3.1.23 \(\int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx\) [23]

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

[Out]

-x/a-3/8*arctanh(cos(x))/b-1/2*(a^2-3*b^2)*arctanh(cos(x))/b^3-(a^4-3*a^2*b^2+3*b^4)*arctanh(cos(x))/b^5+2*(a^
2-b^2)^(5/2)*arctanh((a+b*tan(1/2*x))/(a^2-b^2)^(1/2))/a/b^5+a*cot(x)/b^2+a*(a^2-3*b^2)*cot(x)/b^4+1/3*a*cot(x
)^3/b^2-3/8*cot(x)*csc(x)/b-1/2*(a^2-3*b^2)*cot(x)*csc(x)/b^3-1/4*cot(x)*csc(x)^3/b

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Rubi [A]
time = 0.18, antiderivative size = 186, normalized size of antiderivative = 1.00, 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 = {3983, 2976, 3855, 3852, 8, 3853, 2739, 632, 212} \begin {gather*} \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 \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 (a^2-3 b^2\right ) \cot (x) \csc (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 ^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} \end {gather*}

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 8

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

Rule 212

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

Rule 632

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 2739

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 2976

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 3852

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 3853

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 3855

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

Rule 3983

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])

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 \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.97, size = 312, normalized size = 1.68 \begin {gather*} \frac {-192 b^5 x+384 \left (-a^2+b^2\right )^{5/2} \text {ArcTan}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^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 )+54 a b^4 \csc ^2\left (\frac {x}{2}\right )-3 a b^4 \csc ^4\left (\frac {x}{2}\right )-192 a^5 \log \left (\cos \left (\frac {x}{2}\right )\right )+480 a^3 b^2 \log \left (\cos \left (\frac {x}{2}\right )\right )-360 a b^4 \log \left (\cos \left (\frac {x}{2}\right )\right )+192 a^5 \log \left (\sin \left (\frac {x}{2}\right )\right )-480 a^3 b^2 \log \left (\sin \left (\frac {x}{2}\right )\right )+360 a b^4 \log \left (\sin \left (\frac {x}{2}\right )\right )+24 a^3 b^2 \sec ^2\left (\frac {x}{2}\right )-54 a b^4 \sec ^2\left (\frac {x}{2}\right )+3 a b^4 \sec ^4\left (\frac {x}{2}\right )-64 a^2 b^3 \csc ^3(x) \sin ^4\left (\frac {x}{2}\right )+4 a^2 b^3 \csc ^4\left (\frac {x}{2}\right ) \sin (x)-96 a^4 b \tan \left (\frac {x}{2}\right )+224 a^2 b^3 \tan \left (\frac {x}{2}\right )}{192 a b^5} \end {gather*}

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 [A]
time = 0.26, size = 251, normalized size = 1.35

