Optimal. Leaf size=49 \[ -\frac{2 d (c+d x) \cosh (a+b x)}{b^2}+\frac{2 d^2 \sinh (a+b x)}{b^3}+\frac{(c+d x)^2 \sinh (a+b x)}{b} \]
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Rubi [A] time = 0.0468619, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2637} \[ -\frac{2 d (c+d x) \cosh (a+b x)}{b^2}+\frac{2 d^2 \sinh (a+b x)}{b^3}+\frac{(c+d x)^2 \sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (c+d x)^2 \cosh (a+b x) \, dx &=\frac{(c+d x)^2 \sinh (a+b x)}{b}-\frac{(2 d) \int (c+d x) \sinh (a+b x) \, dx}{b}\\ &=-\frac{2 d (c+d x) \cosh (a+b x)}{b^2}+\frac{(c+d x)^2 \sinh (a+b x)}{b}+\frac{\left (2 d^2\right ) \int \cosh (a+b x) \, dx}{b^2}\\ &=-\frac{2 d (c+d x) \cosh (a+b x)}{b^2}+\frac{2 d^2 \sinh (a+b x)}{b^3}+\frac{(c+d x)^2 \sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.139609, size = 44, normalized size = 0.9 \[ \frac{\sinh (a+b x) \left (b^2 (c+d x)^2+2 d^2\right )-2 b d (c+d x) \cosh (a+b x)}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 147, normalized size = 3. \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2} \left ( \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) -2\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) +2\,\sinh \left ( bx+a \right ) \right ) }{{b}^{2}}}-2\,{\frac{a{d}^{2} \left ( \left ( bx+a \right ) \sinh \left ( bx+a \right ) -\cosh \left ( bx+a \right ) \right ) }{{b}^{2}}}+2\,{\frac{cd \left ( \left ( bx+a \right ) \sinh \left ( bx+a \right ) -\cosh \left ( bx+a \right ) \right ) }{b}}+{\frac{{a}^{2}{d}^{2}\sinh \left ( bx+a \right ) }{{b}^{2}}}-2\,{\frac{dac\sinh \left ( bx+a \right ) }{b}}+{c}^{2}\sinh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19404, size = 182, normalized size = 3.71 \begin{align*} \frac{c^{2} e^{\left (b x + a\right )}}{2 \, b} + \frac{{\left (b x e^{a} - e^{a}\right )} c d e^{\left (b x\right )}}{b^{2}} - \frac{c^{2} e^{\left (-b x - a\right )}}{2 \, b} - \frac{{\left (b x + 1\right )} c d e^{\left (-b x - a\right )}}{b^{2}} + \frac{{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} d^{2} e^{\left (b x\right )}}{2 \, b^{3}} - \frac{{\left (b^{2} x^{2} + 2 \, b x + 2\right )} d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90595, size = 140, normalized size = 2.86 \begin{align*} -\frac{2 \,{\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \sinh \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.753647, size = 112, normalized size = 2.29 \begin{align*} \begin{cases} \frac{c^{2} \sinh{\left (a + b x \right )}}{b} + \frac{2 c d x \sinh{\left (a + b x \right )}}{b} + \frac{d^{2} x^{2} \sinh{\left (a + b x \right )}}{b} - \frac{2 c d \cosh{\left (a + b x \right )}}{b^{2}} - \frac{2 d^{2} x \cosh{\left (a + b x \right )}}{b^{2}} + \frac{2 d^{2} \sinh{\left (a + b x \right )}}{b^{3}} & \text{for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) \cosh{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31982, size = 151, normalized size = 3.08 \begin{align*} \frac{{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{2 \, b^{3}} - \frac{{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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