Optimal. Leaf size=15 \[ a x+\frac{b \tan (e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01241, antiderivative size = 15, normalized size of antiderivative = 1., 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 3767
Rule 8
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 = 0.00264, size = 15, normalized size = 1. \[ a x+\frac{b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{b\tan \left ( fx+e \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.977491, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{b \tan \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.463286, size = 76, normalized size = 5.07 \begin{align*} \frac{a f x \cos \left (f x + e\right ) + b \sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27023, size = 22, normalized size = 1.47 \begin{align*} a x + \frac{b \tan \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]