\(\int \sec (e+f x) (1-2 \sec ^2(e+f x)) \, dx\) [30]

Optimal result

Integrand size = 19, antiderivative size = 17 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec (e+f x) \tan (e+f x)}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {4128} \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\tan (e+f x) \sec (e+f x)}{f} \]

[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 4128

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*} \text {integral}& = -\frac {\sec (e+f x) \tan (e+f x)}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec (e+f x) \tan (e+f x)}{f} \]

[In]

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

[Out]

-((Sec[e + f*x]*Tan[e + f*x])/f)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(-\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}\)	\(18\)
default	\(-\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}\)	\(18\)
parts	\(-\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}\)	\(18\)
parallelrisch	\(-\frac {2 \sin \left (f x +e \right )}{f \left (1+\cos \left (2 f x +2 e \right )\right )}\)	\(25\)
risch	\(\frac {2 i \left ({\mathrm e}^{3 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}\)	\(41\)
norman	\(\frac {-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}\)	\(48\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{2}} \]

[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)

Sympy [F]

\[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=- \int \left (- \sec {\left (e + f x \right )}\right )\, dx - \int 2 \sec ^{3}{\left (e + f x \right )}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=\frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )} f} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {1}{f {\left (\frac {1}{\sin \left (f x + e\right )} - \sin \left (f x + e\right )\right )}} \]

[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)))

Mupad [B] (verification not implemented)

Time = 15.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=\frac {\sin \left (e+f\,x\right )}{f\,\left ({\sin \left (e+f\,x\right )}^2-1\right )} \]

[In]

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

[Out]

sin(e + f*x)/(f*(sin(e + f*x)^2 - 1))