Optimal. Leaf size=61 \[ -\frac{3 i a^3 \sec (c+d x)}{d}-\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^2}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0501246, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3496, 3486, 3770} \[ -\frac{3 i a^3 \sec (c+d x)}{d}-\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^2}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3496
Rule 3486
Rule 3770
Rubi steps
\begin{align*} \int \cos (c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^2}{d}-\left (3 a^2\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac{3 i a^3 \sec (c+d x)}{d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^2}{d}-\left (3 a^3\right ) \int \sec (c+d x) \, dx\\ &=-\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 i a^3 \sec (c+d x)}{d}-\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^2}{d}\\ \end{align*}
Mathematica [B] time = 0.649382, size = 123, normalized size = 2.02 \[ \frac{a^3 \cos ^2(c+d x) (\tan (c+d x)-i)^3 \left ((-\cos (2 c-d x)+i \sin (2 c-d x)) (5 \cos (c+d x)-i \sin (c+d x))+6 (\sin (3 c)+i \cos (3 c)) \cos (c+d x) \tanh ^{-1}\left (\cos (c) \tan \left (\frac{d x}{2}\right )+\sin (c)\right )\right )}{d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 101, normalized size = 1.7 \begin{align*}{\frac{-i{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}-{\frac{i{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }{d}}-{\frac{5\,i{a}^{3}\cos \left ( dx+c \right ) }{d}}-3\,{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.22594, size = 111, normalized size = 1.82 \begin{align*} -\frac{2 i \, a^{3}{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 3 \, a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 6 i \, a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.19292, size = 281, normalized size = 4.61 \begin{align*} \frac{-4 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 6 i \, a^{3} e^{\left (i \, d x + i \, c\right )} - 3 \,{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 3 \,{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.810117, size = 107, normalized size = 1.75 \begin{align*} \frac{3 a^{3} \left (\log{\left (e^{i d x} - i e^{- i c} \right )} - \log{\left (e^{i d x} + i e^{- i c} \right )}\right )}{d} - \frac{2 i a^{3} e^{- i c} e^{i d x}}{d \left (e^{2 i d x} + e^{- 2 i c}\right )} + \begin{cases} - \frac{4 i a^{3} e^{i c} e^{i d x}}{d} & \text{for}\: d \neq 0 \\4 a^{3} x e^{i c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.31265, size = 316, normalized size = 5.18 \begin{align*} \frac{63 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 33 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 63 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 33 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 128 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 192 i \, a^{3} e^{\left (i \, d x + i \, c\right )} + 63 \, a^{3} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 33 \, a^{3} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 63 \, a^{3} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 33 \, a^{3} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right )}{32 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]