3.11 \(\int (a+b \sec ^2(e+f x)) \, dx\)

Optimal. Leaf size=15 \[ a x+\frac{b \tan (e+f x)}{f} \]

[Out]

a*x + (b*Tan[e + f*x])/f

________________________________________________________________________________________

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]

Int[a + b*Sec[e + f*x]^2,x]

[Out]

a*x + (b*Tan[e + f*x])/f

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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]

Integrate[a + b*Sec[e + f*x]^2,x]

[Out]

a*x + (b*Tan[e + f*x])/f

________________________________________________________________________________________

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]

int(a+b*sec(f*x+e)^2,x)

[Out]

a*x+b*tan(f*x+e)/f

________________________________________________________________________________________

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]

integrate(a+b*sec(f*x+e)^2,x, algorithm="maxima")

[Out]

a*x + b*tan(f*x + e)/f

________________________________________________________________________________________

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]

integrate(a+b*sec(f*x+e)^2,x, algorithm="fricas")

[Out]

(a*f*x*cos(f*x + e) + b*sin(f*x + e))/(f*cos(f*x + e))

________________________________________________________________________________________

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]

integrate(a+b*sec(f*x+e)**2,x)

[Out]

Integral(a + b*sec(e + f*x)**2, x)

________________________________________________________________________________________

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]

integrate(a+b*sec(f*x+e)^2,x, algorithm="giac")

[Out]

a*x + b*tan(f*x + e)/f