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

Optimal. Leaf size=11 \[ -\frac {\tan (e+f x)}{f} \]

[Out]

-tan(f*x+e)/f

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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3852, 8} \begin {gather*} -\frac {\tan (e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\tan (e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-Sec[e + f*x]^2,x]

[Out]

-(Tan[e + f*x]/f)

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Maple [A]
time = 0.23, size = 12, normalized size = 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	\(\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-tan(f*x+e)/f

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Maxima [A]
time = 0.29, size = 12, normalized size = 1.09 \begin {gather*} -\frac {\tan \left (f x + e\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-tan(f*x + e)/f

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Fricas [A]
time = 2.96, size = 21, normalized size = 1.91 \begin {gather*} -\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(f*x + e)/(f*cos(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \sec ^{2}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sec(e + f*x)**2, x)

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Giac [A]
time = 0.42, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\tan \left (f x + e\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-tan(f*x + e)/f

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Mupad [B]
time = 2.42, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {tan}\left (e+f\,x\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/cos(e + f*x)^2,x)

[Out]

-tan(e + f*x)/f

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