3.10 \(\int \frac{\cot ^6(x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=49 \[ -\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) (4-3 \csc (x))}{12 a}-\frac{\cot (x) (8-3 \csc (x))}{8 a} \]

[Out]

-(x/a) - (3*ArcTanh[Cos[x]])/(8*a) + (Cot[x]^3*(4 - 3*Csc[x]))/(12*a) - (Cot[x]*(8 - 3*Csc[x]))/(8*a)

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Rubi [A]  time = 0.0807971, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3881, 3770} \[ -\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) (4-3 \csc (x))}{12 a}-\frac{\cot (x) (8-3 \csc (x))}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^6/(a + a*Csc[x]),x]

[Out]

-(x/a) - (3*ArcTanh[Cos[x]])/(8*a) + (Cot[x]^3*(4 - 3*Csc[x]))/(12*a) - (Cot[x]*(8 - 3*Csc[x]))/(8*a)

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rule 3881

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(e*(e*Cot[
c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x]))/(d*m*(m - 1)), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m - 2)
*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cot ^6(x)}{a+a \csc (x)} \, dx &=\frac{\int \cot ^4(x) (-a+a \csc (x)) \, dx}{a^2}\\ &=\frac{\cot ^3(x) (4-3 \csc (x))}{12 a}-\frac{\int \cot ^2(x) (-4 a+3 a \csc (x)) \, dx}{4 a^2}\\ &=\frac{\cot ^3(x) (4-3 \csc (x))}{12 a}-\frac{\cot (x) (8-3 \csc (x))}{8 a}+\frac{\int (-8 a+3 a \csc (x)) \, dx}{8 a^2}\\ &=-\frac{x}{a}+\frac{\cot ^3(x) (4-3 \csc (x))}{12 a}-\frac{\cot (x) (8-3 \csc (x))}{8 a}+\frac{3 \int \csc (x) \, dx}{8 a}\\ &=-\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) (4-3 \csc (x))}{12 a}-\frac{\cot (x) (8-3 \csc (x))}{8 a}\\ \end{align*}

Mathematica [B]  time = 0.0525165, size = 163, normalized size = 3.33 \[ -\frac{x}{a}+\frac{2 \tan \left (\frac{x}{2}\right )}{3 a}-\frac{2 \cot \left (\frac{x}{2}\right )}{3 a}-\frac{\csc ^4\left (\frac{x}{2}\right )}{64 a}+\frac{5 \csc ^2\left (\frac{x}{2}\right )}{32 a}+\frac{\sec ^4\left (\frac{x}{2}\right )}{64 a}-\frac{5 \sec ^2\left (\frac{x}{2}\right )}{32 a}+\frac{3 \log \left (\sin \left (\frac{x}{2}\right )\right )}{8 a}-\frac{3 \log \left (\cos \left (\frac{x}{2}\right )\right )}{8 a}+\frac{\cot \left (\frac{x}{2}\right ) \csc ^2\left (\frac{x}{2}\right )}{24 a}-\frac{\tan \left (\frac{x}{2}\right ) \sec ^2\left (\frac{x}{2}\right )}{24 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^6/(a + a*Csc[x]),x]

[Out]

-(x/a) - (2*Cot[x/2])/(3*a) + (5*Csc[x/2]^2)/(32*a) + (Cot[x/2]*Csc[x/2]^2)/(24*a) - Csc[x/2]^4/(64*a) - (3*Lo
g[Cos[x/2]])/(8*a) + (3*Log[Sin[x/2]])/(8*a) - (5*Sec[x/2]^2)/(32*a) + Sec[x/2]^4/(64*a) + (2*Tan[x/2])/(3*a)
- (Sec[x/2]^2*Tan[x/2])/(24*a)