method	result	size
default	\(-\frac {-\frac {b^{3} \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4}+\frac {2 a \left (\tan ^{3}\left (\frac {x}{2}\right )\right ) b^{2}}{3}-2 a^{2} b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+4 b^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+8 a^{3} \tan \left (\frac {x}{2}\right )-18 a \,b^{2} \tan \left (\frac {x}{2}\right )}{16 b^{4}}-\frac {1}{64 b \tan \left (\frac {x}{2}\right )^{4}}-\frac {4 a^{2}-8 b^{2}}{32 b^{3} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (16 a^{4}-40 a^{2} b^{2}+30 b^{4}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{16 b^{5}}+\frac {a}{24 b^{2} \tan \left (\frac {x}{2}\right )^{3}}+\frac {a \left (4 a^{2}-9 b^{2}\right )}{8 b^{4} \tan \left (\frac {x}{2}\right )}-\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}+\frac {\left (-32 a^{6}+96 a^{4} b^{2}-96 a^{2} b^{4}+32 b^{6}\right ) \arctan \left (\frac {2 b \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{16 a \,b^{5} \sqrt {-a^{2}+b^{2}}}\)	\(251\)
risch	\(-\frac {x}{a}-\frac {72 i a^{3} {\mathrm e}^{4 i x}-56 i a \,b^{2}-12 a^{2} b \,{\mathrm e}^{7 i x}+27 b^{3} {\mathrm e}^{7 i x}+72 i a \,b^{2} {\mathrm e}^{6 i x}-24 i a^{3} {\mathrm e}^{6 i x}+12 a^{2} b \,{\mathrm e}^{5 i x}-3 b^{3} {\mathrm e}^{5 i x}+24 i a^{3}-72 i a^{3} {\mathrm e}^{2 i x}+12 a^{2} b \,{\mathrm e}^{3 i x}-3 b^{3} {\mathrm e}^{3 i x}+152 i a \,b^{2} {\mathrm e}^{2 i x}-168 i a \,b^{2} {\mathrm e}^{4 i x}-12 a^{2} b \,{\mathrm e}^{i x}+27 \,{\mathrm e}^{i x} b^{3}}{12 b^{4} \left ({\mathrm e}^{2 i x}-1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right ) a^{4}}{b^{5}}+\frac {5 \ln \left ({\mathrm e}^{i x}+1\right ) a^{2}}{2 b^{3}}-\frac {15 \ln \left ({\mathrm e}^{i x}+1\right )}{8 b}+\frac {\ln \left ({\mathrm e}^{i x}-1\right ) a^{4}}{b^{5}}-\frac {5 \ln \left ({\mathrm e}^{i x}-1\right ) a^{2}}{2 b^{3}}+\frac {15 \ln \left ({\mathrm e}^{i x}-1\right )}{8 b}-\frac {\sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i b -\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{5}}+\frac {2 \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i x}+\frac {i b -\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{3}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {i b -\sqrt {a^{2}-b^{2}}}{a}\right )}{b a}+\frac {\sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i b +\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{5}}-\frac {2 \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i x}+\frac {i b +\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{3}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {i b +\sqrt {a^{2}-b^{2}}}{a}\right )}{b a}\)	\(554\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/b^4*(-1/4*b^3*tan(1/2*x)^4+2/3*a*tan(1/2*x)^3*b^2-2*a^2*b*tan(1/2*x)^2+4*b^3*tan(1/2*x)^2+8*a^3*tan(1/2*
x)-18*a*b^2*tan(1/2*x))-1/64/b/tan(1/2*x)^4-1/32*(4*a^2-8*b^2)/b^3/tan(1/2*x)^2+1/16/b^5*(16*a^4-40*a^2*b^2+30
*b^4)*ln(tan(1/2*x))+1/24*a/b^2/tan(1/2*x)^3+1/8*a*(4*a^2-9*b^2)/b^4/tan(1/2*x)-2/a*arctan(tan(1/2*x))+1/16*(-
32*a^6+96*a^4*b^2-96*a^2*b^4+32*b^6)/a/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))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (168) = 336\).
time = 4.16, size = 852, normalized size = 4.58 \begin {gather*} \left [-\frac {48 \, b^{5} x \cos \left (x\right )^{4} - 96 \, b^{5} x \cos \left (x\right )^{2} + 48 \, b^{5} x - 6 \, {\left (4 \, a^{3} b^{2} - 9 \, a b^{4}\right )} \cos \left (x\right )^{3} - 24 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a b \sin \left (x\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (x\right ) \sin \left (x\right ) + a \cos \left (x\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) + 6 \, {\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (x\right ) + 3 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{4} - 2 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{4} - 2 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 16 \, {\left ({\left (3 \, a^{4} b - 7 \, a^{2} b^{3}\right )} \cos \left (x\right )^{3} - 3 \, {\left (a^{4} b - 2 \, a^{2} b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{48 \, {\left (a b^{5} \cos \left (x\right )^{4} - 2 \, a b^{5} \cos \left (x\right )^{2} + a b^{5}\right )}}, -\frac {48 \, b^{5} x \cos \left (x\right )^{4} - 96 \, b^{5} x \cos \left (x\right )^{2} + 48 \, b^{5} x - 6 \, {\left (4 \, a^{3} b^{2} - 9 \, a b^{4}\right )} \cos \left (x\right )^{3} - 48 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (x\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (x\right )}\right ) + 6 \, {\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (x\right ) + 3 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{4} - 2 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4} + {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{4} - 2 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 16 \, {\left ({\left (3 \, a^{4} b - 7 \, a^{2} b^{3}\right )} \cos \left (x\right )^{3} - 3 \, {\left (a^{4} b - 2 \, a^{2} b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{48 \, {\left (a b^{5} \cos \left (x\right )^{4} - 2 \, a b^{5} \cos \left (x\right )^{2} + a b^{5}\right )}}\right ] \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)**6/(a + b*csc(x)), x)

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Giac [A]
time = 0.44, size = 299, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 1.59, size = 2500, normalized size = 13.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^6/(a + b/sin(x)),x)

