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

Optimal. Leaf size=186 \[ \frac{a \left (a^2-3 b^2\right ) \cot (x)}{b^4}-\frac{\left (a^2-3 b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}-\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) \tanh ^{-1}(\cos (x))}{b^5}+\frac{2 \left (a^2-b^2\right )^{5/2} \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{a b^5}-\frac{\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 b^3}+\frac{a \cot ^3(x)}{3 b^2}+\frac{a \cot (x)}{b^2}-\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 b}-\frac{\cot (x) \csc ^3(x)}{4 b}-\frac{3 \cot (x) \csc (x)}{8 b} \]

[Out]

-(x/a) - (3*ArcTanh[Cos[x]])/(8*b) - ((a^2 - 3*b^2)*ArcTanh[Cos[x]])/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcT
anh[Cos[x]])/b^5 + (2*(a^2 - b^2)^(5/2)*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b^5) + (a*Cot[x])/b^2 +
(a*(a^2 - 3*b^2)*Cot[x])/b^4 + (a*Cot[x]^3)/(3*b^2) - (3*Cot[x]*Csc[x])/(8*b) - ((a^2 - 3*b^2)*Cot[x]*Csc[x])/
(2*b^3) - (Cot[x]*Csc[x]^3)/(4*b)

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Rubi [A]  time = 0.277752, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {3898, 2897, 3770, 3767, 8, 3768, 2660, 618, 206} \[ \frac{a \left (a^2-3 b^2\right ) \cot (x)}{b^4}-\frac{\left (a^2-3 b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}-\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) \tanh ^{-1}(\cos (x))}{b^5}+\frac{2 \left (a^2-b^2\right )^{5/2} \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{a b^5}-\frac{\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 b^3}+\frac{a \cot ^3(x)}{3 b^2}+\frac{a \cot (x)}{b^2}-\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 b}-\frac{\cot (x) \csc ^3(x)}{4 b}-\frac{3 \cot (x) \csc (x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^6/(a + b*Csc[x]),x]

[Out]

-(x/a) - (3*ArcTanh[Cos[x]])/(8*b) - ((a^2 - 3*b^2)*ArcTanh[Cos[x]])/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcT
anh[Cos[x]])/b^5 + (2*(a^2 - b^2)^(5/2)*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b^5) + (a*Cot[x])/b^2 +
(a*(a^2 - 3*b^2)*Cot[x])/b^4 + (a*Cot[x]^3)/(3*b^2) - (3*Cot[x]*Csc[x])/(8*b) - ((a^2 - 3*b^2)*Cot[x]*Csc[x])/
(2*b^3) - (Cot[x]*Csc[x]^3)/(4*b)

Rule 3898

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

Rule 2897

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x]
/; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && G
tQ[p, 0]))

