3.1.77 \(\int \frac {1}{a-b \cos ^5(x)} \, dx\) [77]

Optimal. Leaf size=494 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]

[Out]

2/5*arctan((a^(1/5)+b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)-b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-b^(1/5))^(1/2)/(a^(1/5
)+b^(1/5))^(1/2)+2/5*arctan((a^(1/5)-(-1)^(1/5)*b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)+(-1)^(1/5)*b^(1/5))^(1/2))/
a^(4/5)/(a^(1/5)-(-1)^(1/5)*b^(1/5))^(1/2)/(a^(1/5)+(-1)^(1/5)*b^(1/5))^(1/2)+2/5*arctan((a^(1/5)+(-1)^(2/5)*b
^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)-(-1)^(2/5)*b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(2/5)*b^(1/5))^(1/2)/(a^(1/
5)+(-1)^(2/5)*b^(1/5))^(1/2)+2/5*arctan((a^(1/5)-(-1)^(3/5)*b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)+(-1)^(3/5)*b^(1
/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(3/5)*b^(1/5))^(1/2)/(a^(1/5)+(-1)^(3/5)*b^(1/5))^(1/2)+2/5*arctan((a^(1/5)+
(-1)^(4/5)*b^(1/5))^(1/2)*tan(1/2*x)/(a^(1/5)-(-1)^(4/5)*b^(1/5))^(1/2))/a^(4/5)/(a^(1/5)-(-1)^(4/5)*b^(1/5))^
(1/2)/(a^(1/5)+(-1)^(4/5)*b^(1/5))^(1/2)

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Rubi [A]
time = 0.47, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3292, 2738, 211} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - b*Cos[x]^5)^(-1),x]

[Out]

(2*ArcTan[(Sqrt[a^(1/5) + b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - b^(1/5)]*Sqrt
[a^(1/5) + b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/
5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1
/5) + (-1)^(2/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(2/5)*
b^(1/5)]*Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^
(1/5) + (-1)^(3/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]
) + (2*ArcTan[(Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]])/(5*a^(4/5)*Sq
rt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*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 3292

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rubi steps

\begin {align*} \int \frac {1}{a-b \cos ^5(x)} \, dx &=\int \left (\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-\sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cos (x)\right )}+\frac {1}{5 a^{4/5} \left (\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cos (x)\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{\sqrt [5]{a}-\sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}+\frac {\int \frac {1}{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b} \cos (x)} \, dx}{5 a^{4/5}}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}-\sqrt [5]{b}+\left (\sqrt [5]{a}+\sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}+\left (\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}+\left (\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}+\left (\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}+\left (\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{5 a^{4/5}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.19, size = 130, normalized size = 0.26 \begin {gather*} -\frac {8}{5} \text {RootSum}\left [b+5 b \text {$\#$1}^2+10 b \text {$\#$1}^4-32 a \text {$\#$1}^5+10 b \text {$\#$1}^6+5 b \text {$\#$1}^8+b \text {$\#$1}^{10}\&,\frac {2 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{b+4 b \text {$\#$1}^2-16 a \text {$\#$1}^3+6 b \text {$\#$1}^4+4 b \text {$\#$1}^6+b \text {$\#$1}^8}\&\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a - b*Cos[x]^5)^(-1),x]

[Out]

(-8*RootSum[b + 5*b*#1^2 + 10*b*#1^4 - 32*a*#1^5 + 10*b*#1^6 + 5*b*#1^8 + b*#1^10 & , (2*ArcTan[Sin[x]/(Cos[x]
- #1)]*#1^3 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3)/(b + 4*b*#1^2 - 16*a*#1^3 + 6*b*#1^4 + 4*b*#1^6 + b*#1^8) &
])/5

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.49, size = 148, normalized size = 0.30

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{10}+\left (5 a -5 b \right ) \textit {\_Z}^{8}+\left (10 a +10 b \right ) \textit {\_Z}^{6}+\left (10 a -10 b \right ) \textit {\_Z}^{4}+\left (5 a +5 b \right ) \textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a +\textit {\_R}^{9} b +4 \textit {\_R}^{7} a -4 \textit {\_R}^{7} b +6 \textit {\_R}^{5} a +6 \textit {\_R}^{5} b +4 \textit {\_R}^{3} a -4 \textit {\_R}^{3} b +\textit {\_R} a +\textit {\_R} b}\right )}{5}\)	\(148\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (1+\left (9765625 a^{10}-9765625 a^{8} b^{2}\right ) \textit {\_Z}^{10}+1953125 a^{8} \textit {\_Z}^{8}+156250 a^{6} \textit {\_Z}^{6}+6250 a^{4} \textit {\_Z}^{4}+125 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (\frac {11718750 i a^{10}}{b}-11718750 i b \,a^{8}\right ) \textit {\_R}^{9}+\left (-\frac {1171875 a^{9}}{b}+1171875 a^{7} b \right ) \textit {\_R}^{8}+\left (\frac {2109375 i a^{8}}{b}+234375 i a^{6} b \right ) \textit {\_R}^{7}+\left (-\frac {218750 a^{7}}{b}-15625 a^{5} b \right ) \textit {\_R}^{6}+\left (\frac {143750 i a^{6}}{b}-3125 i a^{4} b \right ) \textit {\_R}^{5}-\frac {15625 a^{5} \textit {\_R}^{4}}{b}+\frac {4375 i a^{4} \textit {\_R}^{3}}{b}-\frac {500 a^{3} \textit {\_R}^{2}}{b}+\frac {50 i a^{2} \textit {\_R}}{b}-\frac {6 a}{b}\right )\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*cos(x)^5),x,method=_RETURNVERBOSE)

