3.442 \(\int \frac{\tan (x)}{\sqrt [3]{a^3+b^3 \tan ^2(x)}} \, dx\)

Optimal. Leaf size=133 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{a^3-b^3}}+\frac{3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}+\frac{\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}} \]

[Out]

(Sqrt[3]*ArcTan[(1 + (2*(a^3 + b^3*Tan[x]^2)^(1/3))/(a^3 - b^3)^(1/3))/Sqrt[3]])
/(2*(a^3 - b^3)^(1/3)) + Log[Cos[x]]/(2*(a^3 - b^3)^(1/3)) + (3*Log[(a^3 - b^3)^
(1/3) - (a^3 + b^3*Tan[x]^2)^(1/3)])/(4*(a^3 - b^3)^(1/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.289968, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{a^3-b^3}}+\frac{3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}+\frac{\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}} \]

Antiderivative was successfully verified.

[In]  Int[Tan[x]/(a^3 + b^3*Tan[x]^2)^(1/3),x]

[Out]

(Sqrt[3]*ArcTan[(1 + (2*(a^3 + b^3*Tan[x]^2)^(1/3))/(a^3 - b^3)^(1/3))/Sqrt[3]])
/(2*(a^3 - b^3)^(1/3)) + Log[Cos[x]]/(2*(a^3 - b^3)^(1/3)) + (3*Log[(a^3 - b^3)^
(1/3) - (a^3 + b^3*Tan[x]^2)^(1/3)])/(4*(a^3 - b^3)^(1/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.9056, size = 167, normalized size = 1.26 \[ \frac{3 \log{\left (- \sqrt [3]{a - b} \sqrt [3]{a^{2} + a b + b^{2}} + \sqrt [3]{a^{3} + b^{3} \tan ^{2}{\left (x \right )}} \right )}}{4 \sqrt [3]{a - b} \sqrt [3]{a^{2} + a b + b^{2}}} - \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{4 \sqrt [3]{a - b} \sqrt [3]{a^{2} + a b + b^{2}}} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{1}{3} + \frac{2 \sqrt [3]{a^{3} + b^{3} \tan ^{2}{\left (x \right )}}}{3 \sqrt [3]{a - b} \sqrt [3]{a^{2} + a b + b^{2}}}\right ) \right )}}{2 \sqrt [3]{a - b} \sqrt [3]{a^{2} + a b + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(tan(x)/(a**3+b**3*tan(x)**2)**(1/3),x)

[Out]

3*log(-(a - b)**(1/3)*(a**2 + a*b + b**2)**(1/3) + (a**3 + b**3*tan(x)**2)**(1/3
))/(4*(a - b)**(1/3)*(a**2 + a*b + b**2)**(1/3)) - log(tan(x)**2 + 1)/(4*(a - b)
**(1/3)*(a**2 + a*b + b**2)**(1/3)) + sqrt(3)*atan(sqrt(3)*(1/3 + 2*(a**3 + b**3
*tan(x)**2)**(1/3)/(3*(a - b)**(1/3)*(a**2 + a*b + b**2)**(1/3))))/(2*(a - b)**(
1/3)*(a**2 + a*b + b**2)**(1/3))

_______________________________________________________________________________________

Mathematica [C]  time = 1.5005, size = 90, normalized size = 0.68 \[ -\frac{3 \sqrt [3]{\frac{\left (a^3-b^3\right ) \cos (2 x)+a^3+b^3}{b^3}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{\left (b^3-a^3\right ) \cos ^2(x)}{b^3}\right )}{2 \sqrt [3]{\sec ^2(x) \left (\left (a^3-b^3\right ) \cos (2 x)+a^3+b^3\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[Tan[x]/(a^3 + b^3*Tan[x]^2)^(1/3),x]

[Out]

(-3*((a^3 + b^3 + (a^3 - b^3)*Cos[2*x])/b^3)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4
/3, ((-a^3 + b^3)*Cos[x]^2)/b^3])/(2*((a^3 + b^3 + (a^3 - b^3)*Cos[2*x])*Sec[x]^
2)^(1/3))

_______________________________________________________________________________________

Maple [F]  time = 0.152, size = 0, normalized size = 0. \[ \int{\tan \left ( x \right ){\frac{1}{\sqrt [3]{{a}^{3}+{b}^{3} \left ( \tan \left ( x \right ) \right ) ^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tan(x)/(a^3+b^3*tan(x)^2)^(1/3),x)

[Out]

int(tan(x)/(a^3+b^3*tan(x)^2)^(1/3),x)

_______________________________________________________________________________________

Maxima [A]  time = 15.2106, size = 14, normalized size = 0.11 \[ -\frac{2 \, \cos \left (x\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/(b^3*tan(x)^2 + a^3)^(1/3),x, algorithm="maxima")

[Out]

-2*cos(x)/(a*b)

_______________________________________________________________________________________

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(tan(x)/(b^3*tan(x)^2 + a^3)^(1/3),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan{\left (x \right )}}{\sqrt [3]{a^{3} + b^{3} \tan ^{2}{\left (x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/(a**3+b**3*tan(x)**2)**(1/3),x)

[Out]

Integral(tan(x)/(a**3 + b**3*tan(x)**2)**(1/3), x)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/(b^3*tan(x)^2 + a^3)^(1/3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError