3.449 \(\int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx\)

Optimal. Leaf size=95 \[ -\frac{2}{15} \left (2 \cos ^9(x)+1\right )^{5/6}+\frac{\tan ^{-1}\left (\frac{1-\sqrt [3]{2 \cos ^9(x)+1}}{\sqrt{3} \sqrt [6]{2 \cos ^9(x)+1}}\right )}{3 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )-\frac{1}{9} \tanh ^{-1}\left (\sqrt{2 \cos ^9(x)+1}\right ) \]

[Out]

ArcTan[(1 - (1 + 2*Cos[x]^9)^(1/3))/(Sqrt[3]*(1 + 2*Cos[x]^9)^(1/6))]/(3*Sqrt[3]
) + ArcTanh[(1 + 2*Cos[x]^9)^(1/6)]/3 - ArcTanh[Sqrt[1 + 2*Cos[x]^9]]/9 - (2*(1
+ 2*Cos[x]^9)^(5/6))/15

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Rubi [A]  time = 0.472983, antiderivative size = 162, normalized size of antiderivative = 1.71, number of steps used = 14, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{2}{15} \left (2 \cos ^9(x)+1\right )^{5/6}-\frac{1}{18} \log \left (\sqrt [3]{2 \cos ^9(x)+1}-\sqrt [6]{2 \cos ^9(x)+1}+1\right )+\frac{1}{18} \log \left (\sqrt [3]{2 \cos ^9(x)+1}+\sqrt [6]{2 \cos ^9(x)+1}+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{2}{9} \tanh ^{-1}\left (\sqrt [6]{2 \cos ^9(x)+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*Cos[x]^9)^(5/6)*Tan[x],x]

[Out]

ArcTan[(1 - 2*(1 + 2*Cos[x]^9)^(1/6))/Sqrt[3]]/(3*Sqrt[3]) - ArcTan[(1 + 2*(1 +
2*Cos[x]^9)^(1/6))/Sqrt[3]]/(3*Sqrt[3]) + (2*ArcTanh[(1 + 2*Cos[x]^9)^(1/6)])/9
- (2*(1 + 2*Cos[x]^9)^(5/6))/15 - Log[1 - (1 + 2*Cos[x]^9)^(1/6) + (1 + 2*Cos[x]
^9)^(1/3)]/18 + Log[1 + (1 + 2*Cos[x]^9)^(1/6) + (1 + 2*Cos[x]^9)^(1/3)]/18

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*cos(x)**9)**(5/6)*tan(x),x)

[Out]

Timed out

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Mathematica [C]  time = 25.4161, size = 924, normalized size = 9.73 \[ \frac{(126 \cos (x)+84 \cos (3 x)+36 \cos (5 x)+9 \cos (7 x)+\cos (9 x)+128)^{5/6} \tan (x) \left (5\ 2^{5/6} \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{1}{2} \sec ^2(x)^{9/2}\right ) \sqrt [6]{\sec ^8(x) \left (56 \cos (2 x)+28 \cos (4 x)+8 \cos (6 x)+\cos (8 x)+64 \sqrt{\sec ^2(x)}+35\right )}-4 \cos ^{10}(x) \left (\tan ^{10}(x)+5 \tan ^8(x)+10 \tan ^6(x)+10 \tan ^4(x)+5 \tan ^2(x)+2 \sqrt{\sec ^2(x)}+1\right )\right )}{960\ 2^{5/6} \sqrt [6]{\cos ^{10}(x) \left (\tan ^{10}(x)+5 \tan ^8(x)+10 \tan ^6(x)+10 \tan ^4(x)+5 \tan ^2(x)+2 \sqrt{\sec ^2(x)}+1\right )} \left (\frac{-4 \left (10 \sec ^2(x) \tan ^9(x)+40 \sec ^2(x) \tan ^7(x)+60 \sec ^2(x) \tan ^5(x)+40 \sec ^2(x) \tan ^3(x)+10 \sec ^2(x) \tan (x)+2 \sqrt{\sec ^2(x)} \tan (x)\right ) \cos ^{10}(x)+40 \sin (x) \left (\tan ^{10}(x)+5 \tan ^8(x)+10 \tan ^6(x)+10 \tan ^4(x)+5 \tan ^2(x)+2 \sqrt{\sec ^2(x)}+1\right ) \cos ^9(x)+\frac{15 \sqrt [6]{\sec ^8(x) \left (56 \cos (2 x)+28 \cos (4 x)+8 \cos (6 x)+\cos (8 x)+64 \sqrt{\sec ^2(x)}+35\right )} \left (\frac{1}{\sqrt [6]{\frac{1}{2} \sec ^2(x)^{9/2}+1}}-\, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{1}{2} \sec ^2(x)^{9/2}\right )\right ) \tan (x)}{\sqrt [6]{2}}+\frac{5 \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{1}{2} \sec ^2(x)^{9/2}\right ) \left (8 \left (56 \cos (2 x)+28 \cos (4 x)+8 \cos (6 x)+\cos (8 x)+64 \sqrt{\sec ^2(x)}+35\right ) \tan (x) \sec ^8(x)+\left (-112 \sin (2 x)-112 \sin (4 x)-48 \sin (6 x)-8 \sin (8 x)+64 \sqrt{\sec ^2(x)} \tan (x)\right ) \sec ^8(x)\right )}{3 \sqrt [6]{2} \left (\sec ^8(x) \left (56 \cos (2 x)+28 \cos (4 x)+8 \cos (6 x)+\cos (8 x)+64 \sqrt{\sec ^2(x)}+35\right )\right )^{5/6}}}{30 \sqrt [6]{\cos ^{10}(x) \left (\tan ^{10}(x)+5 \tan ^8(x)+10 \tan ^6(x)+10 \tan ^4(x)+5 \tan ^2(x)+2 \sqrt{\sec ^2(x)}+1\right )}}-\frac{\left (5\ 2^{5/6} \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{1}{2} \sec ^2(x)^{9/2}\right ) \sqrt [6]{\sec ^8(x) \left (56 \cos (2 x)+28 \cos (4 x)+8 \cos (6 x)+\cos (8 x)+64 \sqrt{\sec ^2(x)}+35\right )}-4 \cos ^{10}(x) \left (\tan ^{10}(x)+5 \tan ^8(x)+10 \tan ^6(x)+10 \tan ^4(x)+5 \tan ^2(x)+2 \sqrt{\sec ^2(x)}+1\right )\right ) \left (\cos ^{10}(x) \left (10 \sec ^2(x) \tan ^9(x)+40 \sec ^2(x) \tan ^7(x)+60 \sec ^2(x) \tan ^5(x)+40 \sec ^2(x) \tan ^3(x)+10 \sec ^2(x) \tan (x)+2 \sqrt{\sec ^2(x)} \tan (x)\right )-10 \cos ^9(x) \sin (x) \left (\tan ^{10}(x)+5 \tan ^8(x)+10 \tan ^6(x)+10 \tan ^4(x)+5 \tan ^2(x)+2 \sqrt{\sec ^2(x)}+1\right )\right )}{180 \left (\cos ^{10}(x) \left (\tan ^{10}(x)+5 \tan ^8(x)+10 \tan ^6(x)+10 \tan ^4(x)+5 \tan ^2(x)+2 \sqrt{\sec ^2(x)}+1\right )\right )^{7/6}}\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 + 2*Cos[x]^9)^(5/6)*Tan[x],x]

