Optimal. Leaf size=109 \[ \frac{2 a \sinh (c+d x)}{d^3}-\frac{2 a x \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{12 b x^2 \sinh (c+d x)}{d^3}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{24 b \sinh (c+d x)}{d^5}-\frac{24 b x \cosh (c+d x)}{d^4}+\frac{b x^4 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.187407, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5287, 3296, 2637} \[ \frac{2 a \sinh (c+d x)}{d^3}-\frac{2 a x \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{12 b x^2 \sinh (c+d x)}{d^3}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{24 b \sinh (c+d x)}{d^5}-\frac{24 b x \cosh (c+d x)}{d^4}+\frac{b x^4 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5287
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a x^2 \cosh (c+d x)+b x^4 \cosh (c+d x)\right ) \, dx\\ &=a \int x^2 \cosh (c+d x) \, dx+b \int x^4 \cosh (c+d x) \, dx\\ &=\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}-\frac{(2 a) \int x \sinh (c+d x) \, dx}{d}-\frac{(4 b) \int x^3 \sinh (c+d x) \, dx}{d}\\ &=-\frac{2 a x \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}+\frac{(2 a) \int \cosh (c+d x) \, dx}{d^2}+\frac{(12 b) \int x^2 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{2 a x \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{2 a \sinh (c+d x)}{d^3}+\frac{12 b x^2 \sinh (c+d x)}{d^3}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}-\frac{(24 b) \int x \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{24 b x \cosh (c+d x)}{d^4}-\frac{2 a x \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{2 a \sinh (c+d x)}{d^3}+\frac{12 b x^2 \sinh (c+d x)}{d^3}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}+\frac{(24 b) \int \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{24 b x \cosh (c+d x)}{d^4}-\frac{2 a x \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{24 b \sinh (c+d x)}{d^5}+\frac{2 a \sinh (c+d x)}{d^3}+\frac{12 b x^2 \sinh (c+d x)}{d^3}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.129127, size = 74, normalized size = 0.68 \[ \frac{\left (a d^2 \left (d^2 x^2+2\right )+b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \sinh (c+d x)-2 d x \left (a d^2+2 b \left (d^2 x^2+6\right )\right ) \cosh (c+d x)}{d^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 298, normalized size = 2.7 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{b \left ( \left ( dx+c \right ) ^{4}\sinh \left ( dx+c \right ) -4\, \left ( dx+c \right ) ^{3}\cosh \left ( dx+c \right ) +12\, \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -24\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +24\,\sinh \left ( dx+c \right ) \right ) }{{d}^{2}}}-4\,{\frac{cb \left ( \left ( dx+c \right ) ^{3}\sinh \left ( dx+c \right ) -3\, \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) +6\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) -6\,\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}+6\,{\frac{b{c}^{2} \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}}}-4\,{\frac{b{c}^{3} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}+{\frac{b{c}^{4}\sinh \left ( dx+c \right ) }{{d}^{2}}}+a \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 ) -2\,ac \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) +a{c}^{2}\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 = 1.03039, size = 289, normalized size = 2.65 \begin{align*} -\frac{1}{30} \, d{\left (\frac{5 \,{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{4}} + \frac{5 \,{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a e^{\left (-d x - c\right )}}{d^{4}} + \frac{3 \,{\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{6}} + \frac{3 \,{\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac{1}{15} \,{\left (3 \, b x^{5} + 5 \, a x^{3}\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 = 2.02551, size = 174, normalized size = 1.6 \begin{align*} -\frac{2 \,{\left (2 \, b d^{3} x^{3} +{\left (a d^{3} + 12 \, b d\right )} x\right )} \cosh \left (d x + c\right ) -{\left (b d^{4} x^{4} + 2 \, a d^{2} +{\left (a d^{4} + 12 \, b d^{2}\right )} x^{2} + 24 \, b\right )} \sinh \left (d x + c\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.61537, size = 134, normalized size = 1.23 \begin{align*} \begin{cases} \frac{a x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 a x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 a \sinh{\left (c + d x \right )}}{d^{3}} + \frac{b x^{4} \sinh{\left (c + d x \right )}}{d} - \frac{4 b x^{3} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{12 b x^{2} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{24 b x \cosh{\left (c + d x \right )}}{d^{4}} + \frac{24 b \sinh{\left (c + d x \right )}}{d^{5}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{3}}{3} + \frac{b x^{5}}{5}\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 = 1.19823, size = 186, normalized size = 1.71 \begin{align*} \frac{{\left (b d^{4} x^{4} + a d^{4} x^{2} - 4 \, b d^{3} x^{3} - 2 \, a d^{3} x + 12 \, b d^{2} x^{2} + 2 \, a d^{2} - 24 \, b d x + 24 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac{{\left (b d^{4} x^{4} + a d^{4} x^{2} + 4 \, b d^{3} x^{3} + 2 \, a d^{3} x + 12 \, b d^{2} x^{2} + 2 \, a d^{2} + 24 \, b d x + 24 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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