3.78 \(\int \frac{1}{a-b \cos ^6(x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.249201, antiderivative size = 175, normalized size of antiderivative = 1., 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 = {3211, 3181, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-\sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}-\sqrt [3]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \cot (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Cos[x]^6)^(-1),x]

[Out]

-ArcTan[(Sqrt[a^(1/3) - b^(1/3)]*Cot[x])/a^(1/6)]/(3*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 3211

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]

Rule 3181

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 203

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

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{\operatorname{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{\operatorname{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{\operatorname{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*}

Mathematica [C]  time = 0.176012, size = 146, normalized size = 0.83 \[ -\frac{8}{3} \text{RootSum}\left [-64 \text{$\#$1}^3 a+\text{$\#$1}^6 b+6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b+20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b+6 \text{$\#$1} b+b\& ,\frac{2 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right )-i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right )}{-32 \text{$\#$1}^2 a+\text{$\#$1}^5 b+5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b+10 \text{$\#$1}^2 b+5 \text{$\#$1} b+b}\& \right ] \]

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]  time = 0.121, size = 62, normalized size = 0.4 \begin{align*}{\frac{1}{6\,a}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+3\,a{{\it \_Z}}^{2}+a-b \right ) }{\frac{ \left ({{\it \_R}}^{4}+2\,{{\it \_R}}^{2}+1 \right ) \ln \left ( \tan \left ( x \right ) -{\it \_R} \right ) }{{{\it \_R}}^{5}+2\,{{\it \_R}}^{3}+{\it \_R}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{b \cos \left (x\right )^{6} - a}\,{d x} \end{align*}

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{b \cos \left (x\right )^{6} - a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-1/(b*cos(x)^6 - a), x)