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Maple [B]  time = 0.069, size = 108, normalized size = 2.2 \begin{align*}{\frac{1}{64\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{4}}-{\frac{1}{24\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{5}{8\,a}\tan \left ({\frac{x}{2}} \right ) }-2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}-{\frac{1}{64\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-4}}+{\frac{1}{24\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{5}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{3}{8\,a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^6/(a+a*csc(x)),x)

[Out]

1/64/a*tan(1/2*x)^4-1/24/a*tan(1/2*x)^3-1/8/a*tan(1/2*x)^2+5/8/a*tan(1/2*x)-2/a*arctan(tan(1/2*x))-1/64/a/tan(
1/2*x)^4+1/24/a/tan(1/2*x)^3+1/8/a/tan(1/2*x)^2-5/8/a/tan(1/2*x)+3/8/a*ln(tan(1/2*x))

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Maxima [B]  time = 1.46734, size = 181, normalized size = 3.69 \begin{align*} \frac{\frac{120 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{24 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{8 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}}{192 \, a} - \frac{2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} + \frac{3 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{8 \, a} + \frac{{\left (\frac{8 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{24 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{120 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 3\right )}{\left (\cos \left (x\right ) + 1\right )}^{4}}{192 \, a \sin \left (x\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^6/(a+a*csc(x)),x, algorithm="maxima")

[Out]

1/192*(120*sin(x)/(cos(x) + 1) - 24*sin(x)^2/(cos(x) + 1)^2 - 8*sin(x)^3/(cos(x) + 1)^3 + 3*sin(x)^4/(cos(x) +
1)^4)/a - 2*arctan(sin(x)/(cos(x) + 1))/a + 3/8*log(sin(x)/(cos(x) + 1))/a + 1/192*(8*sin(x)/(cos(x) + 1) + 2
4*sin(x)^2/(cos(x) + 1)^2 - 120*sin(x)^3/(cos(x) + 1)^3 - 3)*(cos(x) + 1)^4/(a*sin(x)^4)

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Fricas [B]  time = 0.506739, size = 336, normalized size = 6.86 \begin{align*} -\frac{48 \, x \cos \left (x\right )^{4} - 96 \, x \cos \left (x\right )^{2} + 30 \, \cos \left (x\right )^{3} + 9 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 9 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 16 \,{\left (4 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 48 \, x - 18 \, \cos \left (x\right )}{48 \,{\left (a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^6/(a+a*csc(x)),x, algorithm="fricas")

[Out]

-1/48*(48*x*cos(x)^4 - 96*x*cos(x)^2 + 30*cos(x)^3 + 9*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) - 9*(
cos(x)^4 - 2*cos(x)^2 + 1)*log(-1/2*cos(x) + 1/2) - 16*(4*cos(x)^3 - 3*cos(x))*sin(x) + 48*x - 18*cos(x))/(a*c
os(x)^4 - 2*a*cos(x)^2 + a)

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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)**6/(a+a*csc(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.39928, size = 147, normalized size = 3. \begin{align*} -\frac{x}{a} + \frac{3 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{8 \, a} + \frac{3 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{4} - 8 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{3} - 24 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{2} + 120 \, a^{3} \tan \left (\frac{1}{2} \, x\right )}{192 \, a^{4}} - \frac{150 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 120 \, \tan \left (\frac{1}{2} \, x\right )^{3} - 24 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 8 \, \tan \left (\frac{1}{2} \, x\right ) + 3}{192 \, a \tan \left (\frac{1}{2} \, x\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^6/(a+a*csc(x)),x, algorithm="giac")

[Out]

-x/a + 3/8*log(abs(tan(1/2*x)))/a + 1/192*(3*a^3*tan(1/2*x)^4 - 8*a^3*tan(1/2*x)^3 - 24*a^3*tan(1/2*x)^2 + 120
*a^3*tan(1/2*x))/a^4 - 1/192*(150*tan(1/2*x)^4 + 120*tan(1/2*x)^3 - 24*tan(1/2*x)^2 - 8*tan(1/2*x) + 3)/(a*tan
(1/2*x)^4)