3.308 \(\int \frac{1}{(\csc (x)-\sin (x))^2} \, dx\)

Optimal. Leaf size=8 \[ \frac{\tan ^3(x)}{3} \]

[Out]

Tan[x]^3/3

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Rubi [A]  time = 0.014444, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {30} \[ \frac{\tan ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-2),x]

[Out]

Tan[x]^3/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(\csc (x)-\sin (x))^2} \, dx &=\operatorname{Subst}\left (\int x^2 \, dx,x,\tan (x)\right )\\ &=\frac{\tan ^3(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0029058, size = 8, normalized size = 1. \[ \frac{\tan ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-2),x]

[Out]

Tan[x]^3/3

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Maple [A]  time = 0.036, size = 7, normalized size = 0.9 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(x)-sin(x))^2,x)

[Out]

1/3*tan(x)^3

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Maxima [A]  time = 1.0229, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \tan \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^2,x, algorithm="maxima")

[Out]

1/3*tan(x)^3

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Fricas [B]  time = 1.97841, size = 50, normalized size = 6.25 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^2,x, algorithm="fricas")

[Out]

-1/3*(cos(x)^2 - 1)*sin(x)/cos(x)^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \sin{\left (x \right )} + \csc{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))**2,x)

[Out]

Integral((-sin(x) + csc(x))**(-2), x)

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Giac [A]  time = 1.13383, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \tan \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^2,x, algorithm="giac")

[Out]

1/3*tan(x)^3