3.9 \(\int -\csc ^2(e+f x) \, dx\)

Optimal. Leaf size=10 \[ \frac{\cot (e+f x)}{f} \]

[Out]

Cot[e + f*x]/f

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Rubi [A]  time = 0.0088738, antiderivative size = 10, 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} \[ \frac{\cot (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[-Csc[e + f*x]^2,x]

[Out]

Cot[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 -\csc ^2(e+f x) \, dx &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (e+f x))}{f}\\ &=\frac{\cot (e+f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.0161906, size = 10, normalized size = 1. \[ \frac{\cot (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[-Csc[e + f*x]^2,x]

[Out]

Cot[e + f*x]/f

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Maple [A]  time = 0.018, size = 11, normalized size = 1.1 \begin{align*}{\frac{\cot \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cot(f*x+e)/f

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Maxima [A]  time = 0.962417, size = 16, normalized size = 1.6 \begin{align*} \frac{1}{f \tan \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(f*tan(f*x + e))

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Fricas [A]  time = 1.48445, size = 42, normalized size = 4.2 \begin{align*} \frac{\cos \left (f x + e\right )}{f \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(f*x + e)/(f*sin(f*x + e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(csc(e + f*x)**2, x)

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Giac [A]  time = 1.12689, size = 18, normalized size = 1.8 \begin{align*} \frac{1}{f \tan \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(f*tan(f*x + e))