Optimal. Leaf size=89 \[ \frac{6 a d^2 (c+d x) \sinh (e+f x)}{f^3}-\frac{3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac{a (c+d x)^3 \sinh (e+f x)}{f}+\frac{a (c+d x)^4}{4 d}-\frac{6 a d^3 \cosh (e+f x)}{f^4} \]
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Rubi [A] time = 0.13154, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3317, 3296, 2638} \[ \frac{6 a d^2 (c+d x) \sinh (e+f x)}{f^3}-\frac{3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac{a (c+d x)^3 \sinh (e+f x)}{f}+\frac{a (c+d x)^4}{4 d}-\frac{6 a d^3 \cosh (e+f x)}{f^4} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx &=\int \left (a (c+d x)^3+a (c+d x)^3 \cosh (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+a \int (c+d x)^3 \cosh (e+f x) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+\frac{a (c+d x)^3 \sinh (e+f x)}{f}-\frac{(3 a d) \int (c+d x)^2 \sinh (e+f x) \, dx}{f}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac{a (c+d x)^3 \sinh (e+f x)}{f}+\frac{\left (6 a d^2\right ) \int (c+d x) \cosh (e+f x) \, dx}{f^2}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac{6 a d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac{a (c+d x)^3 \sinh (e+f x)}{f}-\frac{\left (6 a d^3\right ) \int \sinh (e+f x) \, dx}{f^3}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{6 a d^3 \cosh (e+f x)}{f^4}-\frac{3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac{6 a d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac{a (c+d x)^3 \sinh (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.541269, size = 122, normalized size = 1.37 \[ a \left (\frac{(c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+6\right )\right ) \sinh (e+f x)}{f^3}-\frac{3 d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)}{f^4}+\frac{1}{4} x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 482, normalized size = 5.4 \begin{align*}{\frac{1}{f} \left ({\frac{{d}^{3}a \left ( fx+e \right ) ^{4}}{4\,{f}^{3}}}+{\frac{{d}^{3}a \left ( \left ( fx+e \right ) ^{3}\sinh \left ( fx+e \right ) -3\, \left ( fx+e \right ) ^{2}\cosh \left ( fx+e \right ) +6\, \left ( fx+e \right ) \sinh \left ( fx+e \right ) -6\,\cosh \left ( fx+e \right ) \right ) }{{f}^{3}}}-{\frac{{d}^{3}ea \left ( fx+e \right ) ^{3}}{{f}^{3}}}-3\,{\frac{{d}^{3}ea \left ( \left ( fx+e \right ) ^{2}\sinh \left ( fx+e \right ) -2\, \left ( fx+e \right ) \cosh \left ( fx+e \right ) +2\,\sinh \left ( fx+e \right ) \right ) }{{f}^{3}}}+{\frac{c{d}^{2}a \left ( fx+e \right ) ^{3}}{{f}^{2}}}+3\,{\frac{c{d}^{2}a \left ( \left ( fx+e \right ) ^{2}\sinh \left ( fx+e \right ) -2\, \left ( fx+e \right ) \cosh \left ( fx+e \right ) +2\,\sinh \left ( fx+e \right ) \right ) }{{f}^{2}}}+{\frac{3\,{d}^{3}{e}^{2}a \left ( fx+e \right ) ^{2}}{2\,{f}^{3}}}+3\,{\frac{{d}^{3}{e}^{2}a \left ( \left ( fx+e \right ) \sinh \left ( fx+e \right ) -\cosh \left ( fx+e \right ) \right ) }{{f}^{3}}}-3\,{\frac{{d}^{2}eca \left ( fx+e \right ) ^{2}}{{f}^{2}}}-6\,{\frac{{d}^{2}eca \left ( \left ( fx+e \right ) \sinh \left ( fx+e \right ) -\cosh \left ( fx+e \right ) \right ) }{{f}^{2}}}+{\frac{3\,{c}^{2}da \left ( fx+e \right ) ^{2}}{2\,f}}+3\,{\frac{{c}^{2}da \left ( \left ( fx+e \right ) \sinh \left ( fx+e \right ) -\cosh \left ( fx+e \right ) \right ) }{f}}-{\frac{{d}^{3}{e}^{3}a \left ( fx+e \right ) }{{f}^{3}}}-{\frac{{d}^{3}{e}^{3}a\sinh \left ( fx+e \right ) }{{f}^{3}}}+3\,{\frac{{d}^{2}{e}^{2}ca \left ( fx+e \right ) }{{f}^{2}}}+3\,{\frac{{d}^{2}{e}^{2}ca\sinh \left ( fx+e \right ) }{{f}^{2}}}-3\,{\frac{de{c}^{2}a \left ( fx+e \right ) }{f}}-3\,{\frac{de{c}^{2}a\sinh \left ( fx+e \right ) }{f}}+{c}^{3}a \left ( fx+e \right ) +a{c}^{3}\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12106, size = 320, normalized size = 3.6 \begin{align*} \frac{1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac{3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac{3}{2} \, a c^{2} d{\left (\frac{{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac{{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac{3}{2} \, a c d^{2}{\left (\frac{{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac{{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac{1}{2} \, a d^{3}{\left (\frac{{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} - \frac{{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac{a c^{3} \sinh \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01208, size = 365, normalized size = 4.1 \begin{align*} \frac{a d^{3} f^{4} x^{4} + 4 \, a c d^{2} f^{4} x^{3} + 6 \, a c^{2} d f^{4} x^{2} + 4 \, a c^{3} f^{4} x - 12 \,{\left (a d^{3} f^{2} x^{2} + 2 \, a c d^{2} f^{2} x + a c^{2} d f^{2} + 2 \, a d^{3}\right )} \cosh \left (f x + e\right ) + 4 \,{\left (a d^{3} f^{3} x^{3} + 3 \, a c d^{2} f^{3} x^{2} + a c^{3} f^{3} + 6 \, a c d^{2} f + 3 \,{\left (a c^{2} d f^{3} + 2 \, a d^{3} f\right )} x\right )} \sinh \left (f x + e\right )}{4 \, f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.41751, size = 264, normalized size = 2.97 \begin{align*} \begin{cases} a c^{3} x + \frac{a c^{3} \sinh{\left (e + f x \right )}}{f} + \frac{3 a c^{2} d x^{2}}{2} + \frac{3 a c^{2} d x \sinh{\left (e + f x \right )}}{f} - \frac{3 a c^{2} d \cosh{\left (e + f x \right )}}{f^{2}} + a c d^{2} x^{3} + \frac{3 a c d^{2} x^{2} \sinh{\left (e + f x \right )}}{f} - \frac{6 a c d^{2} x \cosh{\left (e + f x \right )}}{f^{2}} + \frac{6 a c d^{2} \sinh{\left (e + f x \right )}}{f^{3}} + \frac{a d^{3} x^{4}}{4} + \frac{a d^{3} x^{3} \sinh{\left (e + f x \right )}}{f} - \frac{3 a d^{3} x^{2} \cosh{\left (e + f x \right )}}{f^{2}} + \frac{6 a d^{3} x \sinh{\left (e + f x \right )}}{f^{3}} - \frac{6 a d^{3} \cosh{\left (e + f x \right )}}{f^{4}} & \text{for}\: f \neq 0 \\\left (a \cosh{\left (e \right )} + a\right ) \left (c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25464, size = 351, normalized size = 3.94 \begin{align*} \frac{1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac{3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac{{\left (a d^{3} f^{3} x^{3} + 3 \, a c d^{2} f^{3} x^{2} + 3 \, a c^{2} d f^{3} x - 3 \, a d^{3} f^{2} x^{2} + a c^{3} f^{3} - 6 \, a c d^{2} f^{2} x - 3 \, a c^{2} d f^{2} + 6 \, a d^{3} f x + 6 \, a c d^{2} f - 6 \, a d^{3}\right )} e^{\left (f x + e\right )}}{2 \, f^{4}} - \frac{{\left (a d^{3} f^{3} x^{3} + 3 \, a c d^{2} f^{3} x^{2} + 3 \, a c^{2} d f^{3} x + 3 \, a d^{3} f^{2} x^{2} + a c^{3} f^{3} + 6 \, a c d^{2} f^{2} x + 3 \, a c^{2} d f^{2} + 6 \, a d^{3} f x + 6 \, a c d^{2} f + 6 \, a d^{3}\right )} e^{\left (-f x - e\right )}}{2 \, f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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