Optimal. Leaf size=68 \[ \frac {\sinh (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac {\sinh (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac {\sinh (c+d x)}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5618, 2637} \[ \frac {\sinh (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac {\sinh (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac {\sinh (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 5618
Rubi steps
\begin {align*} \int \cosh (c+d x) \sinh ^2(a+b x) \, dx &=\int \left (\frac {1}{4} \cosh (2 a-c+(2 b-d) x)-\frac {1}{2} \cosh (c+d x)+\frac {1}{4} \cosh (2 a+c+(2 b+d) x)\right ) \, dx\\ &=\frac {1}{4} \int \cosh (2 a-c+(2 b-d) x) \, dx+\frac {1}{4} \int \cosh (2 a+c+(2 b+d) x) \, dx-\frac {1}{2} \int \cosh (c+d x) \, dx\\ &=\frac {\sinh (2 a-c+(2 b-d) x)}{4 (2 b-d)}-\frac {\sinh (c+d x)}{2 d}+\frac {\sinh (2 a+c+(2 b+d) x)}{4 (2 b+d)}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 74, normalized size = 1.09 \[ \frac {1}{4} \left (\frac {\sinh (2 a+2 b x-c-d x)}{2 b-d}+\frac {\sinh (2 a+2 b x+c+d x)}{2 b+d}-\frac {2 \sinh (c) \cosh (d x)}{d}-\frac {2 \cosh (c) \sinh (d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 114, normalized size = 1.68 \[ \frac {4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - {\left (d^{2} \cosh \left (b x + a\right )^{2} + d^{2} \sinh \left (b x + a\right )^{2} + 4 \, b^{2} - d^{2}\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2} - {\left (4 \, b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 124, normalized size = 1.82 \[ \frac {e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \, {\left (2 \, b + d\right )}} + \frac {e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{8 \, {\left (2 \, b - d\right )}} - \frac {e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \, {\left (2 \, b - d\right )}} - \frac {e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{8 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (d x + c\right )}}{4 \, d} + \frac {e^{\left (-d x - c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 63, normalized size = 0.93 \[ \frac {\sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{8 b -4 d}-\frac {\sinh \left (d x +c \right )}{2 d}+\frac {\sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 b +4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 76, normalized size = 1.12 \[ \frac {d^2\,\left (\mathrm {sinh}\left (c+d\,x\right )-{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\right )-2\,b^2\,\mathrm {sinh}\left (c+d\,x\right )+2\,b\,d\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{4\,b^2\,d-d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.54, size = 411, normalized size = 6.04 \[ \begin {cases} x \sinh ^{2}{\relax (a )} \cosh {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh ^{2}{\left (a - \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} + \frac {x \sinh {\left (a - \frac {d x}{2} \right )} \sinh {\left (c + d x \right )} \cosh {\left (a - \frac {d x}{2} \right )}}{2} + \frac {x \cosh ^{2}{\left (a - \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} - \frac {3 \sinh {\left (a - \frac {d x}{2} \right )} \cosh {\left (a - \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{2 d} - \frac {\sinh {\left (c + d x \right )} \cosh ^{2}{\left (a - \frac {d x}{2} \right )}}{d} & \text {for}\: b = - \frac {d}{2} \\\frac {x \sinh ^{2}{\left (a + \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} - \frac {x \sinh {\left (a + \frac {d x}{2} \right )} \sinh {\left (c + d x \right )} \cosh {\left (a + \frac {d x}{2} \right )}}{2} + \frac {x \cosh ^{2}{\left (a + \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} + \frac {3 \sinh {\left (a + \frac {d x}{2} \right )} \cosh {\left (a + \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{2 d} - \frac {\sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + \frac {d x}{2} \right )}}{d} & \text {for}\: b = \frac {d}{2} \\\left (\frac {x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b}\right ) \cosh {\relax (c )} & \text {for}\: d = 0 \\\frac {2 b^{2} \sinh ^{2}{\left (a + b x \right )} \sinh {\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac {2 b^{2} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b^{2} d - d^{3}} + \frac {2 b d \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac {d^{2} \sinh ^{2}{\left (a + b x \right )} \sinh {\left (c + d x \right )}}{4 b^{2} d - d^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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