3.30 \(\int \sec (e+f x) (1-2 \sec ^2(e+f x)) \, dx\)

Optimal. Leaf size=17 \[ -\frac{\tan (e+f x) \sec (e+f x)}{f} \]

[Out]

-((Sec[e + f*x]*Tan[e + f*x])/f)

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Rubi [A]  time = 0.015395, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {4043} \[ -\frac{\tan (e+f x) \sec (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]*(1 - 2*Sec[e + f*x]^2),x]

[Out]

-((Sec[e + f*x]*Tan[e + f*x])/f)

Rule 4043

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[C*m + A*(m + 1), 0]

Rubi steps

\begin{align*} \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx &=-\frac{\sec (e+f x) \tan (e+f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.0127411, size = 17, normalized size = 1. \[ -\frac{\tan (e+f x) \sec (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]*(1 - 2*Sec[e + f*x]^2),x]

[Out]

-((Sec[e + f*x]*Tan[e + f*x])/f)

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Maple [A]  time = 0.025, size = 18, normalized size = 1.1 \begin{align*} -{\frac{\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(1-2*sec(f*x+e)^2),x)

[Out]

-sec(f*x+e)*tan(f*x+e)/f

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Maxima [A]  time = 0.92257, size = 30, normalized size = 1.76 \begin{align*} \frac{\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(1-2*sec(f*x+e)^2),x, algorithm="maxima")

[Out]

sin(f*x + e)/((sin(f*x + e)^2 - 1)*f)

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Fricas [A]  time = 0.445573, size = 46, normalized size = 2.71 \begin{align*} -\frac{\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(1-2*sec(f*x+e)^2),x, algorithm="fricas")

[Out]

-sin(f*x + e)/(f*cos(f*x + e)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \sec{\left (e + f x \right )}\, dx - \int 2 \sec ^{3}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(1-2*sec(f*x+e)**2),x)

[Out]

-Integral(-sec(e + f*x), x) - Integral(2*sec(e + f*x)**3, x)

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Giac [A]  time = 1.1536, size = 35, normalized size = 2.06 \begin{align*} -\frac{1}{f{\left (\frac{1}{\sin \left (f x + e\right )} - \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(1-2*sec(f*x+e)^2),x, algorithm="giac")

[Out]

-1/(f*(1/sin(f*x + e) - sin(f*x + e)))