Optimal. Leaf size=25 \[ -\frac{i a}{4 f (c-i c \tan (e+f x))^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0725323, 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} \[ -\frac{i a}{4 f (c-i c \tan (e+f x))^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^4} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(c-i c \tan (e+f x))^5} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac{i a}{4 f (c-i c \tan (e+f x))^4}\\ \end{align*}
Mathematica [B] time = 0.55047, size = 74, normalized size = 2.96 \[ \frac{a (-i (2 \sin (e+f x)+3 \sin (3 (e+f x)))+10 \cos (e+f x)+5 \cos (3 (e+f x))) (\sin (5 (e+f x))-i \cos (5 (e+f x)))}{64 c^4 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 22, normalized size = 0.9 \begin{align*}{\frac{-{\frac{i}{4}}a}{f{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}} \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 [B] time = 1.32627, size = 169, normalized size = 6.76 \begin{align*} \frac{-i \, a e^{\left (8 i \, f x + 8 i \, e\right )} - 4 i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 6 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 4 i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{64 \, c^{4} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.13309, size = 168, normalized size = 6.72 \begin{align*} \begin{cases} \frac{- 8192 i a c^{12} f^{3} e^{8 i e} e^{8 i f x} - 32768 i a c^{12} f^{3} e^{6 i e} e^{6 i f x} - 49152 i a c^{12} f^{3} e^{4 i e} e^{4 i f x} - 32768 i a c^{12} f^{3} e^{2 i e} e^{2 i f x}}{524288 c^{16} f^{4}} & \text{for}\: 524288 c^{16} f^{4} \neq 0 \\\frac{x \left (a e^{8 i e} + 3 a e^{6 i e} + 3 a e^{4 i e} + a e^{2 i e}\right )}{8 c^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26034, size = 169, normalized size = 6.76 \begin{align*} -\frac{2 \,{\left (a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 3 i \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 7 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 8 i \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 7 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 i \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{c^{4} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]