Optimal. Leaf size=74 \[ -\frac{2 i a d (c+d x) \sinh (e+f x)}{f^2}+\frac{i a (c+d x)^2 \cosh (e+f x)}{f}+\frac{a (c+d x)^3}{3 d}+\frac{2 i a d^2 \cosh (e+f x)}{f^3} \]
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Rubi [A] time = 0.0970964, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3317, 3296, 2638} \[ -\frac{2 i a d (c+d x) \sinh (e+f x)}{f^2}+\frac{i a (c+d x)^2 \cosh (e+f x)}{f}+\frac{a (c+d x)^3}{3 d}+\frac{2 i a d^2 \cosh (e+f x)}{f^3} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x)^2 (a+i a \sinh (e+f x)) \, dx &=\int \left (a (c+d x)^2+i a (c+d x)^2 \sinh (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^3}{3 d}+(i a) \int (c+d x)^2 \sinh (e+f x) \, dx\\ &=\frac{a (c+d x)^3}{3 d}+\frac{i a (c+d x)^2 \cosh (e+f x)}{f}-\frac{(2 i a d) \int (c+d x) \cosh (e+f x) \, dx}{f}\\ &=\frac{a (c+d x)^3}{3 d}+\frac{i a (c+d x)^2 \cosh (e+f x)}{f}-\frac{2 i a d (c+d x) \sinh (e+f x)}{f^2}+\frac{\left (2 i a d^2\right ) \int \sinh (e+f x) \, dx}{f^2}\\ &=\frac{a (c+d x)^3}{3 d}+\frac{2 i a d^2 \cosh (e+f x)}{f^3}+\frac{i a (c+d x)^2 \cosh (e+f x)}{f}-\frac{2 i a d (c+d x) \sinh (e+f x)}{f^2}\\ \end{align*}
Mathematica [A] time = 0.472278, size = 88, normalized size = 1.19 \[ \frac{a \left (3 i \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)+f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )-6 i d f (c+d x) \sinh (e+f x)\right )}{3 f^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 249, normalized size = 3.4 \begin{align*}{\frac{1}{f} \left ({\frac{a{d}^{2} \left ( fx+e \right ) ^{3}}{3\,{f}^{2}}}+{\frac{i{d}^{2}a \left ( \left ( fx+e \right ) ^{2}\cosh \left ( fx+e \right ) -2\, \left ( fx+e \right ) \sinh \left ( fx+e \right ) +2\,\cosh \left ( fx+e \right ) \right ) }{{f}^{2}}}-{\frac{{d}^{2}ea \left ( fx+e \right ) ^{2}}{{f}^{2}}}-{\frac{2\,i{d}^{2}ea \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{{f}^{2}}}+{\frac{cda \left ( fx+e \right ) ^{2}}{f}}+{\frac{2\,idca \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{f}}+{\frac{{d}^{2}{e}^{2}a \left ( fx+e \right ) }{{f}^{2}}}+{\frac{i{d}^{2}{e}^{2}a\cosh \left ( fx+e \right ) }{{f}^{2}}}-2\,{\frac{deca \left ( fx+e \right ) }{f}}-{\frac{2\,ideca\cosh \left ( fx+e \right ) }{f}}+{c}^{2}a \left ( fx+e \right ) +i{c}^{2}a\cosh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06462, size = 190, normalized size = 2.57 \begin{align*} \frac{1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + i \, a c 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{1}{2} i \, a 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{i \, a c^{2} \cosh \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.64756, size = 402, normalized size = 5.43 \begin{align*} \frac{{\left (3 i \, a d^{2} f^{2} x^{2} + 3 i \, a c^{2} f^{2} + 6 i \, a c d f + 6 i \, a d^{2} +{\left (6 i \, a c d f^{2} + 6 i \, a d^{2} f\right )} x +{\left (3 i \, a d^{2} f^{2} x^{2} + 3 i \, a c^{2} f^{2} - 6 i \, a c d f + 6 i \, a d^{2} +{\left (6 i \, a c d f^{2} - 6 i \, a d^{2} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \,{\left (a d^{2} f^{3} x^{3} + 3 \, a c d f^{3} x^{2} + 3 \, a c^{2} f^{3} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{6 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.53159, size = 350, normalized size = 4.73 \begin{align*} a c^{2} x + a c d x^{2} + \frac{a d^{2} x^{3}}{3} + \begin{cases} \frac{\left (\left (2 i a c^{2} f^{11} e^{2 e} + 4 i a c d f^{11} x e^{2 e} + 4 i a c d f^{10} e^{2 e} + 2 i a d^{2} f^{11} x^{2} e^{2 e} + 4 i a d^{2} f^{10} x e^{2 e} + 4 i a d^{2} f^{9} e^{2 e}\right ) e^{- f x} + \left (2 i a c^{2} f^{11} e^{4 e} + 4 i a c d f^{11} x e^{4 e} - 4 i a c d f^{10} e^{4 e} + 2 i a d^{2} f^{11} x^{2} e^{4 e} - 4 i a d^{2} f^{10} x e^{4 e} + 4 i a d^{2} f^{9} e^{4 e}\right ) e^{f x}\right ) e^{- 3 e}}{4 f^{12}} & \text{for}\: 4 f^{12} e^{3 e} \neq 0 \\\frac{x^{3} \left (i a d^{2} e^{2 e} - i a d^{2}\right ) e^{- e}}{6} + \frac{x^{2} \left (i a c d e^{2 e} - i a c d\right ) e^{- e}}{2} + \frac{x \left (i a c^{2} e^{2 e} - i a c^{2}\right ) e^{- e}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26303, size = 205, normalized size = 2.77 \begin{align*} \frac{1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x - \frac{{\left (-i \, a d^{2} f^{2} x^{2} - 2 i \, a c d f^{2} x - i \, a c^{2} f^{2} + 2 i \, a d^{2} f x + 2 i \, a c d f - 2 i \, a d^{2}\right )} e^{\left (f x + e\right )}}{2 \, f^{3}} + \frac{{\left (i \, a d^{2} f^{2} x^{2} + 2 i \, a c d f^{2} x + i \, a c^{2} f^{2} + 2 i \, a d^{2} f x + 2 i \, a c d f + 2 i \, a d^{2}\right )} e^{\left (-f x - e\right )}}{2 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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