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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]