\(\int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx\) [284]

Optimal result

Integrand size = 21, antiderivative size = 57 \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\frac {\cos ^2(e+f x)^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {5}{2},\sin ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f \sqrt [3]{d \sec (e+f x)}} \]

[Out]

1/3*(cos(f*x+e)^2)^(4/3)*hypergeom([4/3, 3/2],[5/2],sin(f*x+e)^2)*tan(f*x+e)^3/f/(d*sec(f*x+e))^(1/3)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2697} \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\frac {\cos ^2(e+f x)^{4/3} \tan ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {5}{2},\sin ^2(e+f x)\right )}{3 f \sqrt [3]{d \sec (e+f x)}} \]

[In]

Int[Tan[e + f*x]^2/(d*Sec[e + f*x])^(1/3),x]

[Out]

((Cos[e + f*x]^2)^(4/3)*Hypergeometric2F1[4/3, 3/2, 5/2, Sin[e + f*x]^2]*Tan[e + f*x]^3)/(3*f*(d*Sec[e + f*x])
^(1/3))

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^2(e+f x)^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {5}{2},\sin ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f \sqrt [3]{d \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=-\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{6},\frac {5}{6},\sec ^2(e+f x)\right ) \tan (e+f x)}{f \sqrt [3]{d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}} \]

[In]

Integrate[Tan[e + f*x]^2/(d*Sec[e + f*x])^(1/3),x]

[Out]

(-3*Hypergeometric2F1[-1/2, -1/6, 5/6, Sec[e + f*x]^2]*Tan[e + f*x])/(f*(d*Sec[e + f*x])^(1/3)*Sqrt[-Tan[e + f
*x]^2])

Maple [F]

\[\int \frac {\tan ^{2}\left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}d x\]

[In]

int(tan(f*x+e)^2/(d*sec(f*x+e))^(1/3),x)

[Out]

int(tan(f*x+e)^2/(d*sec(f*x+e))^(1/3),x)

Fricas [F]

\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(tan(f*x+e)^2/(d*sec(f*x+e))^(1/3),x, algorithm="fricas")

[Out]

integral((d*sec(f*x + e))^(2/3)*tan(f*x + e)^2/(d*sec(f*x + e)), x)

Sympy [F]

\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int \frac {\tan ^{2}{\left (e + f x \right )}}{\sqrt [3]{d \sec {\left (e + f x \right )}}}\, dx \]

[In]

integrate(tan(f*x+e)**2/(d*sec(f*x+e))**(1/3),x)

[Out]

Integral(tan(e + f*x)**2/(d*sec(e + f*x))**(1/3), x)

Maxima [F]

\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(tan(f*x+e)^2/(d*sec(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

integrate(tan(f*x + e)^2/(d*sec(f*x + e))^(1/3), x)

Giac [F]

\[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(tan(f*x+e)^2/(d*sec(f*x+e))^(1/3),x, algorithm="giac")

[Out]

integrate(tan(f*x + e)^2/(d*sec(f*x + e))^(1/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^2(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]

[In]

int(tan(e + f*x)^2/(d/cos(e + f*x))^(1/3),x)

[Out]

int(tan(e + f*x)^2/(d/cos(e + f*x))^(1/3), x)