 9 \, {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (16 \, a^{3} - 24 \, a^{2} b + 18 \, a b^{2} - 5 \, b^{3}\right )} d x + 3 \, {\left (b^{3} \cosh \left (d x + c\right )^{5} + 6 \, {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (16 \, a^{2} b - 16 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 177, normalized size = 1.38 \[ \frac {b^{3} e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {b^{3} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} + \frac {1}{16} \, {\left (16 \, a^{3} - 24 \, a^{2} b + 18 \, a b^{2} - 5 \, b^{3}\right )} x + \frac {3 \, {\left (2 \, a b^{2} - b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {3 \, {\left (16 \, a^{2} b - 16 \, a b^{2} + 5 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {3 \, {\left (16 \, a^{2} b - 16 \, a b^{2} + 5 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {3 \, {\left (2 \, a b^{2} - b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 131, normalized size = 1.02 \[ \frac {b^{3} \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+3 a \,b^{2} \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{2} b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 197, normalized size = 1.54 \[ \frac {3}{64} \, a b^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {3}{8} \, a^{2} b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{3} x - \frac {1}{384} \, b^{3} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 123, normalized size = 0.96 \[ \frac {\frac {45\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-36\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+36\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {9\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{2}+48\,a^3\,d\,x-15\,b^3\,d\,x+54\,a\,b^2\,d\,x-72\,a^2\,b\,d\,x}{48\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.63, size = 350, normalized size = 2.73 \[ \begin {cases} a^{3} x + \frac {3 a^{2} b x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {3 a^{2} b x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {3 a^{2} b \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {9 a b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {9 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {9 a b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {15 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {9 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {5 b^{3} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b^{3} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b^{3} \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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