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

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

[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 = 0.168556, 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, Rules used = {3230, 266, 51, 55, 617, 204, 31} \[ -\frac{3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}-\frac{3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^4 n}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3-b^3 \cos ^n(x)}+a}{\sqrt{3} a}\right )}{a^4 n}+\frac{\log (\cos (x))}{2 a^4} \]

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)

Rule 3230

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + b*(c*ff*x)^n)^p)/(1 - ff^2*x^2)^(
(m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\tan (x)}{\left (a^3-b^3 \cos ^n(x)\right )^{4/3}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x \left (a^3-b^3 x^n\right )^{4/3}} \, dx,x,\cos (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a^3-b^3 x\right )^{4/3}} \, dx,x,\cos ^n(x)\right )}{n}\\ &=-\frac{3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a^3-b^3 x}} \, dx,x,\cos ^n(x)\right )}{a^3 n}\\ &=-\frac{3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}+\frac{\log (\cos (x))}{2 a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a-x} \, dx,x,\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^4 n}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^2+a x+x^2} \, dx,x,\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^3 n}\\ &=-\frac{3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}+\frac{\log (\cos (x))}{2 a^4}-\frac{3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^4 n}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a^3-b^3 \cos ^n(x)}}{a}\right )}{a^4 n}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a^3-b^3 \cos ^n(x)}}{a}}{\sqrt{3}}\right )}{a^4 n}-\frac{3}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}}+\frac{\log (\cos (x))}{2 a^4}-\frac{3 \log \left (a-\sqrt [3]{a^3-b^3 \cos ^n(x)}\right )}{2 a^4 n}\\ \end{align*}

Mathematica [C]  time = 0.0420222, size = 47, normalized size = 0.42 \[ -\frac{3 \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};1-\frac{b^3 \cos ^n(x)}{a^3}\right )}{a^3 n \sqrt [3]{a^3-b^3 \cos ^n(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.027, size = 137, normalized size = 1.2 \begin{align*}{\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 ) }-{\frac{1}{n{a}^{4}}\ln \left ( -a+\sqrt [3]{{a}^{3}-{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{n}} \right ) }-3\,{\frac{1}{{a}^{3}n\sqrt [3]{{a}^{3}-{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{n}}}} \end{align*}

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

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Maxima [A]  time = 1.42737, size = 184, normalized size = 1.64 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a^3-b^3*cos(x)^n)^(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*c
os(x)^n + a^3)^(1/3)*a^3*n)

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Fricas [A]  time = 2.95077, size = 450, normalized size = 4.02 \begin{align*} -\frac{2 \,{\left (\sqrt{3} b^{3} \cos \left (x\right )^{n} - \sqrt{3} a^{3}\right )} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}}}{3 \, a}\right ) -{\left (b^{3} \cos \left (x\right )^{n} - a^{3}\right )} \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 )} \log \left (-a +{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{1}{3}}\right ) - 6 \,{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{2}{3}} a}{2 \,{\left (a^{4} b^{3} n \cos \left (x\right )^{n} - a^{7} n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(sqrt(3)*b^3*cos(x)^n - sqrt(3)*a^3)*arctan(1/3*(sqrt(3)*a + 2*sqrt(3)*(-b^3*cos(x)^n + a^3)^(1/3))/a)
- (b^3*cos(x)^n - a^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)*log(-a + (-b^3*cos(x)^n + a^3)^(1/3)) - 6*(-b^3*cos(x)^n + a^3)^(2/3)*a)/(a^4*b^3*n*cos(x)^n - a^7*n
)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (x\right )}{{\left (-b^{3} \cos \left (x\right )^{n} + a^{3}\right )}^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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