Optimal. Leaf size=51 \[ \frac{a \sinh (c+d x)}{d}+\frac{2 b \sinh (c+d x)}{d^3}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{b x^2 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.0667648, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5277, 2637, 3296} \[ \frac{a \sinh (c+d x)}{d}+\frac{2 b \sinh (c+d x)}{d^3}-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{b x^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5277
Rule 2637
Rule 3296
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a \cosh (c+d x)+b x^2 \cosh (c+d x)\right ) \, dx\\ &=a \int \cosh (c+d x) \, dx+b \int x^2 \cosh (c+d x) \, dx\\ &=\frac{a \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}-\frac{(2 b) \int x \sinh (c+d x) \, dx}{d}\\ &=-\frac{2 b x \cosh (c+d x)}{d^2}+\frac{a \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{2 b x \cosh (c+d x)}{d^2}+\frac{2 b \sinh (c+d x)}{d^3}+\frac{a \sinh (c+d x)}{d}+\frac{b x^2 \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0667749, size = 40, normalized size = 0.78 \[ \frac{\left (a d^2+b \left (d^2 x^2+2\right )\right ) \sinh (c+d x)-2 b d x \cosh (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 97, normalized size = 1.9 \begin{align*}{\frac{1}{d} \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}}}-2\,{\frac{cb \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}+{\frac{b{c}^{2}\sinh \left ( dx+c \right ) }{{d}^{2}}}+a\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01897, size = 116, normalized size = 2.27 \begin{align*} \frac{a e^{\left (d x + c\right )}}{2 \, d} - \frac{a e^{\left (-d x - c\right )}}{2 \, d} + \frac{{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b e^{\left (d x\right )}}{2 \, d^{3}} - \frac{{\left (d^{2} x^{2} + 2 \, d x + 2\right )} b 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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Fricas [A] time = 1.9949, size = 97, normalized size = 1.9 \begin{align*} -\frac{2 \, b d x \cosh \left (d x + c\right ) -{\left (b d^{2} x^{2} + a d^{2} + 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 = 0.674157, size = 65, normalized size = 1.27 \begin{align*} \begin{cases} \frac{a \sinh{\left (c + d x \right )}}{d} + \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 (a x + \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 = 1.15273, size = 95, normalized size = 1.86 \begin{align*} \frac{{\left (b d^{2} x^{2} + a d^{2} - 2 \, b d x + 2 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{3}} - \frac{{\left (b d^{2} x^{2} + a d^{2} + 2 \, b d x + 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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