Rule 3770

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cot ^6(x)}{a+b \csc (x)} \, dx &=\int \frac{\cos (x) \cot ^5(x)}{b+a \sin (x)} \, dx\\ &=\int \left (-\frac{1}{a}+\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) \csc (x)}{b^5}+\frac{\left (-a^3+3 a b^2\right ) \csc ^2(x)}{b^4}+\frac{\left (a^2-3 b^2\right ) \csc ^3(x)}{b^3}-\frac{a \csc ^4(x)}{b^2}+\frac{\csc ^5(x)}{b}-\frac{\left (a^2-b^2\right )^3}{a b^5 (b+a \sin (x))}\right ) \, dx\\ &=-\frac{x}{a}-\frac{a \int \csc ^4(x) \, dx}{b^2}+\frac{\int \csc ^5(x) \, dx}{b}-\frac{\left (a \left (a^2-3 b^2\right )\right ) \int \csc ^2(x) \, dx}{b^4}+\frac{\left (a^2-3 b^2\right ) \int \csc ^3(x) \, dx}{b^3}-\frac{\left (a^2-b^2\right )^3 \int \frac{1}{b+a \sin (x)} \, dx}{a b^5}+\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) \int \csc (x) \, dx}{b^5}\\ &=-\frac{x}{a}-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) \tanh ^{-1}(\cos (x))}{b^5}-\frac{\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 b^3}-\frac{\cot (x) \csc ^3(x)}{4 b}+\frac{a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{b^2}+\frac{3 \int \csc ^3(x) \, dx}{4 b}+\frac{\left (a \left (a^2-3 b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{b^4}+\frac{\left (a^2-3 b^2\right ) \int \csc (x) \, dx}{2 b^3}-\frac{\left (2 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{b+2 a x+b x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{a b^5}\\ &=-\frac{x}{a}-\frac{\left (a^2-3 b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) \tanh ^{-1}(\cos (x))}{b^5}+\frac{a \cot (x)}{b^2}+\frac{a \left (a^2-3 b^2\right ) \cot (x)}{b^4}+\frac{a \cot ^3(x)}{3 b^2}-\frac{3 \cot (x) \csc (x)}{8 b}-\frac{\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 b^3}-\frac{\cot (x) \csc ^3(x)}{4 b}+\frac{3 \int \csc (x) \, dx}{8 b}+\frac{\left (4 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2-b^2\right )-x^2} \, dx,x,2 a+2 b \tan \left (\frac{x}{2}\right )\right )}{a b^5}\\ &=-\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 b}-\frac{\left (a^2-3 b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) \tanh ^{-1}(\cos (x))}{b^5}+\frac{2 \left (a^2-b^2\right )^{5/2} \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{a b^5}+\frac{a \cot (x)}{b^2}+\frac{a \left (a^2-3 b^2\right ) \cot (x)}{b^4}+\frac{a \cot ^3(x)}{3 b^2}-\frac{3 \cot (x) \csc (x)}{8 b}-\frac{\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 b^3}-\frac{\cot (x) \csc ^3(x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.880693, size = 312, normalized size = 1.68 \[ \frac{384 \left (b^2-a^2\right )^{5/2} \tan ^{-1}\left (\frac{a+b \tan \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )+224 a^2 b^3 \tan \left (\frac{x}{2}\right )+32 a^2 b \left (3 a^2-7 b^2\right ) \cot \left (\frac{x}{2}\right )-24 a^3 b^2 \csc ^2\left (\frac{x}{2}\right )+24 a^3 b^2 \sec ^2\left (\frac{x}{2}\right )-480 a^3 b^2 \log \left (\sin \left (\frac{x}{2}\right )\right )+480 a^3 b^2 \log \left (\cos \left (\frac{x}{2}\right )\right )-64 a^2 b^3 \sin ^4\left (\frac{x}{2}\right ) \csc ^3(x)+4 a^2 b^3 \sin (x) \csc ^4\left (\frac{x}{2}\right )-96 a^4 b \tan \left (\frac{x}{2}\right )+192 a^5 \log \left (\sin \left (\frac{x}{2}\right )\right )-192 a^5 \log \left (\cos \left (\frac{x}{2}\right )\right )-3 a b^4 \csc ^4\left (\frac{x}{2}\right )+54 a b^4 \csc ^2\left (\frac{x}{2}\right )+3 a b^4 \sec ^4\left (\frac{x}{2}\right )-54 a b^4 \sec ^2\left (\frac{x}{2}\right )+360 a b^4 \log \left (\sin \left (\frac{x}{2}\right )\right )-360 a b^4 \log \left (\cos \left (\frac{x}{2}\right )\right )-192 b^5 x}{192 a b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^6/(a + b*Csc[x]),x]

[Out]

