Optimal. Leaf size=171 \[ -\frac{2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac{i f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d^2}-\frac{i f^2 \sinh ^2(c+d x)}{4 a d^3}+\frac{2 f^2 \sinh (c+d x)}{a d^3}-\frac{i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}+\frac{(e+f x)^2 \sinh (c+d x)}{a d}-\frac{i e f x}{2 a d}-\frac{i f^2 x^2}{4 a d} \]
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Rubi [A] time = 0.186152, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {5563, 3296, 2637, 5446, 3310} \[ -\frac{2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac{i f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d^2}-\frac{i f^2 \sinh ^2(c+d x)}{4 a d^3}+\frac{2 f^2 \sinh (c+d x)}{a d^3}-\frac{i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}+\frac{(e+f x)^2 \sinh (c+d x)}{a d}-\frac{i e f x}{2 a d}-\frac{i f^2 x^2}{4 a d} \]
Antiderivative was successfully verified.
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Rule 5563
Rule 3296
Rule 2637
Rule 5446
Rule 3310
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{a}+\frac{\int (e+f x)^2 \cosh (c+d x) \, dx}{a}\\ &=\frac{(e+f x)^2 \sinh (c+d x)}{a d}-\frac{i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}+\frac{(i f) \int (e+f x) \sinh ^2(c+d x) \, dx}{a d}-\frac{(2 f) \int (e+f x) \sinh (c+d x) \, dx}{a d}\\ &=-\frac{2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac{(e+f x)^2 \sinh (c+d x)}{a d}+\frac{i f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d^2}-\frac{i f^2 \sinh ^2(c+d x)}{4 a d^3}-\frac{i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}-\frac{(i f) \int (e+f x) \, dx}{2 a d}+\frac{\left (2 f^2\right ) \int \cosh (c+d x) \, dx}{a d^2}\\ &=-\frac{i e f x}{2 a d}-\frac{i f^2 x^2}{4 a d}-\frac{2 f (e+f x) \cosh (c+d x)}{a d^2}+\frac{2 f^2 \sinh (c+d x)}{a d^3}+\frac{(e+f x)^2 \sinh (c+d x)}{a d}+\frac{i f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d^2}-\frac{i f^2 \sinh ^2(c+d x)}{4 a d^3}-\frac{i (e+f x)^2 \sinh ^2(c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.910861, size = 99, normalized size = 0.58 \[ \frac{-2 i \cosh (2 (c+d x)) \left (2 d^2 (e+f x)^2+f^2\right )+8 \sinh (c+d x) \left (2 \left (d^2 (e+f x)^2+2 f^2\right )+i d f (e+f x) \cosh (c+d x)\right )-32 d f (e+f x) \cosh (c+d x)}{16 a d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.138, size = 241, normalized size = 1.4 \begin{align*}{\frac{-{\frac{i}{16}} \left ( 2\,{f}^{2}{x}^{2}{d}^{2}+4\,{d}^{2}efx+2\,{d}^{2}{e}^{2}-2\,d{f}^{2}x-2\,efd+{f}^{2} \right ){{\rm e}^{2\,dx+2\,c}}}{a{d}^{3}}}+{\frac{ \left ({f}^{2}{x}^{2}{d}^{2}+2\,{d}^{2}efx+{d}^{2}{e}^{2}-2\,d{f}^{2}x-2\,efd+2\,{f}^{2} \right ){{\rm e}^{dx+c}}}{2\,a{d}^{3}}}-{\frac{ \left ({f}^{2}{x}^{2}{d}^{2}+2\,{d}^{2}efx+{d}^{2}{e}^{2}+2\,d{f}^{2}x+2\,efd+2\,{f}^{2} \right ){{\rm e}^{-dx-c}}}{2\,a{d}^{3}}}-{\frac{{\frac{i}{16}} \left ( 2\,{f}^{2}{x}^{2}{d}^{2}+4\,{d}^{2}efx+2\,{d}^{2}{e}^{2}+2\,d{f}^{2}x+2\,efd+{f}^{2} \right ){{\rm e}^{-2\,dx-2\,c}}}{a{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13009, size = 520, normalized size = 3.04 \begin{align*} \frac{{\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} - 2 i \, d e f - i \, f^{2} +{\left (-4 i \, d^{2} e f - 2 i \, d f^{2}\right )} x +{\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} + 2 i \, d e f - i \, f^{2} +{\left (-4 i \, d^{2} e f + 2 i \, d f^{2}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 8 \,{\left (d^{2} f^{2} x^{2} + d^{2} e^{2} - 2 \, d e f + 2 \, f^{2} + 2 \,{\left (d^{2} e f - d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 8 \,{\left (d^{2} f^{2} x^{2} + d^{2} e^{2} + 2 \, d e f + 2 \, f^{2} + 2 \,{\left (d^{2} e f + d