\(\int -\sec ^2(e+f x) \, dx\) [31]

Optimal result

Integrand size = 10, antiderivative size = 11 \[ \int -\sec ^2(e+f x) \, dx=-\frac {\tan (e+f x)}{f} \]

[Out]

-tan(f*x+e)/f

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3852, 8} \[ \int -\sec ^2(e+f x) \, dx=-\frac {\tan (e+f x)}{f} \]

[In]

Int[-Sec[e + f*x]^2,x]

[Out]

-(Tan[e + f*x]/f)

Rule 8

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

Rule 3852

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*} \text {integral}& = \frac {\text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f} \\ & = -\frac {\tan (e+f x)}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int -\sec ^2(e+f x) \, dx=-\frac {\tan (e+f x)}{f} \]

[In]

Integrate[-Sec[e + f*x]^2,x]

[Out]

-(Tan[e + f*x]/f)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(-\frac {\tan \left (f x +e \right )}{f}\)	\(12\)
default	\(-\frac {\tan \left (f x +e \right )}{f}\)	\(12\)
risch	\(-\frac {2 i}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\)	\(20\)
norman	\(\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}\)	\(30\)
parallelrisch	\(\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}\)	\(30\)

[In]

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

[Out]

-tan(f*x+e)/f

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int -\sec ^2(e+f x) \, dx=-\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \]

[In]

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

[Out]

-sin(f*x + e)/(f*cos(f*x + e))

Sympy [F]

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

[In]

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

[Out]

-Integral(sec(e + f*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int -\sec ^2(e+f x) \, dx=-\frac {\tan \left (f x + e\right )}{f} \]

[In]

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

[Out]

-tan(f*x + e)/f

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int -\sec ^2(e+f x) \, dx=-\frac {\tan \left (f x + e\right )}{f} \]

[In]

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

[Out]

-tan(f*x + e)/f

Mupad [B] (verification not implemented)

Time = 15.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int -\sec ^2(e+f x) \, dx=-\frac {\mathrm {tan}\left (e+f\,x\right )}{f} \]

[In]

int(-1/cos(e + f*x)^2,x)

[Out]

-tan(e + f*x)/f