3.9 \(\int \frac{\cot ^5(x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\csc ^3(x)}{3 a}+\frac{\csc ^2(x)}{2 a}+\frac{\csc (x)}{a}+\frac{\log (\sin (x))}{a} \]

[Out]

Csc[x]/a + Csc[x]^2/(2*a) - Csc[x]^3/(3*a) + Log[Sin[x]]/a

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Rubi [A]  time = 0.0489153, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3879, 75} \[ -\frac{\csc ^3(x)}{3 a}+\frac{\csc ^2(x)}{2 a}+\frac{\csc (x)}{a}+\frac{\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^5/(a + a*Csc[x]),x]

[Out]

Csc[x]/a + Csc[x]^2/(2*a) - Csc[x]^3/(3*a) + Log[Sin[x]]/a

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int \frac{\cot ^5(x)}{a+a \csc (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^2 (a+a x)}{x^4} \, dx,x,\sin (x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}-\frac{a^3}{x^3}-\frac{a^3}{x^2}+\frac{a^3}{x}\right ) \, dx,x,\sin (x)\right )}{a^4}\\ &=\frac{\csc (x)}{a}+\frac{\csc ^2(x)}{2 a}-\frac{\csc ^3(x)}{3 a}+\frac{\log (\sin (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0281058, size = 26, normalized size = 0.72 \[ \frac{-\frac{1}{3} \csc ^3(x)+\frac{\csc ^2(x)}{2}+\csc (x)+\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^5/(a + a*Csc[x]),x]

[Out]

(Csc[x] + Csc[x]^2/2 - Csc[x]^3/3 + Log[Sin[x]])/a

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Maple [A]  time = 0.057, size = 35, normalized size = 1. \begin{align*} -{\frac{1}{3\,a \left ( \sin \left ( x \right ) \right ) ^{3}}}+{\frac{1}{2\,a \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{\ln \left ( \sin \left ( x \right ) \right ) }{a}}+{\frac{1}{a\sin \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^5/(a+a*csc(x)),x)

[Out]

-1/3/a/sin(x)^3+1/2/a/sin(x)^2+ln(sin(x))/a+1/a/sin(x)

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Maxima [A]  time = 0.966617, size = 39, normalized size = 1.08 \begin{align*} \frac{\log \left (\sin \left (x\right )\right )}{a} + \frac{6 \, \sin \left (x\right )^{2} + 3 \, \sin \left (x\right ) - 2}{6 \, a \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^5/(a+a*csc(x)),x, algorithm="maxima")

[Out]

log(sin(x))/a + 1/6*(6*sin(x)^2 + 3*sin(x) - 2)/(a*sin(x)^3)

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Fricas [A]  time = 0.503354, size = 136, normalized size = 3.78 \begin{align*} \frac{6 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) + 6 \, \cos \left (x\right )^{2} - 3 \, \sin \left (x\right ) - 4}{6 \,{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^5/(a+a*csc(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{5}{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**5/(a+a*csc(x)),x)

[Out]

Integral(cot(x)**5/(csc(x) + 1), x)/a

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Giac [A]  time = 1.41917, size = 41, normalized size = 1.14 \begin{align*} \frac{\log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} + \frac{6 \, \sin \left (x\right )^{2} + 3 \, \sin \left (x\right ) - 2}{6 \, a \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^5/(a+a*csc(x)),x, algorithm="giac")

[Out]

log(abs(sin(x)))/a + 1/6*(6*sin(x)^2 + 3*sin(x) - 2)/(a*sin(x)^3)