[Out]

tan(x/2)*(a/(8*b^2) + (2*a*(1/(2*b) - a^2/(4*b^3)))/b) - tan(x/2)^2*(1/(4*b) - a^2/(8*b^3)) + tan(x/2)^4/(64*b
) - (2*atan((((((4*(24*a*b^16 + 58*a^3*b^14 - 345*a^5*b^12 + 543*a^7*b^10 - 440*a^9*b^8 + 184*a^11*b^6 - 32*a^
13*b^4))/b^12 + (((((4*(24*a^3*b^16 - 32*a^5*b^14))/b^12 - (4*tan(x/2)*(16*a^2*b^17 - 136*a^4*b^15 + 128*a^6*b
^13))/b^12)*1i)/a - (4*(93*a^3*b^15 - 240*a^5*b^13 + 208*a^7*b^11 - 64*a^9*b^9))/b^12 + (4*tan(x/2)*(62*a^2*b^
16 - 335*a^4*b^14 + 604*a^6*b^12 - 456*a^8*b^10 + 128*a^10*b^8))/b^12)*1i)/a - (4*tan(x/2)*(16*b^17 - 180*a^2*
b^15 + 252*a^4*b^13 - 148*a^6*b^11 + 100*a^8*b^9 - 40*a^10*b^7 + 8*a^12*b^5))/b^12)*1i)/a - (4*(53*a*b^15 - 12
0*a^3*b^13 + 48*a^5*b^11 + 56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5))/b^12 + (4*tan(x/2)*(8*a^16 + 62*b^16 - 410*a
^2*b^14 + 929*a^4*b^12 - 1096*a^6*b^10 + 873*a^8*b^8 - 550*a^10*b^6 + 255*a^12*b^4 - 68*a^14*b^2))/b^12)/a - (
(4*(53*a*b^15 - 120*a^3*b^13 + 48*a^5*b^11 + 56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5))/b^12 + (((4*(24*a*b^16 + 5
8*a^3*b^14 - 345*a^5*b^12 + 543*a^7*b^10 - 440*a^9*b^8 + 184*a^11*b^6 - 32*a^13*b^4))/b^12 + (((4*(93*a^3*b^15
- 240*a^5*b^13 + 208*a^7*b^11 - 64*a^9*b^9))/b^12 + (((4*(24*a^3*b^16 - 32*a^5*b^14))/b^12 - (4*tan(x/2)*(16*
a^2*b^17 - 136*a^4*b^15 + 128*a^6*b^13))/b^12)*1i)/a - (4*tan(x/2)*(62*a^2*b^16 - 335*a^4*b^14 + 604*a^6*b^12
- 456*a^8*b^10 + 128*a^10*b^8))/b^12)*1i)/a - (4*tan(x/2)*(16*b^17 - 180*a^2*b^15 + 252*a^4*b^13 - 148*a^6*b^1
1 + 100*a^8*b^9 - 40*a^10*b^7 + 8*a^12*b^5))/b^12)*1i)/a - (4*tan(x/2)*(8*a^16 + 62*b^16 - 410*a^2*b^14 + 929*
a^4*b^12 - 1096*a^6*b^10 + 873*a^8*b^8 - 550*a^10*b^6 + 255*a^12*b^4 - 68*a^14*b^2))/b^12)/a)/((((((4*(24*a*b^
16 + 58*a^3*b^14 - 345*a^5*b^12 + 543*a^7*b^10 - 440*a^9*b^8 + 184*a^11*b^6 - 32*a^13*b^4))/b^12 + (((((4*(24*
a^3*b^16 - 32*a^5*b^14))/b^12 - (4*tan(x/2)*(16*a^2*b^17 - 136*a^4*b^15 + 128*a^6*b^13))/b^12)*1i)/a - (4*(93*
a^3*b^15 - 240*a^5*b^13 + 208*a^7*b^11 - 64*a^9*b^9))/b^12 + (4*tan(x/2)*(62*a^2*b^16 - 335*a^4*b^14 + 604*a^6
*b^12 - 456*a^8*b^10 + 128*a^10*b^8))/b^12)*1i)/a - (4*tan(x/2)*(16*b^17 - 180*a^2*b^15 + 252*a^4*b^13 - 148*a
^6*b^11 + 100*a^8*b^9 - 40*a^10*b^7 + 8*a^12*b^5))/b^12)*1i)/a - (4*(53*a*b^15 - 120*a^3*b^13 + 48*a^5*b^11 +
56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5))/b^12 + (4*tan(x/2)*(8*a^16 + 62*b^16 - 410*a^2*b^14 + 929*a^4*b^12 - 10
96*a^6*b^10 + 873*a^8*b^8 - 550*a^10*b^6 + 255*a^12*b^4 - 68*a^14*b^2))/b^12)*1i)/a + (((4*(53*a*b^15 - 120*a^
3*b^13 + 48*a^5*b^11 + 56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5))/b^12 + (((4*(24*a*b^16 + 58*a^3*b^14 - 345*a^5*b