[Out]

1/5*sum((_R^8+4*_R^6+6*_R^4+4*_R^2+1)/(_R^9*a+_R^9*b+4*_R^7*a-4*_R^7*b+6*_R^5*a+6*_R^5*b+4*_R^3*a-4*_R^3*b+_R*
a+_R*b)*ln(tan(1/2*x)-_R),_R=RootOf((a+b)*_Z^10+(5*a-5*b)*_Z^8+(10*a+10*b)*_Z^6+(10*a-10*b)*_Z^4+(5*a+5*b)*_Z^
2+a-b))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)^5),x, algorithm="maxima")

[Out]

-integrate(1/(b*cos(x)^5 - a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)^5),x, algorithm="fricas")

[Out]

Exception raised: RuntimeError >> no explicit roots found

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a - b \cos ^{5}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)**5),x)

[Out]

Integral(1/(a - b*cos(x)**5), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)^5),x, algorithm="giac")

[Out]

integrate(-1/(b*cos(x)^5 - a), x)

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Mupad [B]
time = 7.83, size = 1518, normalized size = 3.07 \begin {gather*} \sum _{k=1}^{10}\ln \left (-\frac {b^7\,\left (a+b\right )\,\left (-7\,\mathrm {cot}\left (\frac {x}{2}\right )+\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )\,a\,56+\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )\,b+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^3\,a^3\,5800+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^5\,a^5\,225000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^7\,a^7\,3875000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^9\,a^9\,25000000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^2\,a^2\,\mathrm {cot}\left (\frac {x}{2}\right )\,735-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^4\,a^4\,\mathrm {cot}\left (\frac {x}{2}\right )\,28875-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^6\,a^6\,\mathrm {cot}\left (\frac {x}{2}\right )\,503125-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^8\,a^8\,\mathrm {cot}\left (\frac {x}{2}\right )\,3281250+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^3\,a^2\,b\,800+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^5\,a^4\,b\,100000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^7\,a^6\,b\,4000000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^9\,a^8\,b\,50000000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^7\,a^5\,b^2\,125000+{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^9\,a^7\,b^2\,25000000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^2\,a\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,35-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^4\,a^3\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,7000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^6\,a^5\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,350000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^8\,a^7\,b\,\mathrm {cot}\left (\frac {x}{2}\right )\,5000000-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^6\,a^4\,b^2\,\mathrm {cot}\left (\frac {x}{2}\right )\,3125-{\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right )}^8\,a^6\,b^2\,\mathrm {cot}\left (\frac {x}{2}\right )\,1718750\right )\,10995116277760}{\mathrm {cot}\left (\frac {x}{2}\right )}\right )\,\mathrm {root}\left (9765625\,a^8\,b^2\,d^{10}-9765625\,a^{10}\,d^{10}-1953125\,a^8\,d^8-156250\,a^6\,d^6-6250\,a^4\,d^4-125\,a^2\,d^2-1,d,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*cos(x)^5),x)

[Out]

symsum(log(-(10995116277760*b^7*(a + b)*(56*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 -
156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)*a - 7*cot(x/2) + root(9765625*a^8*b^2*d^10 - 9765625*a^
10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)*b + 5800*root(9765625*a^8*b
^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^3*a^3 +
225000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*
a^2*d^2 - 1, d, k)^5*a^5 + 3875000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^
6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^7*a^7 + 25000000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 -
1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^9*a^9 - 735*root(9765625*a^8*b^2*d^1
0 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^2*a^2*cot(x/2
) - 28875*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 12
5*a^2*d^2 - 1, d, k)^4*a^4*cot(x/2) - 503125*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 -
156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^6*a^6*cot(x/2) - 3281250*root(9765625*a^8*b^2*d^10 - 9
765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^8*a^8*cot(x/2) + 8
00*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d
^2 - 1, d, k)^3*a^2*b + 100000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^
6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^5*a^4*b + 4000000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 19
53125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^7*a^6*b + 50000000*root(9765625*a^8*b^2
*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^9*a^8*b +
125000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*
a^2*d^2 - 1, d, k)^7*a^5*b^2 + 25000000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 1562
50*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^9*a^7*b^2 - 35*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^1
0 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^2*a*b*cot(x/2) - 7000*root(976562
5*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^
4*a^3*b*cot(x/2) - 350000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6
250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^6*a^5*b*cot(x/2) - 5000000*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10
- 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^8*a^7*b*cot(x/2) - 3125*root(976562
5*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^
6*a^4*b^2*cot(x/2) - 1718750*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6
- 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)^8*a^6*b^2*cot(x/2)))/cot(x/2))*root(9765625*a^8*b^2*d^10 - 9765625*a^1
0*d^10 - 1953125*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k), k, 1, 10)

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