3.76 \(\int \frac{1}{a+b \cos ^8(x)} \, dx\)

Optimal. Leaf size=245 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}} \]

[Out]

ArcTan[(Sqrt[(-a)^(1/4) - b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - b^(1/4)]) + ArcTan[(Sqr
t[(-a)^(1/4) - I*b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) + ArcTan[(Sqrt[(-a)^
(1/4) + I*b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) + ArcTan[(Sqrt[(-a)^(1/4) +
b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)])

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Rubi [A]  time = 0.486312, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3211, 3181, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{a \sqrt [4]{b}+(-a)^{5/4}} \cot (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt{a \sqrt [4]{b}+(-a)^{5/4}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[x]^8)^(-1),x]

[Out]

ArcTan[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) + ArcTan[
(Sqrt[(-a)^(1/4) + I*b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) + ArcTan[(Sqrt[(
-a)^(1/4) + b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)]) + ArcTan[(Sqrt[(-a)^(5/4) +
a*b^(1/4)]*Cot[x])/(-a)^(5/8)]/(4*(-a)^(3/8)*Sqrt[(-a)^(5/4) + a*b^(1/4)])

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 ^8(x)} \, dx &=\frac{\int \frac{1}{1-\frac{\sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1-\frac{i \sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1+\frac{i \sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1+\frac{\sqrt [4]{b} \cos ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\left (1-\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\left (1-\frac{i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\left (1+\frac{i \sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\left (1+\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}\right ) x^2} \, dx,x,\cot (x)\right )}{4 a}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}} \cot (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{(-a)^{5/4}+a \sqrt [4]{b}} \cot (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt{(-a)^{5/4}+a \sqrt [4]{b}}}\\ \end{align*}

Mathematica [C]  time = 0.260426, size = 172, normalized size = 0.7 \[ 8 \text{RootSum}\left [256 \text{$\#$1}^4 a+\text{$\#$1}^8 b+8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b+56 \text{$\#$1}^5 b+70 \text{$\#$1}^4 b+56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b+8 \text{$\#$1} b+b\& ,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left (\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right )-i \text{$\#$1}^3 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right )}{128 \text{$\#$1}^3 a+\text{$\#$1}^7 b+7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b+35 \text{$\#$1}^4 b+35 \text{$\#$1}^3 b+21 \text{$\#$1}^2 b+7 \text{$\#$1} b+b}\& \right ] \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Cos[x]^8)^(-1),x]

[Out]

8*RootSum[b + 8*b*#1 + 28*b*#1^2 + 56*b*#1^3 + 256*a*#1^4 + 70*b*#1^4 + 56*b*#1^5 + 28*b*#1^6 + 8*b*#1^7 + b*#
1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(b + 7*b*#1 + 21*b*#1
^2 + 128*a*#1^3 + 35*b*#1^3 + 35*b*#1^4 + 21*b*#1^5 + 7*b*#1^6 + b*#1^7) & ]

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Maple [C]  time = 0.091, size = 76, normalized size = 0.3 \begin{align*}{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+4\,a{{\it \_Z}}^{6}+6\,a{{\it \_Z}}^{4}+4\,a{{\it \_Z}}^{2}+a+b \right ) }{\frac{ \left ({{\it \_R}}^{6}+3\,{{\it \_R}}^{4}+3\,{{\it \_R}}^{2}+1 \right ) \ln \left ( \tan \left ( x \right ) -{\it \_R} \right ) }{{{\it \_R}}^{7}+3\,{{\it \_R}}^{5}+3\,{{\it \_R}}^{3}+{\it \_R}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cos(x)^8),x)

[Out]

1/8/a*sum((_R^6+3*_R^4+3*_R^2+1)/(_R^7+3*_R^5+3*_R^3+_R)*ln(tan(x)-_R),_R=RootOf(_Z^8*a+4*_Z^6*a+6*_Z^4*a+4*_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 )^{8} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(x)^8),x, algorithm="maxima")

[Out]

integrate(1/(b*cos(x)^8 + 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)^8),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)**8),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 )^{8} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(x)^8),x, algorithm="giac")

[Out]

integrate(1/(b*cos(x)^8 + a), x)