^12 + 543*a^7*b^10 - 440*a^9*b^8 + 184*a^11*b^6 - 32*a^13*b^4))/b^12 + (((4*(93*a^3*b^15 - 240*a^5*b^13 + 208*
a^7*b^11 - 64*a^9*b^9))/b^12 + (((4*(24*a^3*b^16 - 32*a^5*b^14))/b^12 - (4*tan(x/2)*(16*a^2*b^17 - 136*a^4*b^1
5 + 128*a^6*b^13))/b^12)*1i)/a - (4*tan(x/2)*(62*a^2*b^16 - 335*a^4*b^14 + 604*a^6*b^12 - 456*a^8*b^10 + 128*a
^10*b^8))/b^12)*1i)/a - (4*tan(x/2)*(16*b^17 - 180*a^2*b^15 + 252*a^4*b^13 - 148*a^6*b^11 + 100*a^8*b^9 - 40*a
^10*b^7 + 8*a^12*b^5))/b^12)*1i)/a - (4*tan(x/2)*(8*a^16 + 62*b^16 - 410*a^2*b^14 + 929*a^4*b^12 - 1096*a^6*b^
10 + 873*a^8*b^8 - 550*a^10*b^6 + 255*a^12*b^4 - 68*a^14*b^2))/b^12)*1i)/a + (8*(15*a*b^14 + 8*a^15 - 20*a^3*b
^12 - 97*a^5*b^10 + 305*a^7*b^8 - 366*a^9*b^6 + 223*a^11*b^4 - 68*a^13*b^2))/b^12 - (8*tan(x/2)*(4*a^2*b^13 -
4*b^15 + 12*a^4*b^11 - 20*a^6*b^9 + 8*a^8*b^7))/b^12)))/a - (a*tan(x/2)^3)/(24*b^2) - (tan(x/2)^3*(18*a*b^2 -
8*a^3) + tan(x/2)^2*(2*a^2*b - 4*b^3) + b^3/4 - (2*a*b^2*tan(x/2))/3)/(16*b^4*tan(x/2)^4) + (log(tan(x/2))*(a^
4 + (15*b^4)/8 - (5*a^2*b^2)/2))/b^5 - (atan(((((a + b)^5*(a - b)^5)^(1/2)*((4*tan(x/2)*(8*a^16 + 62*b^16 - 41
0*a^2*b^14 + 929*a^4*b^12 - 1096*a^6*b^10 + 873*a^8*b^8 - 550*a^10*b^6 + 255*a^12*b^4 - 68*a^14*b^2))/b^12 - (
4*(53*a*b^15 - 120*a^3*b^13 + 48*a^5*b^11 + 56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5))/b^12 + (((a + b)^5*(a - b)^
5)^(1/2)*((4*(24*a*b^16 + 58*a^3*b^14 - 345*a^5*b^12 + 543*a^7*b^10 - 440*a^9*b^8 + 184*a^11*b^6 - 32*a^13*b^4
))/b^12 - (4*tan(x/2)*(16*b^17 - 180*a^2*b^15 + 252*a^4*b^13 - 148*a^6*b^11 + 100*a^8*b^9 - 40*a^10*b^7 + 8*a^
12*b^5))/b^12 + (((a + b)^5*(a - b)^5)^(1/2)*((4*tan(x/2)*(62*a^2*b^16 - 335*a^4*b^14 + 604*a^6*b^12 - 456*a^8
*b^10 + 128*a^10*b^8))/b^12 - (4*(93*a^3*b^15 - 240*a^5*b^13 + 208*a^7*b^11 - 64*a^9*b^9))/b^12 + (((4*(24*a^3
*b^16 - 32*a^5*b^14))/b^12 - (4*tan(x/2)*(16*a^2*b^17 - 136*a^4*b^15 + 128*a^6*b^13))/b^12)*((a + b)^5*(a - b)
^5)^(1/2))/(a*b^5)))/(a*b^5)))/(a*b^5))*1i)/(a*b^5) - (((a + b)^5*(a - b)^5)^(1/2)*((4*(53*a*b^15 - 120*a^3*b^
13 + 48*a^5*b^11 + 56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5))/b^12 - (4*tan(x/2)*(8*a^16 + 62*b^16 - 410*a^2*b^14
+ 929*a^4*b^12 - 1096*a^6*b^10 + 873*a^8*b^8 - 550*a^10*b^6 + 255*a^12*b^4 - 68*a^14*b^2))/b^12 + (((a + b)^5*
(a - b)^5)^(1/2)*((4*(24*a*b^16 + 58*a^3*b^14 - 345*a^5*b^12 + 543*a^7*b^10 - 440*a^9*b^8 + 184*a^11*b^6 - 32*
a^13*b^4))/b^12 - (4*tan(x/2)*(16*b^17 - 180*a^2*b^15 + 252*a^4*b^13 - 148*a^6*b^11 + 100*a^8*b^9 - 40*a^10*b^
7 + 8*a^12*b^5))/b^12 + (((a + b)^5*(a - b)^5)^...

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