(-192*b^5*x + 384*(-a^2 + b^2)^(5/2)*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]] + 32*a^2*b*(3*a^2 - 7*b^2)*Cot[
x/2] - 24*a^3*b^2*Csc[x/2]^2 + 54*a*b^4*Csc[x/2]^2 - 3*a*b^4*Csc[x/2]^4 - 192*a^5*Log[Cos[x/2]] + 480*a^3*b^2*
Log[Cos[x/2]] - 360*a*b^4*Log[Cos[x/2]] + 192*a^5*Log[Sin[x/2]] - 480*a^3*b^2*Log[Sin[x/2]] + 360*a*b^4*Log[Si
n[x/2]] + 24*a^3*b^2*Sec[x/2]^2 - 54*a*b^4*Sec[x/2]^2 + 3*a*b^4*Sec[x/2]^4 - 64*a^2*b^3*Csc[x]^3*Sin[x/2]^4 +
4*a^2*b^3*Csc[x/2]^4*Sin[x] - 96*a^4*b*Tan[x/2] + 224*a^2*b^3*Tan[x/2])/(192*a*b^5)

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Maple [B]  time = 0.075, size = 363, normalized size = 2. \begin{align*}{\frac{1}{64\,b} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{4}}-{\frac{a}{24\,{b}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{{a}^{2}}{8\,{b}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{4\,b} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{{a}^{3}}{2\,{b}^{4}}\tan \left ({\frac{x}{2}} \right ) }+{\frac{9\,a}{8\,{b}^{2}}\tan \left ({\frac{x}{2}} \right ) }-2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}-{\frac{1}{64\,b} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-4}}-{\frac{{a}^{2}}{8\,{b}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{4\,b} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{4}}{{b}^{5}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{5\,{a}^{2}}{2\,{b}^{3}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }+{\frac{15}{8\,b}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }+{\frac{a}{24\,{b}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{{a}^{3}}{2\,{b}^{4}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{9\,a}{8\,{b}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-2\,{\frac{{a}^{5}}{{b}^{5}\sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,b\tan \left ( x/2 \right ) +2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) }+6\,{\frac{{a}^{3}}{{b}^{3}\sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,b\tan \left ( x/2 \right ) +2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) }-6\,{\frac{a}{b\sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,b\tan \left ( x/2 \right ) +2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) }+2\,{\frac{b}{a\sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,b\tan \left ( x/2 \right ) +2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64/b*tan(1/2*x)^4-1/24/b^2*tan(1/2*x)^3*a+1/8/b^3*tan(1/2*x)^2*a^2-1/4/b*tan(1/2*x)^2-1/2/b^4*a^3*tan(1/2*x)
+9/8/b^2*a*tan(1/2*x)-2/a*arctan(tan(1/2*x))-1/64/b/tan(1/2*x)^4-1/8/b^3/tan(1/2*x)^2*a^2+1/4/b/tan(1/2*x)^2+1
/b^5*ln(tan(1/2*x))*a^4-5/2/b^3*ln(tan(1/2*x))*a^2+15/8/b*ln(tan(1/2*x))+1/24*a/b^2/tan(1/2*x)^3+1/2*a^3/b^4/t
an(1/2*x)-9/8*a/b^2/tan(1/2*x)-2*a^5/b^5/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*x)+2*a)/(-a^2+b^2)^(1/2))+6*
a^3/b^3/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*x)+2*a)/(-a^2+b^2)^(1/2))-6*a/b/(-a^2+b^2)^(1/2)*arctan(1/2*(
2*b*tan(1/2*x)+2*a)/(-a^2+b^2)^(1/2))+2*b/a/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*x)+2*a)/(-a^2+b^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.70655, size = 2061, normalized size = 11.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(48*b^5*x*cos(x)^4 - 96*b^5*x*cos(x)^2 + 48*b^5*x - 6*(4*a^3*b^2 - 9*a*b^4)*cos(x)^3 - 24*((a^4 - 2*a^2
*b^2 + b^4)*cos(x)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(x)^2)*sqrt(a^2 - b^2)*log(((a^2 -
2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 -
2*a*b*sin(x) - a^2 - b^2)) + 6*(4*a^3*b^2 - 7*a*b^4)*cos(x) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a
^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) - 3*(8*a^5 - 2
0*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*
log(-1/2*cos(x) + 1/2) + 16*((3*a^4*b - 7*a^2*b^3)*cos(x)^3 - 3*(a^4*b - 2*a^2*b^3)*cos(x))*sin(x))/(a*b^5*cos
(x)^4 - 2*a*b^5*cos(x)^2 + a*b^5), -1/48*(48*b^5*x*cos(x)^4 - 96*b^5*x*cos(x)^2 + 48*b^5*x - 6*(4*a^3*b^2 - 9*
a*b^4)*cos(x)^3 - 48*((a^4 - 2*a^2*b^2 + b^4)*cos(x)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos
(x)^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) + 6*(4*a^3*b^2 - 7*a*b^4
)*cos(x) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2
+ 15*a*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) - 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b
^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2) + 16*((3*a^4*b - 7*a^2*b^3)*
cos(x)^3 - 3*(a^4*b - 2*a^2*b^3)*cos(x))*sin(x))/(a*b^5*cos(x)^4 - 2*a*b^5*cos(x)^2 + a*b^5)]

