Optimal. Leaf size=98 \[ -\frac {f \cosh (c+d x)}{a d^2}+\frac {i f \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac {i (e+f x) \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x) \sinh (c+d x)}{a d}-\frac {i f x}{4 a d} \]
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Rubi [A] time = 0.10, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5563, 3296, 2638, 5446, 2635, 8} \[ -\frac {f \cosh (c+d x)}{a d^2}+\frac {i f \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac {i (e+f x) \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x) \sinh (c+d x)}{a d}-\frac {i f x}{4 a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 3296
Rule 5446
Rule 5563
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \, dx}{a}\\ &=\frac {(e+f x) \sinh (c+d x)}{a d}-\frac {i (e+f x) \sinh ^2(c+d x)}{2 a d}+\frac {(i f) \int \sinh ^2(c+d x) \, dx}{2 a d}-\frac {f \int \sinh (c+d x) \, dx}{a d}\\ &=-\frac {f \cosh (c+d x)}{a d^2}+\frac {(e+f x) \sinh (c+d x)}{a d}+\frac {i f \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {i (e+f x) \sinh ^2(c+d x)}{2 a d}-\frac {(i f) \int 1 \, dx}{4 a d}\\ &=-\frac {i f x}{4 a d}-\frac {f \cosh (c+d x)}{a d^2}+\frac {(e+f x) \sinh (c+d x)}{a d}+\frac {i f \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {i (e+f x) \sinh ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 60, normalized size = 0.61 \[ \frac {d (e+f x) (4 \sinh (c+d x)-i \cosh (2 (c+d x)))+i f (\sinh (c+d x)+4 i) \cosh (c+d x)}{4 a d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 92, normalized size = 0.94 \[ \frac {{\left (-2 i \, d f x - 2 i \, d e + {\left (-2 i \, d f x - 2 i \, d e + i \, f\right )} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, {\left (d f x + d e - f\right )} e^{\left (3 \, d x + 3 \, c\right )} - 8 \, {\left (d f x + d e + f\right )} e^{\left (d x + c\right )} - i \, f\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 246, normalized size = 2.51 \[ \frac {-2 i \, d f x e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d f x e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d f x e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d f x e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d f x e^{\left (d x + 2 \, c\right )} - 2 \, d f x e^{c} - 2 i \, d e^{\left (5 \, d x + 6 \, c + 1\right )} + i \, f e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d e^{\left (4 \, d x + 5 \, c + 1\right )} - 7 \, f e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d e^{\left (3 \, d x + 4 \, c + 1\right )} + 8 i \, f e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d e^{\left (2 \, d x + 3 \, c + 1\right )} - 8 \, f e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d e^{\left (d x + 2 \, c + 1\right )} + 7 i \, f e^{\left (d x + 2 \, c\right )} - 2 \, d e^{\left (c + 1\right )} - f e^{c}}{16 \, {\left (a d^{2} e^{\left (3 \, d x + 4 \, c\right )} - i \, a d^{2} e^{\left (2 \, d x + 3 \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 120, normalized size = 1.22 \[ -\frac {i f \left (\frac {\left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-\frac {i c f \left (\cosh ^{2}\left (d x +c \right )\right )}{2}+\frac {i d e \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+c f \sinh \left (d x +c \right )-\sinh \left (d x +c \right ) d e}{d^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 144, normalized size = 1.47 \[ -{\mathrm {e}}^{-c-d\,x}\,\left (\frac {f+d\,e}{2\,a\,d^2}+\frac {f\,x}{2\,a\,d}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (f+2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}+\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (f-2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}-\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )-{\mathrm {e}}^{c+d\,x}\,\left (\frac {f-d\,e}{2\,a\,d^2}-\frac {f\,x}{2\,a\,d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 323, normalized size = 3.30 \[ \begin {cases} \frac {\left (\left (- 512 a^{3} d^{7} e e^{2 c} - 512 a^{3} d^{7} f x e^{2 c} - 512 a^{3} d^{6} f e^{2 c}\right ) e^{- d x} + \left (512 a^{3} d^{7} e e^{4 c} + 512 a^{3} d^{7} f x e^{4 c} - 512 a^{3} d^{6} f e^{4 c}\right ) e^{d x} + \left (- 128 i a^{3} d^{7} e e^{c} - 128 i a^{3} d^{7} f x e^{c} - 64 i a^{3} d^{6} f e^{c}\right ) e^{- 2 d x} + \left (- 128 i a^{3} d^{7} e e^{5 c} - 128 i a^{3} d^{7} f x e^{5 c} + 64 i a^{3} d^{6} f e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{1024 a^{4} d^{8}} & \text {for}\: 1024 a^{4} d^{8} e^{3 c} \neq 0 \\\frac {x^{2} \left (- i f e^{4 c} + 2 f e^{3 c} + 2 f e^{c} + i f\right ) e^{- 2 c}}{8 a} + \frac {x \left (- i e e^{4 c} + 2 e e^{3 c} + 2 e e^{c} + i e\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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