\(\int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx\) [289]

Optimal result

Integrand size = 21, antiderivative size = 57 \[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\frac {\cos ^2(e+f x)^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {7}{3},\frac {5}{2},\frac {7}{2},\sin ^2(e+f x)\right ) \tan ^5(e+f x)}{5 f \sqrt [3]{d \sec (e+f x)}} \]

[Out]

1/5*(cos(f*x+e)^2)^(7/3)*hypergeom([7/3, 5/2],[7/2],sin(f*x+e)^2)*tan(f*x+e)^5/f/(d*sec(f*x+e))^(1/3)

Rubi [A] (verified)

Time = 0.06 (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} \[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\frac {\cos ^2(e+f x)^{7/3} \tan ^5(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {7}{3},\frac {5}{2},\frac {7}{2},\sin ^2(e+f x)\right )}{5 f \sqrt [3]{d \sec (e+f x)}} \]

[In]

Int[Tan[e + f*x]^4/(d*Sec[e + f*x])^(1/3),x]

[Out]

((Cos[e + f*x]^2)^(7/3)*Hypergeometric2F1[7/3, 5/2, 7/2, Sin[e + f*x]^2]*Tan[e + f*x]^5)/(5*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)^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {7}{3},\frac {5}{2},\frac {7}{2},\sin ^2(e+f x)\right ) \tan ^5(e+f x)}{5 f \sqrt [3]{d \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=-\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{6},\frac {5}{6},\sec ^2(e+f x)\right ) \tan ^3(e+f x)}{f \sqrt [3]{d \sec (e+f x)} \left (-\tan ^2(e+f x)\right )^{3/2}} \]

[In]

Integrate[Tan[e + f*x]^4/(d*Sec[e + f*x])^(1/3),x]

[Out]

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

Maple [F]

\[\int \frac {\tan ^{4}\left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}d x\]

[In]

int(tan(f*x+e)^4/(d*sec(f*x+e))^(1/3),x)

[Out]

int(tan(f*x+e)^4/(d*sec(f*x+e))^(1/3),x)

Fricas [F]

\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(tan(f*x+e)^4/(d*sec(f*x+e))^(1/3),x, algorithm="fricas")

[Out]

integral((d*sec(f*x + e))^(2/3)*tan(f*x + e)^4/(d*sec(f*x + e)), x)

Sympy [F]

\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int \frac {\tan ^{4}{\left (e + f x \right )}}{\sqrt [3]{d \sec {\left (e + f x \right )}}}\, dx \]

[In]

integrate(tan(f*x+e)**4/(d*sec(f*x+e))**(1/3),x)

[Out]

Integral(tan(e + f*x)**4/(d*sec(e + f*x))**(1/3), x)

Maxima [F]

\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(tan(f*x+e)^4/(d*sec(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

integrate(tan(f*x + e)^4/(d*sec(f*x + e))^(1/3), x)

Giac [F]

\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(tan(f*x+e)^4/(d*sec(f*x+e))^(1/3),x, algorithm="giac")

[Out]

integrate(tan(f*x + e)^4/(d*sec(f*x + e))^(1/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]

[In]

int(tan(e + f*x)^4/(d/cos(e + f*x))^(1/3),x)

[Out]

int(tan(e + f*x)^4/(d/cos(e + f*x))^(1/3), x)