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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+b*csc(x)),x)

[Out]

Timed out

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Giac [A]  time = 1.25028, size = 404, normalized size = 2.17 \begin{align*} -\frac{x}{a} + \frac{3 \, b^{3} \tan \left (\frac{1}{2} \, x\right )^{4} - 8 \, a b^{2} \tan \left (\frac{1}{2} \, x\right )^{3} + 24 \, a^{2} b \tan \left (\frac{1}{2} \, x\right )^{2} - 48 \, b^{3} \tan \left (\frac{1}{2} \, x\right )^{2} - 96 \, a^{3} \tan \left (\frac{1}{2} \, x\right ) + 216 \, a b^{2} \tan \left (\frac{1}{2} \, x\right )}{192 \, b^{4}} + \frac{{\left (8 \, a^{4} - 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{8 \, b^{5}} - \frac{2 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (\frac{1}{2} \, x\right ) + a}{\sqrt{-a^{2} + b^{2}}}\right )\right )}}{\sqrt{-a^{2} + b^{2}} a b^{5}} - \frac{400 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{4} - 1000 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, x\right )^{4} + 750 \, b^{4} \tan \left (\frac{1}{2} \, x\right )^{4} - 96 \, a^{3} b \tan \left (\frac{1}{2} \, x\right )^{3} + 216 \, a b^{3} \tan \left (\frac{1}{2} \, x\right )^{3} + 24 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, x\right )^{2} - 48 \, b^{4} \tan \left (\frac{1}{2} \, x\right )^{2} - 8 \, a b^{3} \tan \left (\frac{1}{2} \, x\right ) + 3 \, b^{4}}{192 \, b^{5} \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+b*csc(x)),x, algorithm="giac")

[Out]

-x/a + 1/192*(3*b^3*tan(1/2*x)^4 - 8*a*b^2*tan(1/2*x)^3 + 24*a^2*b*tan(1/2*x)^2 - 48*b^3*tan(1/2*x)^2 - 96*a^3
*tan(1/2*x) + 216*a*b^2*tan(1/2*x))/b^4 + 1/8*(8*a^4 - 20*a^2*b^2 + 15*b^4)*log(abs(tan(1/2*x)))/b^5 - 2*(a^6
- 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))
/(sqrt(-a^2 + b^2)*a*b^5) - 1/192*(400*a^4*tan(1/2*x)^4 - 1000*a^2*b^2*tan(1/2*x)^4 + 750*b^4*tan(1/2*x)^4 - 9
6*a^3*b*tan(1/2*x)^3 + 216*a*b^3*tan(1/2*x)^3 + 24*a^2*b^2*tan(1/2*x)^2 - 48*b^4*tan(1/2*x)^2 - 8*a*b^3*tan(1/
2*x) + 3*b^4)/(b^5*tan(1/2*x)^4)