Optimal. Leaf size=35 \[ \frac {A \tan (e+f x)}{a c f}+\frac {B \sec (e+f x)}{a c f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2967, 2669, 3767, 8} \[ \frac {A \tan (e+f x)}{a c f}+\frac {B \sec (e+f x)}{a c f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2669
Rule 2967
Rule 3767
Rubi steps
\begin {align*} \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))} \, dx &=\frac {\int \sec ^2(e+f x) (A+B \sin (e+f x)) \, dx}{a c}\\ &=\frac {B \sec (e+f x)}{a c f}+\frac {A \int \sec ^2(e+f x) \, dx}{a c}\\ &=\frac {B \sec (e+f x)}{a c f}-\frac {A \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a c f}\\ &=\frac {B \sec (e+f x)}{a c f}+\frac {A \tan (e+f x)}{a c f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 35, normalized size = 1.00 \[ \frac {A \tan (e+f x)}{a c f}+\frac {B \sec (e+f x)}{a c f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 28, normalized size = 0.80 \[ \frac {A \sin \left (f x + e\right ) + B}{a c f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 41, normalized size = 1.17 \[ -\frac {2 \, {\left (A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a c f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.37, size = 57, normalized size = 1.63 \[ \frac {-\frac {2 \left (\frac {A}{2}+\frac {B}{2}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {A}{2}-\frac {B}{2}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a c f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 35, normalized size = 1.00 \[ \frac {\frac {A \tan \left (f x + e\right )}{a c} + \frac {B}{a c \cos \left (f x + e\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 12.55, size = 39, normalized size = 1.11 \[ -\frac {2\,\left (B+A\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a\,c\,f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.53, size = 83, normalized size = 2.37 \[ \begin {cases} - \frac {2 A \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - a c f} - \frac {2 B}{a c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - a c f} & \text {for}\: f \neq 0 \\\frac {x \left (A + B \sin {\relax (e )}\right )}{\left (a \sin {\relax (e )} + a\right ) \left (- c \sin {\relax (e )} + c\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________