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

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

[Out]

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

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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]

Int[a + b*Sec[e + f*x]^2,x]

[Out]

a*x + (b*Tan[e + f*x])/f

Rule 8

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

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]

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*}

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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]

Integrate[a + b*Sec[e + f*x]^2,x]

[Out]

a*x + (b*Tan[e + f*x])/f

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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]

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))

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giac [A]  time = 0.87, 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]

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

[Out]

a*x + b*tan(f*x + e)/f

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maple [A]  time = 0.87, 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]

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

[Out]

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

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maxima [A]  time = 0.33, 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]

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

[Out]

a*x + b*tan(f*x + e)/f

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mupad [B]  time = 4.51, 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]

int(a + b/cos(e + f*x)^2,x)

[Out]

(b*tan(e + f*x) + a*f*x)/f

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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]

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

[Out]

Integral(a + b*sec(e + f*x)**2, x)

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