3.294 \(\int (\cot (x)+\csc (x))^4 \, dx\)

Optimal. Leaf size=30 \[ x-\frac{2 \sin ^3(x)}{3 (1-\cos (x))^3}+\frac{2 \sin (x)}{1-\cos (x)} \]

[Out]

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

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Rubi [A]  time = 0.101051, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4392, 2670, 2680, 8} \[ x-\frac{2 \sin ^3(x)}{3 (1-\cos (x))^3}+\frac{2 \sin (x)}{1-\cos (x)} \]

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 2670

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a/g)^
(2*m), Int[(g*Cos[e + f*x])^(2*m + p)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 -
b^2, 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]

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 (\cot (x)+\csc (x))^4 \, dx &=\int (1+\cos (x))^4 \csc ^4(x) \, 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 = 0.0451802, size = 30, normalized size = 1. \[ x+\frac{8}{3} \cot \left (\frac{x}{2}\right )-\frac{2}{3} \cot \left (\frac{x}{2}\right ) \csc ^2\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^4,x]

[Out]

x + (8*Cot[x/2])/3 - (2*Cot[x/2]*Csc[x/2]^2)/3

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Maple [B]  time = 0.039, size = 68, normalized size = 2.3 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{3}}{3}}+\cot \left ( x \right ) +x-{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{4}}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{4}}{3\,\sin \left ( x \right ) }}+{\frac{ \left ( 8+4\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}}-2\,{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{ \left ( \sin \left ( x \right ) \right ) ^{3}}}-{\frac{4}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}}+ \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((cot(x)+csc(x))^4,x)

[Out]

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

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Maxima [B]  time = 1.47857, size = 76, normalized size = 2.53 \begin{align*} -2 \, \cot \left (x\right )^{3} + x + \frac{4 \,{\left (3 \, \sin \left (x\right )^{2} - 1\right )}}{3 \, \sin \left (x\right )^{3}} - \frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} + \frac{3 \, \tan \left (x\right )^{2} - 1}{3 \, \tan \left (x\right )^{3}} - \frac{4}{3 \, \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cot(x)^3 + x + 4/3*(3*sin(x)^2 - 1)/sin(x)^3 - 1/3*(3*tan(x)^2 + 1)/tan(x)^3 + 1/3*(3*tan(x)^2 - 1)/tan(x)^
3 - 4/3/sin(x)^3

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Fricas [A]  time = 2.09382, size = 109, normalized size = 3.63 \begin{align*} \frac{8 \, \cos \left (x\right )^{2} + 3 \,{\left (x \cos \left (x\right ) - x\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) - 4}{3 \,{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(8*cos(x)^2 + 3*(x*cos(x) - x)*sin(x) + 4*cos(x) - 4)/((cos(x) - 1)*sin(x))

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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((cot(x)+csc(x))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.13628, size = 27, normalized size = 0.9 \begin{align*} x + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 1\right )}}{3 \, \tan \left (\frac{1}{2} \, x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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