Optimal. Leaf size=61 \[ \frac {(4 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac {1}{8} x (4 a-3 b)+\frac {b \sinh ^3(c+d x) \cosh (c+d x)}{4 d} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3014, 2635, 8} \[ \frac {(4 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac {1}{8} x (4 a-3 b)+\frac {b \sinh ^3(c+d x) \cosh (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3014
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {1}{4} (-4 a+3 b) \int \sinh ^2(c+d x) \, dx\\ &=\frac {(4 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {1}{8} (4 a-3 b) \int 1 \, dx\\ &=-\frac {1}{8} (4 a-3 b) x+\frac {(4 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 47, normalized size = 0.77 \[ \frac {-4 (4 a-3 b) (c+d x)+8 (a-b) \sinh (2 (c+d x))+b \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 64, normalized size = 1.05 \[ \frac {b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - {\left (4 \, a - 3 \, b\right )} d x + {\left (b \cosh \left (d x + c\right )^{3} + 4 \, {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 79, normalized size = 1.30 \[ -\frac {1}{8} \, {\left (4 \, a - 3 \, b\right )} x + \frac {b e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {{\left (a - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} - \frac {{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac {b e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 66, normalized size = 1.08 \[ \frac {b \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 97, normalized size = 1.59 \[ \frac {1}{64} \, b {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{8} \, a {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 50, normalized size = 0.82 \[ \frac {\frac {a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{32}}{d}-\frac {a\,x}{2}+\frac {3\,b\,x}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 158, normalized size = 2.59 \[ \begin {cases} \frac {a x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {3 b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right ) \sinh ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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