Optimal. Leaf size=64 \[ -\frac{a \cosh (c+d x)}{d^2}+\frac{a x \sinh (c+d x)}{d}+\frac{2 b \sinh (c+d x)}{d^3}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{b x^2 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.113946, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6742, 3296, 2638, 2637} \[ -\frac{a \cosh (c+d x)}{d^2}+\frac{a x \sinh (c+d x)}{d}+\frac{2 b \sinh (c+d x)}{d^3}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{b x^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3296
Rule 2638
Rule 2637
Rubi steps
\begin{align*} \int x (a+b x) \cosh (c+d x) \, dx &=\int \left (a x \cosh (c+d x)+b x^2 \cosh (c+d x)\right ) \, dx\\ &=a \int x \cosh (c+d x) \, dx+b \int x^2 \cosh (c+d x) \, dx\\ &=\frac{a x \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}-\frac{a \int \sinh (c+d x) \, dx}{d}-\frac{(2 b) \int x \sinh (c+d x) \, dx}{d}\\ &=-\frac{a \cosh (c+d x)}{d^2}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{a x \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}+\frac{(2 b) \int \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{a \cosh (c+d x)}{d^2}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{2 b \sinh (c+d x)}{d^3}+\frac{a x \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0879951, size = 45, normalized size = 0.7 \[ \frac{\left (a d^2 x+b \left (d^2 x^2+2\right )\right ) \sinh (c+d x)-d (a+2 b x) \cosh (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 122, normalized size = 1.9 \begin{align*}{\frac{1}{{d}^{2}} \left ({\frac{b \left ( \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +2\,\sinh \left ( dx+c \right ) \right ) }{d}}-2\,{\frac{cb \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}}+a \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) +{\frac{b{c}^{2}\sinh \left ( dx+c \right ) }{d}}-ca\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13037, size = 216, normalized size = 3.38 \begin{align*} -\frac{1}{12} \, d{\left (\frac{3 \,{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{3}} + \frac{3 \,{\left (d^{2} x^{2} + 2 \, d x + 2\right )} a e^{\left (-d x - c\right )}}{d^{3}} + \frac{2 \,{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{4}} + \frac{2 \,{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} b e^{\left (-d x - c\right )}}{d^{4}}\right )} + \frac{1}{6} \,{\left (2 \, b x^{3} + 3 \, a x^{2}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9195, size = 111, normalized size = 1.73 \begin{align*} -\frac{{\left (2 \, b d x + a d\right )} \cosh \left (d x + c\right ) -{\left (b d^{2} x^{2} + a d^{2} x + 2 \, b\right )} \sinh \left (d x + c\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.95005, size = 82, normalized size = 1.28 \begin{align*} \begin{cases} \frac{a x \sinh{\left (c + d x \right )}}{d} - \frac{a \cosh{\left (c + d x \right )}}{d^{2}} + \frac{b x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 b x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 b \sinh{\left (c + d x \right )}}{d^{3}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{2}}{2} + \frac{b x^{3}}{3}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22437, size = 107, normalized size = 1.67 \begin{align*} \frac{{\left (b d^{2} x^{2} + a d^{2} x - 2 \, b d x - a d + 2 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{3}} - \frac{{\left (b d^{2} x^{2} + a d^{2} x + 2 \, b d x + a d + 2 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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