3.74 \(\int \frac {1}{a+b \cos ^5(x)} \, dx\)

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

[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 = 0.92, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3213, 2659, 205} \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-\sqrt [5]{-1} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+\sqrt [5]{-1} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{2/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{2/5} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{3/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{3/5} \sqrt [5]{b}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \tan \left (\frac {x}{2}\right )}{\sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}}\right )}{5 a^{4/5} \sqrt {\sqrt [5]{a}-(-1)^{4/5} \sqrt [5]{b}} \sqrt {\sqrt [5]{a}+(-1)^{4/5} \sqrt [5]{b}}} \]

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 205

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

Rule 2659

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 3213

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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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]  time = 0.25, size = 130, normalized size = 0.26 \[ \frac {8}{5} \text {RootSum}\left [\text {$\#$1}^{10} b+5 \text {$\#$1}^8 b+10 \text {$\#$1}^6 b+32 \text {$\#$1}^5 a+10 \text {$\#$1}^4 b+5 \text {$\#$1}^2 b+b\& ,\frac {2 \text {$\#$1}^3 \tan ^{-1}\left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-i \text {$\#$1}^3 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (x)+1\right )}{\text {$\#$1}^8 b+4 \text {$\#$1}^6 b+6 \text {$\#$1}^4 b+16 \text {$\#$1}^3 a+4 \text {$\#$1}^2 b+b}\& \right ] \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \cos \relax (x)^{5} + a}\,{d x} \]

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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maple [C]  time = 0.10, size = 150, normalized size = 0.30 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a -b \right ) \textit {\_Z}^{10}+\left (5 a +5 b \right ) \textit {\_Z}^{8}+\left (10 a -10 b \right ) \textit {\_Z}^{6}+\left (10 a +10 b \right ) \textit {\_Z}^{4}+\left (5 a -5 b \right ) \textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (\textit {\_R}^{8}+4 \textit {\_R}^{6}+6 \textit {\_R}^{4}+4 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{9} a -\textit {\_R}^{9} b +4 \textit {\_R}^{7} a +4 \textit {\_R}^{7} b +6 \textit {\_R}^{5} a -6 \textit {\_R}^{5} b +4 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +\textit {\_R} a -\textit {\_R} b}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cos(x)^5),x)

[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 \[ \int \frac {1}{b \cos \relax (x)^{5} + a}\,{d x} \]

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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mupad [B]  time = 8.82, size = 1520, normalized size = 3.08 \[ \text {result too large to display} \]

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)*(7*cot(x/2) - 56*root(9765625*a^8*b^2*d^10 - 9765625*a^10*d^10 - 19531
25*a^8*d^8 - 156250*a^6*d^6 - 6250*a^4*d^4 - 125*a^2*d^2 - 1, d, k)*a + 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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \cos ^{5}{\relax (x )}}\, dx \]

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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