Optimal. Leaf size=66 \[ \frac{a \sinh (c+d x)}{d}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{6 b x \sinh (c+d x)}{d^3}-\frac{6 b \cosh (c+d x)}{d^4}+\frac{b x^3 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.103047, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5277, 2637, 3296, 2638} \[ \frac{a \sinh (c+d x)}{d}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{6 b x \sinh (c+d x)}{d^3}-\frac{6 b \cosh (c+d x)}{d^4}+\frac{b x^3 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5277
Rule 2637
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \left (a+b x^3\right ) \cosh (c+d x) \, dx &=\int \left (a \cosh (c+d x)+b x^3 \cosh (c+d x)\right ) \, dx\\ &=a \int \cosh (c+d x) \, dx+b \int x^3 \cosh (c+d x) \, dx\\ &=\frac{a \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}-\frac{(3 b) \int x^2 \sinh (c+d x) \, dx}{d}\\ &=-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{a \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}+\frac{(6 b) \int x \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{a \sinh (c+d x)}{d}+\frac{6 b x \sinh (c+d x)}{d^3}+\frac{b x^3 \sinh (c+d x)}{d}-\frac{(6 b) \int \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{6 b \cosh (c+d x)}{d^4}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{a \sinh (c+d x)}{d}+\frac{6 b x \sinh (c+d x)}{d^3}+\frac{b x^3 \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0808538, size = 49, normalized size = 0.74 \[ \frac{d \left (a d^2+b x \left (d^2 x^2+6\right )\right ) \sinh (c+d x)-3 b \left (d^2 x^2+2\right ) \cosh (c+d x)}{d^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 158, normalized size = 2.4 \begin{align*}{\frac{1}{d} \left ({\frac{b \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}^{3}}}-3\,{\frac{cb \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}^{3}}}+3\,{\frac{b{c}^{2} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{3}}}-{\frac{b{c}^{3}\sinh \left ( dx+c \right ) }{{d}^{3}}}+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 = 1.02864, size = 140, normalized size = 2.12 \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^{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 )}}{2 \, d^{4}} - \frac{{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} b e^{\left (-d x - c\right )}}{2 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81258, size = 119, normalized size = 1.8 \begin{align*} -\frac{3 \,{\left (b d^{2} x^{2} + 2 \, b\right )} \cosh \left (d x + c\right ) -{\left (b d^{3} x^{3} + a d^{3} + 6 \, b d x\right )} \sinh \left (d x + c\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.40829, size = 82, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a \sinh{\left (c + d x \right )}}{d} + \frac{b x^{3} \sinh{\left (c + d x \right )}}{d} - \frac{3 b x^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{6 b x \sinh{\left (c + d x \right )}}{d^{3}} - \frac{6 b \cosh{\left (c + d x \right )}}{d^{4}} & \text{for}\: d \neq 0 \\\left (a x + \frac{b x^{4}}{4}\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 = 1.22369, size = 119, normalized size = 1.8 \begin{align*} \frac{{\left (b d^{3} x^{3} - 3 \, b d^{2} x^{2} + a d^{3} + 6 \, b d x - 6 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{4}} - \frac{{\left (b d^{3} x^{3} + 3 \, b d^{2} x^{2} + a d^{3} + 6 \, b d x + 6 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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