3.1.78 \(\int \frac {1}{a-b \cos ^6(x)} \, dx\) [78]

Optimal. Leaf size=175 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]

[Out]

-1/3*arctan(cot(x)*(a^(1/3)-b^(1/3))^(1/2)/a^(1/6))/a^(5/6)/(a^(1/3)-b^(1/3))^(1/2)-1/3*arctan(cot(x)*(a^(1/3)
+(-1)^(1/3)*b^(1/3))^(1/2)/a^(1/6))/a^(5/6)/(a^(1/3)+(-1)^(1/3)*b^(1/3))^(1/2)-1/3*arctan(cot(x)*(a^(1/3)-(-1)
^(2/3)*b^(1/3))^(1/2)/a^(1/6))/a^(5/6)/(a^(1/3)-(-1)^(2/3)*b^(1/3))^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3290, 3260, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - b*Cos[x]^6)^(-1),x]

[Out]

-1/3*ArcTan[(Sqrt[a^(1/3) - b^(1/3)]*Cot[x])/a^(1/6)]/(a^(5/6)*Sqrt[a^(1/3) - b^(1/3)]) - ArcTan[(Sqrt[a^(1/3)
+ (-1)^(1/3)*b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) - ArcTan[(Sqrt[a^(1/3)
- (-1)^(2/3)*b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)])

Rule 209

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

Rule 3260

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 3290

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^(4*(k/n))*Rt[-a/b, n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]

Rubi steps

\begin {align*} \int \frac {1}{a-b \cos ^6(x)} \, dx &=\frac {\int \frac {1}{1-\frac {\sqrt [3]{b} \cos ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1+\frac {\sqrt [3]{-1} \sqrt [3]{b} \cos ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}+\frac {\int \frac {1}{1-\frac {(-1)^{2/3} \sqrt [3]{b} \cos ^2(x)}{\sqrt [3]{a}}} \, dx}{3 a}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {\sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\cot (x)\right )}{3 a}-\frac {\text {Subst}\left (\int \frac {1}{1+\left (1+\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\cot (x)\right )}{3 a}-\frac {\text {Subst}\left (\int \frac {1}{1+\left (1-\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) x^2} \, dx,x,\cot (x)\right )}{3 a}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{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 = 146, normalized size = 0.83 \begin {gather*} -\frac {8}{3} \text {RootSum}\left [b+6 b \text {$\#$1}+15 b \text {$\#$1}^2-64 a \text {$\#$1}^3+20 b \text {$\#$1}^3+15 b \text {$\#$1}^4+6 b \text {$\#$1}^5+b \text {$\#$1}^6\&,\frac {2 \text {ArcTan}\left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{b+5 b \text {$\#$1}-32 a \text {$\#$1}^2+10 b \text {$\#$1}^2+10 b \text {$\#$1}^3+5 b \text {$\#$1}^4+b \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a - b*Cos[x]^6)^(-1),x]

[Out]

(-8*RootSum[b + 6*b*#1 + 15*b*#1^2 - 64*a*#1^3 + 20*b*#1^3 + 15*b*#1^4 + 6*b*#1^5 + b*#1^6 & , (2*ArcTan[Sin[2
*x]/(Cos[2*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2)/(b + 5*b*#1 - 32*a*#1^2 + 10*b*#1^2 + 10*b*#
1^3 + 5*b*#1^4 + b*#1^5) & ])/3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.40, size = 62, normalized size = 0.35

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+3 a \,\textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{5}+2 \textit {\_R}^{3}+\textit {\_R}}}{6 a}\)	\(62\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (1+\left (46656 a^{6}-46656 a^{5} b \right ) \textit {\_Z}^{6}+3888 a^{4} \textit {\_Z}^{4}+108 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {15552 i a^{6}}{b}-15552 i a^{5}\right ) \textit {\_R}^{5}+\left (-\frac {2592 a^{5}}{b}+2592 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {864 i a^{4}}{b}+432 i a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {144 a^{3}}{b}-72 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {12 i a^{2}}{b}-12 i a \right ) \textit {\_R} -\frac {2 a}{b}+1\right )\)	\(147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*cos(x)^6),x,method=_RETURNVERBOSE)

