Optimal. Leaf size=449 \[ -\frac{3 i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{i \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d} \]
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Rubi [A] time = 1.78286, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 53, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3728, 3303, 3299, 3302, 3312, 4406, 4428} \[ -\frac{3 i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{i \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3728
Rule 3303
Rule 3299
Rule 3302
Rule 3312
Rule 4406
Rule 4428
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+i a \tan (e+f x))^3} \, dx &=\int \left (\frac{1}{8 a^3 (c+d x)}+\frac{3 \cos (2 e+2 f x)}{8 a^3 (c+d x)}+\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 (c+d x)}+\frac{\cos ^3(2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \sin (2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 (c+d x)}+\frac{i \sin ^3(2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \sin (4 e+4 f x)}{8 a^3 (c+d x)}-\frac{3 \sin (2 e+2 f x) \sin (4 e+4 f x)}{16 a^3 (c+d x)}\right ) \, dx\\ &=\frac{\log (c+d x)}{8 a^3 d}+\frac{i \int \frac{\sin ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\sin (2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\sin (4 e+4 f x)}{c+d x} \, dx}{8 a^3}+\frac{\int \frac{\cos ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{3 \int \frac{\sin (2 e+2 f x) \sin (4 e+4 f x)}{c+d x} \, dx}{16 a^3}+\frac{3 \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac{3 \int \frac{\cos ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{3 \int \frac{\sin ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}\\ &=\frac{\log (c+d x)}{8 a^3 d}+\frac{i \int \left (\frac{3 \sin (2 e+2 f x)}{4 (c+d x)}-\frac{\sin (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac{(3 i) \int \left (\frac{\sin (2 e+2 f x)}{4 (c+d x)}+\frac{\sin (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}+\frac{\int \left (\frac{3 \cos (2 e+2 f x)}{4 (c+d x)}+\frac{\cos (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac{3 \int \left (\frac{\cos (2 e+2 f x)}{2 (c+d x)}-\frac{\cos (6 e+6 f x)}{2 (c+d x)}\right ) \, dx}{16 a^3}-\frac{3 \int \left (\frac{1}{2 (c+d x)}-\frac{\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}+\frac{3 \int \left (\frac{1}{2 (c+d x)}+\frac{\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}-\frac{\left (3 i \cos \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 i \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}+\frac{\left (3 \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 i \sin \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 i \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}\\ &=\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac{i \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{32 a^3}-\frac{(3 i) \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+\frac{\int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+\frac{3 \int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+2 \frac{3 \int \frac{\cos (4 e+4 f x)}{c+d x} \, dx}{16 a^3}\\ &=\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac{\left (i \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (3 i \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac{\cos \left (6 e-\frac{6 c f}{d}\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac{\left (3 \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (i \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (3 i \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (3 \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+2 \left (\frac{\left (3 \cos \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{16 a^3}-\frac{\left (3 \sin \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{16 a^3}\right )\\ &=\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\cos \left (6 e-\frac{6 c f}{d}\right ) \text{Ci}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}-\frac{i \text{Ci}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}+2 \left (\frac{3 \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Ci}\left (\frac{4 c f}{d}+4 f x\right )}{16 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{16 a^3 d}\right )-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.653526, size = 336, normalized size = 0.75 \[ \frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\left (\cos \left (e-\frac{4 c f}{d}\right )-i \sin \left (e-\frac{4 c f}{d}\right )\right ) \left (-i \text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right ) \sin \left (2 e-\frac{2 c f}{d}\right )+\text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right ) \cos \left (2 e-\frac{2 c f}{d}\right )+3 \text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cos \left (2 e-\frac{2 c f}{d}\right )+i \sin \left (2 e-\frac{2 c f}{d}\right )\right )+3 \text{CosIntegral}\left (\frac{4 f (c+d x)}{d}\right )+3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )-\sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )-3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )-i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )-3 i \text{Si}\left (\frac{4 f (c+d x)}{d}\right )\right )+i \sin (3 e) \log (f (c+d x))+\cos (3 e) \log (f (c+d x))\right )}{8 d (a+i a \tan (e+f x))^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.305, size = 163, normalized size = 0.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{8\,{a}^{3}d}}-{\frac{1}{8\,{a}^{3}d}{{\rm e}^{{\frac{6\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,6\,ifx+6\,ie+{\frac{6\,i \left ( cf-de \right ) }{d}} \right ) }-{\frac{3}{8\,{a}^{3}d}{{\rm e}^{{\frac{4\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,4\,ifx+4\,ie+{\frac{4\,i \left ( cf-de \right ) }{d}} \right ) }-{\frac{3}{8\,{a}^{3}d}{{\rm e}^{{\frac{2\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,2\,ifx+2\,ie+{\frac{2\,i \left ( cf-de \right ) }{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37683, size = 365, normalized size = 0.81 \begin{align*} -\frac{f \cos \left (-\frac{6 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac{6 i \,{\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) + 3 \, f \cos \left (-\frac{4 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac{4 i \,{\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) + 3 \, f \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) + 3 i \, f E_{1}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + 3 i \, f E_{1}\left (\frac{4 i \,{\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) \sin \left (-\frac{4 \,{\left (d e - c f\right )}}{d}\right ) + i \, f E_{1}\left (\frac{6 i \,{\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) \sin \left (-\frac{6 \,{\left (d e - c f\right )}}{d}\right ) - f \log \left ({\left (f x + e\right )} d - d e + c f\right )}{8 \, a^{3} d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66196, size = 284, normalized size = 0.63 \begin{align*} \frac{3 \,{\rm Ei}\left (\frac{-2 i \, d f x - 2 i \, c f}{d}\right ) e^{\left (\frac{-2 i \, d e + 2 i \, c f}{d}\right )} + 3 \,{\rm Ei}\left (\frac{-4 i \, d f x - 4 i \, c f}{d}\right ) e^{\left (\frac{-4 i \, d e + 4 i \, c f}{d}\right )} +{\rm Ei}\left (\frac{-6 i \, d f x - 6 i \, c f}{d}\right ) e^{\left (\frac{-6 i \, d e + 6 i \, c f}{d}\right )} + \log \left (\frac{d x + c}{d}\right )}{8 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25765, size = 1142, normalized size = 2.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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