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[C
os[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))

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Rubi [A]  time = 0.155863, 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, Rules used = {3670, 444, 55, 617, 204, 31} \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{a^3-b^3}}+\frac{3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}+\frac{\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}} \]

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[C
os[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))

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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

Mathematica [A]  time = 0.220266, size = 105, normalized size = 0.79 \[ \frac{2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}+1}{\sqrt{3}}\right )+3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )+2 \log (\cos (x))}{4 \sqrt [3]{a^3-b^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \begin{align*} \int{\tan \left ( x \right ){\frac{1}{\sqrt [3]{{a}^{3}+{b}^{3} \left ( \tan \left ( x \right ) \right ) ^{2}}}}}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (x\right )}{{\left (b^{3} \tan \left (x\right )^{2} + a^{3}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [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*tan(x)^2)^(1/3),x, algorithm="fricas")

[Out]

Timed out

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

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)

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Giac [A]  time = 1.2334, size = 251, normalized size = 1.89 \begin{align*} \frac{3 \,{\left (a^{3} - b^{3}\right )}^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b^{3} \tan \left (x\right )^{2} + a^{3}\right )}^{\frac{1}{3}} +{\left (a^{3} - b^{3}\right )}^{\frac{1}{3}}\right )}}{3 \,{\left (a^{3} - b^{3}\right )}^{\frac{1}{3}}}\right )}{2 \,{\left (\sqrt{3} a^{3} - \sqrt{3} b^{3}\right )}} - \frac{\log \left ({\left (b^{3} \tan \left (x\right )^{2} + a^{3}\right )}^{\frac{2}{3}} +{\left (b^{3} \tan \left (x\right )^{2} + a^{3}\right )}^{\frac{1}{3}}{\left (a^{3} - b^{3}\right )}^{\frac{1}{3}} +{\left (a^{3} - b^{3}\right )}^{\frac{2}{3}}\right )}{4 \,{\left (a^{3} - b^{3}\right )}^{\frac{1}{3}}} + \frac{\log \left ({\left |{\left (b^{3} \tan \left (x\right )^{2} + a^{3}\right )}^{\frac{1}{3}} -{\left (a^{3} - b^{3}\right )}^{\frac{1}{3}} \right |}\right )}{2 \,{\left (a^{3} - b^{3}\right )}^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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