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

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

[Out]

-(Tan[e + f*x]/f)

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Rubi [A]  time = 0.0126015, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3767, 8} \[ -\frac{\tan (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[-Sec[e + f*x]^2,x]

[Out]

-(Tan[e + f*x]/f)

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]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0040374, size = 11, normalized size = 1. \[ -\frac{\tan (e+f x)}{f} \]

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 = 0.011, size = 12, normalized size = 1.1 \begin{align*} -{\frac{\tan \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-tan(f*x+e)/f

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Maxima [A]  time = 0.917863, size = 15, normalized size = 1.36 \begin{align*} -\frac{\tan \left (f x + e\right )}{f} \end{align*}

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 = 0.445583, size = 43, normalized size = 3.91 \begin{align*} -\frac{\sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \end{align*}

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

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 = 1.19418, size = 16, normalized size = 1.45 \begin{align*} -\frac{\tan \left (f x + e\right )}{f} \end{align*}

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