Optimal. Leaf size=15 \[ a x+\frac {b \tan (e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3767, 8} \[ a x+\frac {b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \left (a+b \sec ^2(e+f x)\right ) \, dx &=a x+b \int \sec ^2(e+f x) \, dx\\ &=a x-\frac {b \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f}\\ &=a x+\frac {b \tan (e+f x)}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ a x+\frac {b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.48, size = 31, normalized size = 2.07 \[ \frac {a f x \cos \left (f x + e\right ) + b \sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 16, normalized size = 1.07 \[ a x + \frac {b \tan \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.75, size = 16, normalized size = 1.07 \[ a x +\frac {b \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 15, normalized size = 1.00 \[ a x + \frac {b \tan \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.42, size = 17, normalized size = 1.13 \[ \frac {b\,\mathrm {tan}\left (e+f\,x\right )+a\,f\,x}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________