3.301 \(\int \frac{1}{(\cot (x)+\csc (x))^4} \, dx\)

Optimal. Leaf size=26 \[ x+\frac{2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac{2 \sin (x)}{\cos (x)+1} \]

[Out]

x - (2*Sin[x])/(1 + Cos[x]) + (2*Sin[x]^3)/(3*(1 + Cos[x])^3)

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Rubi [A]  time = 0.0711771, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4392, 2680, 8} \[ x+\frac{2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac{2 \sin (x)}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x] + Csc[x])^(-4),x]

[Out]

x - (2*Sin[x])/(1 + Cos[x]) + (2*Sin[x]^3)/(3*(1 + Cos[x])^3)

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0140729, size = 30, normalized size = 1.15 \[ x-\frac{8}{3} \tan \left (\frac{x}{2}\right )+\frac{2}{3} \tan \left (\frac{x}{2}\right ) \sec ^2\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^(-4),x]

[Out]

x - (8*Tan[x/2])/3 + (2*Sec[x/2]^2*Tan[x/2])/3

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Maple [A]  time = 0.073, size = 17, normalized size = 0.7 \begin{align*}{\frac{2}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}-2\,\tan \left ( x/2 \right ) +x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)+csc(x))^4,x)

[Out]

2/3*tan(1/2*x)^3-2*tan(1/2*x)+x

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Maxima [A]  time = 1.47989, size = 47, normalized size = 1.81 \begin{align*} -\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{2 \, \sin \left (x\right )^{3}}{3 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} + 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.93125, size = 123, normalized size = 4.73 \begin{align*} \frac{3 \, x \cos \left (x\right )^{2} + 6 \, x \cos \left (x\right ) - 4 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 3 \, x}{3 \,{\left (\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^4,x, algorithm="fricas")

[Out]

1/3*(3*x*cos(x)^2 + 6*x*cos(x) - 4*(2*cos(x) + 1)*sin(x) + 3*x)/(cos(x)^2 + 2*cos(x) + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.21487, size = 22, normalized size = 0.85 \begin{align*} \frac{2}{3} \, \tan \left (\frac{1}{2} \, x\right )^{3} + x - 2 \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x))^4,x, algorithm="giac")

[Out]

2/3*tan(1/2*x)^3 + x - 2*tan(1/2*x)