3.448 \(\int \frac{\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx\)

Optimal. Leaf size=112 \[ \frac{\log (\cos (x))}{2 a^4}-\frac{3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}-\frac{3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^4 n}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3-b^3 \cos ^n(x)}+a}{\sqrt{3} a}\right )}{a^4 n} \]

[Out]

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

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Rubi [A]  time = 0.288089, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{\log (\cos (x))}{2 a^4}-\frac{3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}-\frac{3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^4 n}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3-b^3 \cos ^n(x)}+a}{\sqrt{3} a}\right )}{a^4 n} \]

Antiderivative was successfully verified.

[In]  Int[Tan[x]/(a^3 - b^3*Cos[x]^n)^(4/3),x]

[Out]

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

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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(tan(x)/(a**3-b**3*cos(x)**n)**(4/3),x)

[Out]

Timed out

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Mathematica [C]  time = 0.321408, size = 71, normalized size = 0.63 \[ \frac{3 \left (\sqrt [3]{1-\frac{a^3 \cos ^{-n}(x)}{b^3}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{a^3 \cos ^{-n}(x)}{b^3}\right )-1\right )}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Tan[x]/(a^3 - b^3*Cos[x]^n)^(4/3),x]

[Out]

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

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Maple [A]  time = 0.042, size = 137, normalized size = 1.2 \[{\frac{1}{2\,n{a}^{4}}\ln \left ( \left ({a}^{3}-{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{n} \right ) ^{{\frac{2}{3}}}+a\sqrt [3]{{a}^{3}-{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{n}}+{a}^{2} \right ) }-{\frac{\sqrt{3}}{n{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( a+2\,\sqrt [3]{{a}^{3}-{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{n}} \right ) } \right ) }-3\,{\frac{1}{{a}^{3}n\sqrt [3]{{a}^{3}-{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{n}}}}-{\frac{1}{n{a}^{4}}\ln \left ( -a+\sqrt [3]{{a}^{3}-{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tan(x)/(a^3-b^3*cos(x)^n)^(4/3),x)

[Out]

1/2/n/a^4*ln((a^3-b^3*cos(x)^n)^(2/3)+a*(a^3-b^3*cos(x)^n)^(1/3)+a^2)-arctan(1/3
*(a+2*(a^3-b^3*cos(x)^n)^(1/3))/a*3^(1/2))*3^(1/2)/a^4/n-3/a^3/n/(a^3-b^3*cos(x)
^n)^(1/3)-1/n/a^4*ln(-a+(a^3-b^3*cos(x)^n)^(1/3))

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Maxima [A]  time = 1.79094, size = 184, normalized size = 1.64 \[ -\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{4} n} + \frac{\log \left (a^{2} +{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{4} n} - \frac{\log \left (-a +{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}}\right )}{a^{4} n} - \frac{3}{{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}} a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/(-b^3*cos(x)^n + a^3)^(4/3),x, algorithm="maxima")

[Out]

-sqrt(3)*arctan(1/3*sqrt(3)*(a + 2*(-b^3*cos(x)^n + a^3)^(1/3))/a)/(a^4*n) + 1/2
*log(a^2 + (-b^3*cos(x)^n + a^3)^(1/3)*a + (-b^3*cos(x)^n + a^3)^(2/3))/(a^4*n)
- log(-a + (-b^3*cos(x)^n + a^3)^(1/3))/(a^4*n) - 3/((-b^3*cos(x)^n + a^3)^(1/3)
*a^3*n)

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Fricas [A]  time = 0.26732, size = 224, normalized size = 2. \[ -\frac{2 \, \sqrt{3}{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right ) -{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}} \log \left (a^{2} +{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{2}{3}}\right ) + 2 \,{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}} \log \left (-a +{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}}\right ) + 6 \, a}{2 \,{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}} a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/(-b^3*cos(x)^n + a^3)^(4/3),x, algorithm="fricas")

[Out]

-1/2*(2*sqrt(3)*(-b^3*cos(x)^n + a^3)^(1/3)*arctan(1/3*sqrt(3)*(a + 2*(-b^3*cos(
x)^n + a^3)^(1/3))/a) - (-b^3*cos(x)^n + a^3)^(1/3)*log(a^2 + (-b^3*cos(x)^n + a
^3)^(1/3)*a + (-b^3*cos(x)^n + a^3)^(2/3)) + 2*(-b^3*cos(x)^n + a^3)^(1/3)*log(-
a + (-b^3*cos(x)^n + a^3)^(1/3)) + 6*a)/((-b^3*cos(x)^n + a^3)^(1/3)*a^4*n)

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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(tan(x)/(a**3-b**3*cos(x)**n)**(4/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan \left (x\right )}{{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)/(-b^3*cos(x)^n + a^3)^(4/3),x, algorithm="giac")

[Out]

integrate(tan(x)/(-b^3*cos(x)^n + a^3)^(4/3), x)