3.3.86 \(\int (d \sec (e+f x))^{4/3} \tan ^4(e+f x) \, dx\) [286]

Optimal. Leaf size=57 \[ \frac {\cos ^2(e+f x)^{19/6} \, _2F_1\left (\frac {5}{2},\frac {19}{6};\frac {7}{2};\sin ^2(e+f x)\right ) (d \sec (e+f x))^{4/3} \tan ^5(e+f x)}{5 f} \]

[Out]

1/5*(cos(f*x+e)^2)^(19/6)*hypergeom([5/2, 19/6],[7/2],sin(f*x+e)^2)*(d*sec(f*x+e))^(4/3)*tan(f*x+e)^5/f

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2697} \begin {gather*} \frac {\cos ^2(e+f x)^{19/6} \tan ^5(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left (\frac {5}{2},\frac {19}{6};\frac {7}{2};\sin ^2(e+f x)\right )}{5 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[e + f*x]^2)^(19/6)*Hypergeometric2F1[5/2, 19/6, 7/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]
^5)/(5*f)

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*} \int (d \sec (e+f x))^{4/3} \tan ^4(e+f x) \, dx &=\frac {\cos ^2(e+f x)^{19/6} \, _2F_1\left (\frac {5}{2},\frac {19}{6};\frac {7}{2};\sin ^2(e+f x)\right ) (d \sec (e+f x))^{4/3} \tan ^5(e+f x)}{5 f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.68, size = 92, normalized size = 1.61 \begin {gather*} \frac {3 d \sqrt [3]{d \sec (e+f x)} \left (27 \sin (e+f x)-18 \sqrt [6]{\cos ^2(e+f x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right ) \sin (e+f x)+\sec (e+f x) \left (-16+7 \sec ^2(e+f x)\right ) \tan (e+f x)\right )}{91 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*d*(d*Sec[e + f*x])^(1/3)*(27*Sin[e + f*x] - 18*(Cos[e + f*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, Sin[
e + f*x]^2]*Sin[e + f*x] + Sec[e + f*x]*(-16 + 7*Sec[e + f*x]^2)*Tan[e + f*x]))/(91*f)

________________________________________________________________________________________

Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (d \sec \left (f x +e \right )\right )^{\frac {4}{3}} \left (\tan ^{4}\left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \sec {\left (e + f x \right )}\right )^{\frac {4}{3}} \tan ^{4}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{4/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________