Optimal. Leaf size=91 \[ \frac{\sinh (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sinh (a+x (b-d)-c)}{8 (b-d)}+\frac{3 \sinh (a+x (b+d)+c)}{8 (b+d)}+\frac{\sinh (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
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Rubi [A] time = 0.0693878, antiderivative size = 91, 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 = {5614, 2637} \[ \frac{\sinh (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sinh (a+x (b-d)-c)}{8 (b-d)}+\frac{3 \sinh (a+x (b+d)+c)}{8 (b+d)}+\frac{\sinh (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
Antiderivative was successfully verified.
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Rule 5614
Rule 2637
Rubi steps
\begin{align*} \int \cosh (a+b x) \cosh ^3(c+d x) \, dx &=\int \left (\frac{1}{8} \cosh (a-3 c+(b-3 d) x)+\frac{3}{8} \cosh (a-c+(b-d) x)+\frac{3}{8} \cosh (a+c+(b+d) x)+\frac{1}{8} \cosh (a+3 c+(b+3 d) x)\right ) \, dx\\ &=\frac{1}{8} \int \cosh (a-3 c+(b-3 d) x) \, dx+\frac{1}{8} \int \cosh (a+3 c+(b+3 d) x) \, dx+\frac{3}{8} \int \cosh (a-c+(b-d) x) \, dx+\frac{3}{8} \int \cosh (a+c+(b+d) x) \, dx\\ &=\frac{\sinh (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac{3 \sinh (a-c+(b-d) x)}{8 (b-d)}+\frac{3 \sinh (a+c+(b+d) x)}{8 (b+d)}+\frac{\sinh (a+3 c+(b+3 d) x)}{8 (b+3 d)}\\ \end{align*}
Mathematica [A] time = 0.427321, size = 85, normalized size = 0.93 \[ \frac{1}{8} \left (\frac{\sinh (a+b x-3 c-3 d x)}{b-3 d}+\frac{3 \sinh (a+b x-c-d x)}{b-d}+\frac{\sinh (a+b x+3 c+3 d x)}{b+3 d}+\frac{3 \sinh (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 84, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{8\,b-24\,d}}+{\frac{3\,\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{3\,\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\sinh \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{8\,b+24\,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 = 1.90776, size = 520, normalized size = 5.71 \begin{align*} \frac{3 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - 3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (d x + c\right )^{3} +{\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \,{\left (3 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} +{\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} -{\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 46.3635, size = 926, normalized size = 10.18 \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 = 1.2004, size = 242, normalized size = 2.66 \begin{align*} \frac{e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{16 \,{\left (b + 3 \, 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 (b x - 3 \, d x + a - 3 \, c\right )}}{16 \,{\left (b - 3 \, d\right )}} - \frac{e^{\left (-b x + 3 \, d x - a + 3 \, c\right )}}{16 \,{\left (b - 3 \, 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 (-b x - 3 \, d x - a - 3 \, c\right )}}{16 \,{\left (b + 3 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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