Optimal. Leaf size=25 \[ \frac{i c}{2 f (a+i a \tan (e+f x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.074293, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ \frac{i c}{2 f (a+i a \tan (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{c-i c \tan (e+f x)}{(a+i a \tan (e+f x))^2} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(a+i a \tan (e+f x))^3} \, dx\\ &=-\frac{(i c) \operatorname{Subst}\left (\int \frac{1}{(a+x)^3} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{i c}{2 f (a+i a \tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 0.795433, size = 45, normalized size = 1.8 \[ \frac{(\tan (e+f x)-3 i) (c-i c \tan (e+f x))}{8 a^2 f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.024, size = 22, normalized size = 0.9 \begin{align*}{\frac{-{\frac{i}{2}}c}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.09435, size = 92, normalized size = 3.68 \begin{align*} \frac{{\left (2 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{8 \, a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.411576, size = 102, normalized size = 4.08 \begin{align*} \begin{cases} \frac{\left (8 i a^{2} c f e^{4 i e} e^{- 2 i f x} + 4 i a^{2} c f e^{2 i e} e^{- 4 i f x}\right ) e^{- 6 i e}}{32 a^{4} f^{2}} & \text{for}\: 32 a^{4} f^{2} e^{6 i e} \neq 0 \\\frac{x \left (c e^{2 i e} + c\right ) e^{- 4 i e}}{2 a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.37751, size = 88, normalized size = 3.52 \begin{align*} -\frac{2 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - i \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{2} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]