[Out]

1/6/a*sum((_R^4+2*_R^2+1)/(_R^5+2*_R^3+_R)*ln(tan(x)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+3*_Z^2*a+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)^6),x, algorithm="maxima")

[Out]

-integrate(1/(b*cos(x)^6 - a), x)

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Fricas [C] Result contains complex when optimal does not.
time = 1.97, size = 16679, normalized size = 95.31 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)^6),x, algorithm="fricas")

[Out]

1/72*sqrt(1/2)*sqrt((-I*sqrt(3) + 1)*(1/(a^4 - a^3*b) - 1/(a^2 - a*b)^2)/(-1/93312/(a^6 - a^5*b) + 1/31104/((a
^4 - a^3*b)*(a^2 - a*b)) - 1/46656/(a^2 - a*b)^3 + 1/93312*b/((a - b)^2*a^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1
/93312/(a^6 - a^5*b) + 1/31104/((a^4 - a^3*b)*(a^2 - a*b)) - 1/46656/(a^2 - a*b)^3 + 1/93312*b/((a - b)^2*a^5)
)^(1/3) - 72/(a^2 - a*b))*log(1/5184*(a^5 - a^4*b - 2*(a^5 - a^4*b)*cos(x)^2)*((-I*sqrt(3) + 1)*(1/(a^4 - a^3*
b) - 1/(a^2 - a*b)^2)/(-1/93312/(a^6 - a^5*b) + 1/31104/((a^4 - a^3*b)*(a^2 - a*b)) - 1/46656/(a^2 - a*b)^3 +
1/93312*b/((a - b)^2*a^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a^6 - a^5*b) + 1/31104/((a^4 - a^3*b)*(a^2
- a*b)) - 1/46656/(a^2 - a*b)^3 + 1/93312*b/((a - b)^2*a^5))^(1/3) - 72/(a^2 - a*b))^2 - (2*a - b)*cos(x)^2 +
1/15552*sqrt(1/2)*((a^6 - a^5*b)*((-I*sqrt(3) + 1)*(1/(a^4 - a^3*b) - 1/(a^2 - a*b)^2)/(-1/93312/(a^6 - a^5*b)
+ 1/31104/((a^4 - a^3*b)*(a^2 - a*b)) - 1/46656/(a^2 - a*b)^3 + 1/93312*b/((a - b)^2*a^5))^(1/3) - 1296*(I*sq
rt(3) + 1) ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a - b \cos ^{6}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)**6),x)

[Out]

Integral(1/(a - b*cos(x)**6), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)^6),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 3.12, size = 184, normalized size = 1.05 \begin {gather*} \sum _{k=1}^6\ln \left (-{\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^2\,a^3\,b^3\,\left ({\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )}^2\,a^2\,36+1\right )\,\left (\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,a^2\,d^2-1,d,k\right )\,a\,\mathrm {tan}\left (x\right )\,6-1\right )\,36\right )\,\mathrm {root}\left (46656\,a^5\,b\,d^6-46656\,a^6\,d^6-3888\,a^4\,d^4-108\,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)^6),x)

[Out]

symsum(log(-36*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^2*a^3*b^3*(36*root
(46656*a^5*b*d^6 - 46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)^2*a^2 + 1)*(6*root(46656*a^5*b*d^6 -
46656*a^6*d^6 - 3888*a^4*d^4 - 108*a^2*d^2 - 1, d, k)*a*tan(x) - 1))*root(46656*a^5*b*d^6 - 46656*a^6*d^6 - 38
88*a^4*d^4 - 108*a^2*d^2 - 1, d, k), k, 1, 6)

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