Optimal. Leaf size=46 \[ \frac {a (A+B \tan (e+f x))^2}{2 c^2 f (B+i A) (1-i \tan (e+f x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {3588, 37} \[ \frac {a (A+B \tan (e+f x))^2}{2 c^2 f (B+i A) (1-i \tan (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(c-i c x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a (A+B \tan (e+f x))^2}{2 (i A+B) c^2 f (1-i \tan (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.01, size = 62, normalized size = 1.35 \[ \frac {a (\cos (3 (e+f x))+i \sin (3 (e+f x))) ((B-3 i A) \cos (e+f x)-(A+3 i B) \sin (e+f x))}{8 c^2 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 45, normalized size = 0.98 \[ \frac {{\left (-i \, A - B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-2 i \, A + 2 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.26, size = 84, normalized size = 1.83 \[ -\frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + i \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{c^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.23, size = 46, normalized size = 1.00 \[ \frac {a \left (\frac {i B}{\tan \left (f x +e \right )+i}-\frac {-i A -B}{2 \left (\tan \left (f x +e \right )+i\right )^{2}}\right )}{f \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.49, size = 51, normalized size = 1.11 \[ \frac {\frac {a\,\left (-B+A\,1{}\mathrm {i}\right )}{2}+B\,a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{c^2\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.38, size = 155, normalized size = 3.37 \[ \begin {cases} \frac {\left (- 8 i A a c^{2} f e^{2 i e} + 8 B a c^{2} f e^{2 i e}\right ) e^{2 i f x} + \left (- 4 i A a c^{2} f e^{4 i e} - 4 B a c^{2} f e^{4 i e}\right ) e^{4 i f x}}{32 c^{4} f^{2}} & \text {for}\: 32 c^{4} f^{2} \neq 0 \\\frac {x \left (A a e^{4 i e} + A a e^{2 i e} - i B a e^{4 i e} + i B a e^{2 i e}\right )}{2 c^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________