Optimal. Leaf size=97 \[ -\frac{3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac{\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac{\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]
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Rubi [A] time = 0.0805228, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5618, 2638} \[ -\frac{3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac{\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac{\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]
Antiderivative was successfully verified.
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Rule 5618
Rule 2638
Rubi steps
\begin{align*} \int \cosh (c+d x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{8} \sinh (a-c+(b-d) x)+\frac{1}{8} \sinh (3 a-c+(3 b-d) x)-\frac{3}{8} \sinh (a+c+(b+d) x)+\frac{1}{8} \sinh (3 a+c+(3 b+d) x)\right ) \, dx\\ &=\frac{1}{8} \int \sinh (3 a-c+(3 b-d) x) \, dx+\frac{1}{8} \int \sinh (3 a+c+(3 b+d) x) \, dx-\frac{3}{8} \int \sinh (a-c+(b-d) x) \, dx-\frac{3}{8} \int \sinh (a+c+(b+d) x) \, dx\\ &=-\frac{3 \cosh (a-c+(b-d) x)}{8 (b-d)}+\frac{\cosh (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac{3 \cosh (a+c+(b+d) x)}{8 (b+d)}+\frac{\cosh (3 a+c+(3 b+d) x)}{8 (3 b+d)}\\ \end{align*}
Mathematica [A] time = 0.495546, size = 90, normalized size = 0.93 \[ \frac{1}{8} \left (-\frac{3 \cosh (a+b x-c-d x)}{b-d}+\frac{\cosh (3 a+3 b x-c-d x)}{3 b-d}+\frac{\cosh (3 a+3 b x+c+d x)}{3 b+d}-\frac{3 \cosh (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 90, normalized size = 0.9 \begin{align*} -{\frac{3\,\cosh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{\cosh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}-{\frac{3\,\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\cosh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14674, size = 562, normalized size = 5.79 \begin{align*} \frac{9 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 3 \,{\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} -{\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) -{\left ({\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{2} d - d^{3} -{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 46.8247, size = 942, normalized size = 9.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17978, size = 247, normalized size = 2.55 \begin{align*} \frac{e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{16 \,{\left (3 \, b + d\right )}} + \frac{e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{16 \,{\left (3 \, b - d\right )}} - \frac{3 \, e^{\left (b x + d x + a + c\right )}}{16 \,{\left (b + d\right )}} - \frac{3 \, e^{\left (b x - d x + a - c\right )}}{16 \,{\left (b - d\right )}} - \frac{3 \, e^{\left (-b x + d x - a + c\right )}}{16 \,{\left (b - d\right )}} - \frac{3 \, e^{\left (-b x - d x - a - c\right )}}{16 \,{\left (b + d\right )}} + \frac{e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{16 \,{\left (3 \, b - d\right )}} + \frac{e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{16 \,{\left (3 \, b + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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