Optimal. Leaf size=75 \[ -\frac{d \sinh ^3(a+b x)}{9 b^2}+\frac{2 d \sinh (a+b x)}{3 b^2}-\frac{2 (c+d x) \cosh (a+b x)}{3 b}+\frac{(c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
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Rubi [A] time = 0.0576496, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3310, 3296, 2637} \[ -\frac{d \sinh ^3(a+b x)}{9 b^2}+\frac{2 d \sinh (a+b x)}{3 b^2}-\frac{2 (c+d x) \cosh (a+b x)}{3 b}+\frac{(c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3310
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (c+d x) \sinh ^3(a+b x) \, dx &=\frac{(c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sinh ^3(a+b x)}{9 b^2}-\frac{2}{3} \int (c+d x) \sinh (a+b x) \, dx\\ &=-\frac{2 (c+d x) \cosh (a+b x)}{3 b}+\frac{(c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sinh ^3(a+b x)}{9 b^2}+\frac{(2 d) \int \cosh (a+b x) \, dx}{3 b}\\ &=-\frac{2 (c+d x) \cosh (a+b x)}{3 b}+\frac{2 d \sinh (a+b x)}{3 b^2}+\frac{(c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d \sinh ^3(a+b x)}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.183453, size = 59, normalized size = 0.79 \[ \frac{-27 b (c+d x) \cosh (a+b x)+3 b (c+d x) \cosh (3 (a+b x))+d (27 \sinh (a+b x)-\sinh (3 (a+b x)))}{36 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 115, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ( -{\frac{ \left ( 2\,bx+2\,a \right ) \cosh \left ( bx+a \right ) }{3}}+{\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{7\,\sinh \left ( bx+a \right ) }{9}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{9}} \right ) }-{\frac{da\cosh \left ( bx+a \right ) }{b} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) }+c \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22695, size = 190, normalized size = 2.53 \begin{align*} \frac{1}{72} \, d{\left (\frac{{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} - \frac{27 \,{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac{27 \,{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} + \frac{{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac{1}{24} \, c{\left (\frac{e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac{9 \, e^{\left (b x + a\right )}}{b} - \frac{9 \, e^{\left (-b x - a\right )}}{b} + \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62549, size = 255, normalized size = 3.4 \begin{align*} \frac{3 \,{\left (b d x + b c\right )} \cosh \left (b x + a\right )^{3} + 9 \,{\left (b d x + b c\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - d \sinh \left (b x + a\right )^{3} - 27 \,{\left (b d x + b c\right )} \cosh \left (b x + a\right ) - 3 \,{\left (d \cosh \left (b x + a\right )^{2} - 9 \, d\right )} \sinh \left (b x + a\right )}{36 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.8197, size = 126, normalized size = 1.68 \begin{align*} \begin{cases} \frac{c \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 c \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{d x \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 d x \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac{7 d \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{2 d \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sinh ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18635, size = 132, normalized size = 1.76 \begin{align*} \frac{{\left (3 \, b d x + 3 \, b c - d\right )} e^{\left (3 \, b x + 3 \, a\right )}}{72 \, b^{2}} - \frac{3 \,{\left (b d x + b c - d\right )} e^{\left (b x + a\right )}}{8 \, b^{2}} - \frac{3 \,{\left (b d x + b c + d\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} + \frac{{\left (3 \, b d x + 3 \, b c + d\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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