3.1.10 \(\int \frac {\cot ^6(x)}{a+a \csc (x)} \, dx\) [10]

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-3/8*arctanh(cos(x))/a+1/12*cot(x)^3*(4-3*csc(x))/a-1/8*cot(x)*(8-3*csc(x))/a

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Rubi [A]
time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {3973, 3966, 3855} \begin {gather*} -\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 {gather*}

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 3855

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

Rule 3966

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 3973

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]

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(49)=98\).
time = 0.05, size = 163, normalized size = 3.33 \begin {gather*} -\frac {x}{a}-\frac {2 \cot \left (\frac {x}{2}\right )}{3 a}+\frac {5 \csc ^2\left (\frac {x}{2}\right )}{32 a}+\frac {\cot \left (\frac {x}{2}\right ) \csc ^2\left (\frac {x}{2}\right )}{24 a}-\frac {\csc ^4\left (\frac {x}{2}\right )}{64 a}-\frac {3 \log \left (\cos \left (\frac {x}{2}\right )\right )}{8 a}+\frac {3 \log \left (\sin \left (\frac {x}{2}\right )\right )}{8 a}-\frac {5 \sec ^2\left (\frac {x}{2}\right )}{32 a}+\frac {\sec ^4\left (\frac {x}{2}\right )}{64 a}+\frac {2 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {\sec ^2\left (\frac {x}{2}\right ) \tan \left (\frac {x}{2}\right )}{24 a} \end {gather*}

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 [A]
time = 0.10, size = 83, normalized size = 1.69

method	result	size
default	\(\frac {\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4}-\frac {2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+10 \tan \left (\frac {x}{2}\right )-32 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {1}{4 \tan \left (\frac {x}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {x}{2}\right )^{2}}-\frac {10}{\tan \left (\frac {x}{2}\right )}+6 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{16 a}\)	\(83\)
risch	\(-\frac {x}{a}-\frac {48 i {\mathrm e}^{6 i x}+15 \,{\mathrm e}^{7 i x}-96 i {\mathrm e}^{4 i x}+9 \,{\mathrm e}^{5 i x}+80 i {\mathrm e}^{2 i x}+9 \,{\mathrm e}^{3 i x}-32 i+15 \,{\mathrm e}^{i x}}{12 a \left ({\mathrm e}^{2 i x}-1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{8 a}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/a*(1/4*tan(1/2*x)^4-2/3*tan(1/2*x)^3-2*tan(1/2*x)^2+10*tan(1/2*x)-32*arctan(tan(1/2*x))-1/4/tan(1/2*x)^4+
2/3/tan(1/2*x)^3+2/tan(1/2*x)^2-10/tan(1/2*x)+6*ln(tan(1/2*x)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (43) = 86\).
time = 0.46, size = 134, normalized size = 2.73 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (43) = 86\).
time = 3.36, size = 104, normalized size = 2.12 \begin {gather*} -\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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cot ^{6}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)**6/(csc(x) + 1), x)/a

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (43) = 86\).
time = 0.42, size = 109, normalized size = 2.22 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.33, size = 123, normalized size = 2.51 \begin {gather*} \frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a}+\frac {2\,\mathrm {atan}\left (\frac {4}{4\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {3}{2}}-\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{2\,\left (4\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {3}{2}\right )}\right )}{a}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{64\,a}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{8\,a}+\frac {-10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}-\frac {1}{4}}{16\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^6/(a + a/sin(x)),x)

[Out]

(5*tan(x/2))/(8*a) + (2*atan(4/(4*tan(x/2) + 3/2) - (3*tan(x/2))/(2*(4*tan(x/2) + 3/2))))/a - tan(x/2)^2/(8*a)
- tan(x/2)^3/(24*a) + tan(x/2)^4/(64*a) + (3*log(tan(x/2)))/(8*a) + ((2*tan(x/2))/3 + 2*tan(x/2)^2 - 10*tan(x
/2)^3 - 1/4)/(16*a*tan(x/2)^4)

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