Optimal. Leaf size=12 \[ \frac {a c \tan (e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3522, 3767, 8} \[ \frac {a c \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3522
Rule 3767
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x)) \, dx &=(a c) \int \sec ^2(e+f x) \, dx\\ &=-\frac {(a c) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f}\\ &=\frac {a c \tan (e+f x)}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \frac {a c \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.43, size = 19, normalized size = 1.58 \[ \frac {2 i \, a c}{f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 13, normalized size = 1.08 \[ \frac {a c \tan \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 13, normalized size = 1.08 \[ \frac {a c \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.15, size = 12, normalized size = 1.00 \[ \frac {a c \tan \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.69, size = 12, normalized size = 1.00 \[ \frac {a\,c\,\mathrm {tan}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.17, size = 27, normalized size = 2.25 \[ - \frac {2 i a c}{- f e^{2 i e} e^{2 i f x} - f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________