Optimal. Leaf size=82 \[ \frac{2 i f (e+f x) \sinh (c+d x)}{a d^2}-\frac{2 i f^2 \cosh (c+d x)}{a d^3}-\frac{i (e+f x)^2 \cosh (c+d x)}{a d}+\frac{(e+f x)^3}{3 a f} \]
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Rubi [A] time = 0.126473, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {5563, 32, 3296, 2638} \[ \frac{2 i f (e+f x) \sinh (c+d x)}{a d^2}-\frac{2 i f^2 \cosh (c+d x)}{a d^3}-\frac{i (e+f x)^2 \cosh (c+d x)}{a d}+\frac{(e+f x)^3}{3 a f} \]
Antiderivative was successfully verified.
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Rule 5563
Rule 32
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \int (e+f x)^2 \sinh (c+d x) \, dx}{a}+\frac{\int (e+f x)^2 \, dx}{a}\\ &=\frac{(e+f x)^3}{3 a f}-\frac{i (e+f x)^2 \cosh (c+d x)}{a d}+\frac{(2 i f) \int (e+f x) \cosh (c+d x) \, dx}{a d}\\ &=\frac{(e+f x)^3}{3 a f}-\frac{i (e+f x)^2 \cosh (c+d x)}{a d}+\frac{2 i f (e+f x) \sinh (c+d x)}{a d^2}-\frac{\left (2 i f^2\right ) \int \sinh (c+d x) \, dx}{a d^2}\\ &=\frac{(e+f x)^3}{3 a f}-\frac{2 i f^2 \cosh (c+d x)}{a d^3}-\frac{i (e+f x)^2 \cosh (c+d x)}{a d}+\frac{2 i f (e+f x) \sinh (c+d x)}{a d^2}\\ \end{align*}
Mathematica [A] time = 0.484384, size = 78, normalized size = 0.95 \[ \frac{-3 i \cosh (c+d x) \left (d^2 (e+f x)^2+2 f^2\right )+6 i d f (e+f x) \sinh (c+d x)+d^3 x \left (3 e^2+3 e f x+f^2 x^2\right )}{3 a d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 223, normalized size = 2.7 \begin{align*} -{\frac{1}{{d}^{3}a} \left ( i{f}^{2} \left ( \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) +2\,\cosh \left ( dx+c \right ) \right ) -2\,ic{f}^{2} \left ( \left ( dx+c \right ) \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) +2\,ifed \left ( \left ( dx+c \right ) \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) +i{c}^{2}{f}^{2}\cosh \left ( dx+c \right ) -2\,icdfe\cosh \left ( dx+c \right ) +i{d}^{2}{e}^{2}\cosh \left ( dx+c \right ) -{\frac{{f}^{2} \left ( dx+c \right ) ^{3}}{3}}+c{f}^{2} \left ( dx+c \right ) ^{2}-efd \left ( dx+c \right ) ^{2}-{c}^{2}{f}^{2} \left ( dx+c \right ) +2\,cdfe \left ( dx+c \right ) -{d}^{2}{e}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59838, size = 366, normalized size = 4.46 \begin{align*} \frac{1}{2} \, e f{\left (\frac{4 \, x e^{\left (d x + c\right )}}{a d e^{\left (d x + c\right )} - i \, a d} + \frac{-2 i \, d^{2} x^{2} e^{c} - 2 i \, d x e^{c} -{\left (2 i \, d x e^{\left (3 \, c\right )} - 2 i \, e^{\left (3 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 2 \,{\left (d^{2} x^{2} e^{\left (2 \, c\right )} - 3 \, d x e^{\left (2 \, c\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - 2 \,{\left (d x + 1\right )} e^{\left (-d x\right )} - 2 i \, e^{c}}{a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}}\right )} + \frac{1}{4} \, e^{2}{\left (\frac{4 \,{\left (d x + c\right )}}{a d} - \frac{2 i \, e^{\left (d x + c\right )}}{a d} - \frac{2 i \, e^{\left (-d x - c\right )}}{a d}\right )} + \frac{{\left (4 \, d^{3} x^{3} e^{c} -{\left (6 i \, d^{2} x^{2} e^{\left (2 \, c\right )} - 12 i \, d x e^{\left (2 \, c\right )} + 12 i \, e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} -{\left (6 i \, d^{2} x^{2} + 12 i \, d x + 12 i\right )} e^{\left (-d x\right )}\right )} f^{2} e^{\left (-c\right )}}{12 \, a d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1649, size = 373, normalized size = 4.55 \begin{align*} \frac{{\left (-3 i \, d^{2} f^{2} x^{2} - 3 i \, d^{2} e^{2} - 6 i \, d e f - 6 i \, f^{2} +{\left (-6 i \, d^{2} e f - 6 i \, d f^{2}\right )} x +{\left (-3 i \, d^{2} f^{2} x^{2} - 3 i \, d^{2} e^{2} + 6 i \, d e f - 6 i \, f^{2} +{\left (-6 i \, d^{2} e f + 6 i \, d f^{2}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-d x - c\right )}}{6 \, a d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.94693, size = 376, normalized size = 4.59 \begin{align*} \begin{cases} \frac{\left (\left (- 2 i a^{5} d^{11} e^{2} e^{2 c} - 4 i a^{5} d^{11} e f x e^{2 c} - 2 i a^{5} d^{11} f^{2} x^{2} e^{2 c} - 4 i a^{5} d^{10} e f e^{2 c} - 4 i a^{5} d^{10} f^{2} x e^{2 c} - 4 i a^{5} d^{9} f^{2} e^{2 c}\right ) e^{- d x} + \left (- 2 i a^{5} d^{11} e^{2} e^{4 c} - 4 i a^{5} d^{11} e f x e^{4 c} - 2 i a^{5} d^{11} f^{2} x^{2} e^{4 c} + 4 i a^{5} d^{10} e f e^{4 c} + 4 i a^{5} d^{10} f^{2} x e^{4 c} - 4 i a^{5} d^{9} f^{2} e^{4 c}\right ) e^{d x}\right ) e^{- 3 c}}{4 a^{6} d^{12}} & \text{for}\: 4 a^{6} d^{12} e^{3 c} \neq 0 \\- \frac{x^{3} \left (i f^{2} e^{2 c} - i f^{2}\right ) e^{- c}}{6 a} - \frac{x^{2} \left (i e f e^{2 c} - i e f\right ) e^{- c}}{2 a} - \frac{x \left (i e^{2} e^{2 c} - i e^{2}\right ) e^{- c}}{2 a} & \text{otherwise} \end{cases} + \frac{e^{2} x}{a} + \frac{e f x^{2}}{a} + \frac{f^{2} x^{3}}{3 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2607, size = 648, normalized size = 7.9 \begin{align*} \frac{2 \, d^{3} f^{2} x^{3} e^{\left (2 \, d x + 3 \, c\right )} - 2 i \, d^{3} f^{2} x^{3} e^{\left (d x + 2 \, c\right )} - 3 i \, d^{2} f^{2} x^{2} e^{\left (3 \, d x + 4 \, c\right )} + 6 \, d^{3} f x^{2} e^{\left (2 \, d x + 3 \, c + 1\right )} - 3 \, d^{2} f^{2} x^{2} e^{\left (2 \, d x + 3 \, c\right )} - 6 i \, d^{3} f x^{2} e^{\left (d x + 2 \, c + 1\right )} - 3 i \, d^{2} f^{2} x^{2} e^{\left (d x + 2 \, c\right )} - 3 \, d^{2} f^{2} x^{2} e^{c} - 6 i \, d^{2} f x e^{\left (3 \, d x + 4 \, c + 1\right )} + 6 i \, d f^{2} x e^{\left (3 \, d x + 4 \, c\right )} + 6 \, d^{3} x e^{\left (2 \, d x + 3 \, c + 2\right )} - 6 \, d^{2} f x e^{\left (2 \, d x + 3 \, c + 1\right )} + 6 \, d f^{2} x e^{\left (2 \, d x + 3 \, c\right )} - 6 i \, d^{3} x e^{\left (d x + 2 \, c + 2\right )} - 6 i \, d^{2} f x e^{\left (d x + 2 \, c + 1\right )} - 6 i \, d f^{2} x e^{\left (d x + 2 \, c\right )} - 6 \, d^{2} f x e^{\left (c + 1\right )} - 6 \, d f^{2} x e^{c} - 3 i \, d^{2} e^{\left (3 \, d x + 4 \, c + 2\right )} + 6 i \, d f e^{\left (3 \, d x + 4 \, c + 1\right )} - 6 i \, f^{2} e^{\left (3 \, d x + 4 \, c\right )} - 3 \, d^{2} e^{\left (2 \, d x + 3 \, c + 2\right )} + 6 \, d f e^{\left (2 \, d x + 3 \, c + 1\right )} - 6 \, f^{2} e^{\left (2 \, d x + 3 \, c\right )} - 3 i \, d^{2} e^{\left (d x + 2 \, c + 2\right )} - 6 i \, d f e^{\left (d x + 2 \, c + 1\right )} - 6 i \, f^{2} e^{\left (d x + 2 \, c\right )} - 3 \, d^{2} e^{\left (c + 2\right )} - 6 \, d f e^{\left (c + 1\right )} - 6 \, f^{2} e^{c}}{6 \,{\left (a d^{3} e^{\left (2 \, d x + 3 \, c\right )} - i \, a d^{3} e^{\left (d x + 2 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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