[Out]

((128 + 126*Cos[x] + 84*Cos[3*x] + 36*Cos[5*x] + 9*Cos[7*x] + Cos[9*x])^(5/6)*Ta
n[x]*(5*2^(5/6)*Hypergeometric2F1[1/6, 1/6, 7/6, -(Sec[x]^2)^(9/2)/2]*(Sec[x]^8*
(35 + 56*Cos[2*x] + 28*Cos[4*x] + 8*Cos[6*x] + Cos[8*x] + 64*Sqrt[Sec[x]^2]))^(1
/6) - 4*Cos[x]^10*(1 + 2*Sqrt[Sec[x]^2] + 5*Tan[x]^2 + 10*Tan[x]^4 + 10*Tan[x]^6
+ 5*Tan[x]^8 + Tan[x]^10)))/(960*2^(5/6)*(Cos[x]^10*(1 + 2*Sqrt[Sec[x]^2] + 5*T
an[x]^2 + 10*Tan[x]^4 + 10*Tan[x]^6 + 5*Tan[x]^8 + Tan[x]^10))^(1/6)*(-((5*2^(5/
6)*Hypergeometric2F1[1/6, 1/6, 7/6, -(Sec[x]^2)^(9/2)/2]*(Sec[x]^8*(35 + 56*Cos[
2*x] + 28*Cos[4*x] + 8*Cos[6*x] + Cos[8*x] + 64*Sqrt[Sec[x]^2]))^(1/6) - 4*Cos[x
]^10*(1 + 2*Sqrt[Sec[x]^2] + 5*Tan[x]^2 + 10*Tan[x]^4 + 10*Tan[x]^6 + 5*Tan[x]^8
+ Tan[x]^10))*(Cos[x]^10*(10*Sec[x]^2*Tan[x] + 2*Sqrt[Sec[x]^2]*Tan[x] + 40*Sec
[x]^2*Tan[x]^3 + 60*Sec[x]^2*Tan[x]^5 + 40*Sec[x]^2*Tan[x]^7 + 10*Sec[x]^2*Tan[x
]^9) - 10*Cos[x]^9*Sin[x]*(1 + 2*Sqrt[Sec[x]^2] + 5*Tan[x]^2 + 10*Tan[x]^4 + 10*
Tan[x]^6 + 5*Tan[x]^8 + Tan[x]^10)))/(180*(Cos[x]^10*(1 + 2*Sqrt[Sec[x]^2] + 5*T
an[x]^2 + 10*Tan[x]^4 + 10*Tan[x]^6 + 5*Tan[x]^8 + Tan[x]^10))^(7/6)) + ((15*(Se
c[x]^8*(35 + 56*Cos[2*x] + 28*Cos[4*x] + 8*Cos[6*x] + Cos[8*x] + 64*Sqrt[Sec[x]^
2]))^(1/6)*(-Hypergeometric2F1[1/6, 1/6, 7/6, -(Sec[x]^2)^(9/2)/2] + (1 + (Sec[x
]^2)^(9/2)/2)^(-1/6))*Tan[x])/2^(1/6) - 4*Cos[x]^10*(10*Sec[x]^2*Tan[x] + 2*Sqrt
[Sec[x]^2]*Tan[x] + 40*Sec[x]^2*Tan[x]^3 + 60*Sec[x]^2*Tan[x]^5 + 40*Sec[x]^2*Ta
n[x]^7 + 10*Sec[x]^2*Tan[x]^9) + 40*Cos[x]^9*Sin[x]*(1 + 2*Sqrt[Sec[x]^2] + 5*Ta
n[x]^2 + 10*Tan[x]^4 + 10*Tan[x]^6 + 5*Tan[x]^8 + Tan[x]^10) + (5*Hypergeometric
2F1[1/6, 1/6, 7/6, -(Sec[x]^2)^(9/2)/2]*(8*Sec[x]^8*(35 + 56*Cos[2*x] + 28*Cos[4
*x] + 8*Cos[6*x] + Cos[8*x] + 64*Sqrt[Sec[x]^2])*Tan[x] + Sec[x]^8*(-112*Sin[2*x
] - 112*Sin[4*x] - 48*Sin[6*x] - 8*Sin[8*x] + 64*Sqrt[Sec[x]^2]*Tan[x])))/(3*2^(
1/6)*(Sec[x]^8*(35 + 56*Cos[2*x] + 28*Cos[4*x] + 8*Cos[6*x] + Cos[8*x] + 64*Sqrt
[Sec[x]^2]))^(5/6)))/(30*(Cos[x]^10*(1 + 2*Sqrt[Sec[x]^2] + 5*Tan[x]^2 + 10*Tan[
x]^4 + 10*Tan[x]^6 + 5*Tan[x]^8 + Tan[x]^10))^(1/6))))

