3.11 \(\int \frac{\cot ^7(x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=58 \[ -\frac{\csc ^5(x)}{5 a}+\frac{\csc ^4(x)}{4 a}+\frac{2 \csc ^3(x)}{3 a}-\frac{\csc ^2(x)}{a}-\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]

[Out]

-(Csc[x]/a) - Csc[x]^2/a + (2*Csc[x]^3)/(3*a) + Csc[x]^4/(4*a) - Csc[x]^5/(5*a) - Log[Sin[x]]/a

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Rubi [A]  time = 0.0581809, antiderivative size = 58, 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, 88} \[ -\frac{\csc ^5(x)}{5 a}+\frac{\csc ^4(x)}{4 a}+\frac{2 \csc ^3(x)}{3 a}-\frac{\csc ^2(x)}{a}-\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^7/(a + a*Csc[x]),x]

[Out]

-(Csc[x]/a) - Csc[x]^2/a + (2*Csc[x]^3)/(3*a) + Csc[x]^4/(4*a) - Csc[x]^5/(5*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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\cot ^7(x)}{a+a \csc (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^3 (a+a x)^2}{x^6} \, dx,x,\sin (x)\right )}{a^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}-\frac{a^5}{x^5}-\frac{2 a^5}{x^4}+\frac{2 a^5}{x^3}+\frac{a^5}{x^2}-\frac{a^5}{x}\right ) \, dx,x,\sin (x)\right )}{a^6}\\ &=-\frac{\csc (x)}{a}-\frac{\csc ^2(x)}{a}+\frac{2 \csc ^3(x)}{3 a}+\frac{\csc ^4(x)}{4 a}-\frac{\csc ^5(x)}{5 a}-\frac{\log (\sin (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.049801, size = 39, normalized size = 0.67 \[ -\frac{\frac{\csc ^5(x)}{5}-\frac{\csc ^4(x)}{4}-\frac{2 \csc ^3(x)}{3}+\csc ^2(x)+\csc (x)+\log (\sin (x))}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^7/(a + a*Csc[x]),x]

[Out]

-((Csc[x] + Csc[x]^2 - (2*Csc[x]^3)/3 - Csc[x]^4/4 + Csc[x]^5/5 + Log[Sin[x]])/a)

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Maple [A]  time = 0.073, size = 55, normalized size = 1. \begin{align*}{\frac{2}{3\,a \left ( \sin \left ( x \right ) \right ) ^{3}}}-{\frac{1}{a \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{1}{4\,a \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{\ln \left ( \sin \left ( x \right ) \right ) }{a}}-{\frac{1}{a\sin \left ( x \right ) }}-{\frac{1}{5\,a \left ( \sin \left ( x \right ) \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^7/(a+a*csc(x)),x)

[Out]

2/3/a/sin(x)^3-1/a/sin(x)^2+1/4/a/sin(x)^4-ln(sin(x))/a-1/a/sin(x)-1/5/a/sin(x)^5

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Maxima [A]  time = 0.960434, size = 57, normalized size = 0.98 \begin{align*} -\frac{\log \left (\sin \left (x\right )\right )}{a} - \frac{60 \, \sin \left (x\right )^{4} + 60 \, \sin \left (x\right )^{3} - 40 \, \sin \left (x\right )^{2} - 15 \, \sin \left (x\right ) + 12}{60 \, a \sin \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^7/(a+a*csc(x)),x, algorithm="maxima")

[Out]

-log(sin(x))/a - 1/60*(60*sin(x)^4 + 60*sin(x)^3 - 40*sin(x)^2 - 15*sin(x) + 12)/(a*sin(x)^5)

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Fricas [A]  time = 0.508639, size = 224, normalized size = 3.86 \begin{align*} -\frac{60 \, \cos \left (x\right )^{4} + 60 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) - 80 \, \cos \left (x\right )^{2} - 15 \,{\left (4 \, \cos \left (x\right )^{2} - 3\right )} \sin \left (x\right ) + 32}{60 \,{\left (a \cos \left (x\right )^{4} - 2 \, 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)^7/(a+a*csc(x)),x, algorithm="fricas")

[Out]

-1/60*(60*cos(x)^4 + 60*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*sin(x))*sin(x) - 80*cos(x)^2 - 15*(4*cos(x)^2 - 3)
*sin(x) + 32)/((a*cos(x)^4 - 2*a*cos(x)^2 + a)*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)**7/(a+a*csc(x)),x)

[Out]

Timed out

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Giac [A]  time = 1.38837, size = 58, normalized size = 1. \begin{align*} -\frac{\log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} - \frac{60 \, \sin \left (x\right )^{4} + 60 \, \sin \left (x\right )^{3} - 40 \, \sin \left (x\right )^{2} - 15 \, \sin \left (x\right ) + 12}{60 \, a \sin \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^7/(a+a*csc(x)),x, algorithm="giac")

[Out]

-log(abs(sin(x)))/a - 1/60*(60*sin(x)^4 + 60*sin(x)^3 - 40*sin(x)^2 - 15*sin(x) + 12)/(a*sin(x)^5)