f^{2}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.36033, size = 644, normalized size = 3.77 \begin{align*} \begin{cases} \frac{\left (\left (- 512 a^{7} d^{17} e^{2} e^{6 c} - 1024 a^{7} d^{17} e f x e^{6 c} - 512 a^{7} d^{17} f^{2} x^{2} e^{6 c} - 1024 a^{7} d^{16} e f e^{6 c} - 1024 a^{7} d^{16} f^{2} x e^{6 c} - 1024 a^{7} d^{15} f^{2} e^{6 c}\right ) e^{- d x} + \left (512 a^{7} d^{17} e^{2} e^{8 c} + 1024 a^{7} d^{17} e f x e^{8 c} + 512 a^{7} d^{17} f^{2} x^{2} e^{8 c} - 1024 a^{7} d^{16} e f e^{8 c} - 1024 a^{7} d^{16} f^{2} x e^{8 c} + 1024 a^{7} d^{15} f^{2} e^{8 c}\right ) e^{d x} + \left (- 128 i a^{7} d^{17} e^{2} e^{5 c} - 256 i a^{7} d^{17} e f x e^{5 c} - 128 i a^{7} d^{17} f^{2} x^{2} e^{5 c} - 128 i a^{7} d^{16} e f e^{5 c} - 128 i a^{7} d^{16} f^{2} x e^{5 c} - 64 i a^{7} d^{15} f^{2} e^{5 c}\right ) e^{- 2 d x} + \left (- 128 i a^{7} d^{17} e^{2} e^{9 c} - 256 i a^{7} d^{17} e f x e^{9 c} - 128 i a^{7} d^{17} f^{2} x^{2} e^{9 c} + 128 i a^{7} d^{16} e f e^{9 c} + 128 i a^{7} d^{16} f^{2} x e^{9 c} - 64 i a^{7} d^{15} f^{2} e^{9 c}\right ) e^{2 d x}\right ) e^{- 7 c}}{1024 a^{8} d^{18}} & \text{for}\: 1024 a^{8} d^{18} e^{7 c} \neq 0 \\- \frac{x^{3} \left (i f^{2} e^{4 c} - 2 f^{2} e^{3 c} - 2 f^{2} e^{c} - i f^{2}\right ) e^{- 2 c}}{12 a} - \frac{x^{2} \left (i e f e^{4 c} - 2 e f e^{3 c} - 2 e f e^{c} - i e f\right ) e^{- 2 c}}{4 a} - \frac{x \left (i e^{2} e^{4 c} - 2 e^{2} e^{3 c} - 2 e^{2} e^{c} - i e^{2}\right ) e^{- 2 c}}{4 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22192, size = 764, normalized size = 4.47 \begin{align*} \frac{-2 i \, d^{2} f^{2} x^{2} e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d^{2} f^{2} x^{2} e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d^{2} f^{2} x^{2} e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d^{2} f^{2} x^{2} e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d^{2} f^{2} x^{2} e^{\left (d x + 2 \, c\right )} - 2 \, d^{2} f^{2} x^{2} e^{c} - 4 i \, d^{2} f x e^{\left (5 \, d x + 6 \, c + 1\right )} + 2 i \, d f^{2} x e^{\left (5 \, d x + 6 \, c\right )} + 12 \, d^{2} f x e^{\left (4 \, d x + 5 \, c + 1\right )} - 14 \, d f^{2} x e^{\left (4 \, d x + 5 \, c\right )} - 16 i \, d^{2} f x e^{\left (3 \, d x + 4 \, c + 1\right )} + 16 i \, d f^{2} x e^{\left (3 \, d x + 4 \, c\right )} - 16 \, d^{2} f x e^{\left (2 \, d x + 3 \, c + 1\right )} - 16 \, d f^{2} x e^{\left (2 \, d x + 3 \, c\right )} + 12 i \, d^{2} f x e^{\left (d x + 2 \, c + 1\right )} + 14 i \, d f^{2} x e^{\left (d x + 2 \, c\right )} - 4 \, d^{2} f x e^{\left (c + 1\right )} - 2 \, d f^{2} x e^{c} - 2 i \, d^{2} e^{\left (5 \, d x + 6 \, c + 2\right )} + 2 i \, d f e^{\left (5 \, d x + 6 \, c + 1\right )} - i \, f^{2} e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d^{2} e^{\left (4 \, d x + 5 \, c + 2\right )} - 14 \, d f e^{\left (4 \, d x + 5 \, c + 1\right )} + 15 \, f^{2} e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d^{2} e^{\left (3 \, d x + 4 \, c + 2\right )} + 16 i \, d f e^{\left (3 \, d x + 4 \, c + 1\right )} - 16 i \, f^{2} e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d^{2} e^{\left (2 \, d x + 3 \, c + 2\right )} - 16 \, d f e^{\left (2 \, d x + 3 \, c + 1\right )} - 16 \, f^{2} e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d^{2} e^{\left (d x + 2 \, c + 2\right )} + 14 i \, d f e^{\left (d x + 2 \, c + 1\right )} + 15 i \, f^{2} e^{\left (d x + 2 \, c\right )} - 2 \, d^{2} e^{\left (c + 2\right )} - 2 \, d f e^{\left (c + 1\right )} - f^{2} e^{c}}{16 \, a d^{3} e^{\left (3 \, d x + 4 \, c\right )} - 16 i \, a d^{3} e^{\left (2 \, d x + 3 \, c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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