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Maple [F]  time = 0.145, size = 0, normalized size = 0. \[ \int \left ( 1+2\, \left ( \cos \left ( x \right ) \right ) ^{9} \right ) ^{{\frac{5}{6}}}\tan \left ( x \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*cos(x)^9)^(5/6)*tan(x),x)

[Out]

int((1+2*cos(x)^9)^(5/6)*tan(x),x)

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Maxima [A]  time = 1.51956, size = 196, normalized size = 2.06 \[ -\frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right )}\right ) - \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} - 1\right )}\right ) - \frac{2}{15} \,{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{5}{6}} + \frac{1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{3}} +{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right ) - \frac{1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{3}} -{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right ) + \frac{1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right ) - \frac{1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*cos(x)^9 + 1)^(5/6)*tan(x),x, algorithm="maxima")

[Out]

-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) + 1)) - 1/9*sqrt(3)*ar
ctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) - 1)) - 2/15*(2*cos(x)^9 + 1)^(5/6) +
1/18*log((2*cos(x)^9 + 1)^(1/3) + (2*cos(x)^9 + 1)^(1/6) + 1) - 1/18*log((2*cos
(x)^9 + 1)^(1/3) - (2*cos(x)^9 + 1)^(1/6) + 1) + 1/9*log((2*cos(x)^9 + 1)^(1/6)
+ 1) - 1/9*log((2*cos(x)^9 + 1)^(1/6) - 1)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*cos(x)^9 + 1)^(5/6)*tan(x),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*cos(x)**9)**(5/6)*tan(x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.241561, size = 197, normalized size = 2.07 \[ -\frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right )}\right ) - \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} - 1\right )}\right ) - \frac{2}{15} \,{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{5}{6}} + \frac{1}{18} \,{\rm ln}\left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{3}} +{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right ) - \frac{1}{18} \,{\rm ln}\left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{3}} -{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right ) + \frac{1}{9} \,{\rm ln}\left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} + 1\right ) - \frac{1}{9} \,{\rm ln}\left ({\left |{\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac{1}{6}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*cos(x)^9 + 1)^(5/6)*tan(x),x, algorithm="giac")

[Out]

-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) + 1)) - 1/9*sqrt(3)*ar
ctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) - 1)) - 2/15*(2*cos(x)^9 + 1)^(5/6) +
1/18*ln((2*cos(x)^9 + 1)^(1/3) + (2*cos(x)^9 + 1)^(1/6) + 1) - 1/18*ln((2*cos(x
)^9 + 1)^(1/3) - (2*cos(x)^9 + 1)^(1/6) + 1) + 1/9*ln((2*cos(x)^9 + 1)^(1/6) + 1
) - 1/9*ln(abs((2*cos(x)^9 + 1)^(1/6) - 1))