Optimal. Leaf size=26 \[ \frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.020528, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3486, 2637} \[ \frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3486
Rule 2637
Rubi steps
\begin{align*} \int \cos (c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{i a \cos (c+d x)}{d}+a \int \cos (c+d x) \, dx\\ &=-\frac{i a \cos (c+d x)}{d}+\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0198565, size = 51, normalized size = 1.96 \[ \frac{i a \sin (c) \sin (d x)}{d}-\frac{i a \cos (c) \cos (d x)}{d}+\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 24, normalized size = 0.9 \begin{align*}{\frac{-ia\cos \left ( dx+c \right ) +a\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10954, size = 30, normalized size = 1.15 \begin{align*} \frac{-i \, a \cos \left (d x + c\right ) + a \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.13935, size = 32, normalized size = 1.23 \begin{align*} -\frac{i \, a e^{\left (i \, d x + i \, c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.231601, size = 26, normalized size = 1. \begin{align*} \begin{cases} - \frac{i a e^{i c} e^{i d x}}{d} & \text{for}\: d \neq 0 \\a 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 = 1.15785, size = 113, normalized size = 4.35 \begin{align*} -\frac{4 i \, a e^{\left (i \, d x + i \, c\right )} + a \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + a \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - a \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - a \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]