### 3.105 $$\int \csc ^4(x) \, dx$$

Optimal. Leaf size=13 $-\frac{1}{3} \cot ^3(x)-\cot (x)$

[Out]

-Cot[x] - Cot[x]^3/3

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Rubi [A]  time = 0.0065964, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3767} $-\frac{1}{3} \cot ^3(x)-\cot (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Csc[x]^4,x]

[Out]

-Cot[x] - Cot[x]^3/3

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]

Rubi steps

\begin{align*} \int \csc ^4(x) \, dx &=-\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )\\ &=-\cot (x)-\frac{\cot ^3(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0023914, size = 17, normalized size = 1.31 $-\frac{2 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csc[x]^4,x]

[Out]

(-2*Cot[x])/3 - (Cot[x]*Csc[x]^2)/3

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Maple [A]  time = 0.026, size = 12, normalized size = 0.9 \begin{align*} \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( x \right ) \right ) ^{2}}{3}} \right ) \cot \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x)^4,x)

[Out]

(-2/3-1/3*csc(x)^2)*cot(x)

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Maxima [A]  time = 0.929989, size = 19, normalized size = 1.46 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^4,x, algorithm="maxima")

[Out]

-1/3*(3*tan(x)^2 + 1)/tan(x)^3

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Fricas [B]  time = 1.96192, size = 73, normalized size = 5.62 \begin{align*} -\frac{2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^4,x, algorithm="fricas")

[Out]

-1/3*(2*cos(x)^3 - 3*cos(x))/((cos(x)^2 - 1)*sin(x))

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Sympy [A]  time = 0.057034, size = 20, normalized size = 1.54 \begin{align*} - \frac{2 \cos{\left (x \right )}}{3 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)**4,x)

[Out]

-2*cos(x)/(3*sin(x)) - cos(x)/(3*sin(x)**3)

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Giac [A]  time = 1.05834, size = 19, normalized size = 1.46 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^4,x, algorithm="giac")

[Out]

-1/3*(3*tan(x)^2 + 1)/tan(x)^3