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

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

[Out]

-ArcTan[(Sqrt[(-a)^(1/4) - b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - b^(1/4)]) - ArcTan[(Sq
rt[(-a)^(1/4) - I*b^(1/4)]*Tan[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)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTan[(Sqrt[(-a)^(1/4)
+ b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)])

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Rubi [A]  time = 0.527758, 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} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+i \sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}} \tan (x)}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a \sqrt [4]{b}+(-a)^{5/4}} \tan (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt{a \sqrt [4]{b}+(-a)^{5/4}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[x]^8)^(-1),x]

[Out]

-ArcTan[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Tan[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)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTan[(Sqrt[
(-a)^(1/4) + b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)]) - ArcTan[(Sqrt[(-a)^(5/4)
+ a*b^(1/4)]*Tan[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 \sin ^8(x)} \, dx &=\frac{\int \frac{1}{1-\frac{\sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1-\frac{i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1+\frac{i \sqrt [4]{b} \sin ^2(x)}{\sqrt [4]{-a}}} \, dx}{4 a}+\frac{\int \frac{1}{1+\frac{\sqrt [4]{b} \sin ^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,\tan (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,\tan (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,\tan (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,\tan (x)\right )}{4 a}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt [4]{-a}-i \sqrt [4]{b}} \tan (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}} \tan (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}} \tan (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}} \tan (x)}{(-a)^{5/8}}\right )}{4 (-a)^{3/8} \sqrt{(-a)^{5/4}+a \sqrt [4]{b}}}\\ \end{align*}

Mathematica [C]  time = 0.265741, size = 174, normalized size = 0.71 \[ 8 \text{RootSum}\left [256 \text{$\#$1}^4 a+\text{$\#$1}^8 b-8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b-56 \text{$\#$1}^5 b+70 \text{$\#$1}^4 b-56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b-8 \text{$\#$1} b+b\& ,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left (\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right )-i \text{$\#$1}^3 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right )}{128 \text{$\#$1}^3 a+\text{$\#$1}^7 b-7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b-35 \text{$\#$1}^4 b+35 \text{$\#$1}^3 b-21 \text{$\#$1}^2 b+7 \text{$\#$1} b-b}\& \right ] \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sin[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 = 0.118, size = 85, normalized size = 0.4 \begin{align*}{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a+b \right ){{\it \_Z}}^{8}+4\,a{{\it \_Z}}^{6}+6\,a{{\it \_Z}}^{4}+4\,a{{\it \_Z}}^{2}+a \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}a+{{\it \_R}}^{7}b+3\,{{\it \_R}}^{5}a+3\,{{\it \_R}}^{3}a+{\it \_R}\,a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \sin \left (x\right )^{8} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*sin(x)^8 + a), x)