3.3.81 \(\int (d \sec (e+f x))^{4/3} \tan ^2(e+f x) \, dx\) [281]

Optimal. Leaf size=57 \[ \frac {\cos ^2(e+f x)^{13/6} \, _2F_1\left (\frac {3}{2},\frac {13}{6};\frac {5}{2};\sin ^2(e+f x)\right ) (d \sec (e+f x))^{4/3} \tan ^3(e+f x)}{3 f} \]

[Out]

1/3*(cos(f*x+e)^2)^(13/6)*hypergeom([3/2, 13/6],[5/2],sin(f*x+e)^2)*(d*sec(f*x+e))^(4/3)*tan(f*x+e)^3/f

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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)^{13/6} \tan ^3(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left (\frac {3}{2},\frac {13}{6};\frac {5}{2};\sin ^2(e+f x)\right )}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^2,x]

[Out]

((Cos[e + f*x]^2)^(13/6)*Hypergeometric2F1[3/2, 13/6, 5/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]
^3)/(3*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 ^2(e+f x) \, dx &=\frac {\cos ^2(e+f x)^{13/6} \, _2F_1\left (\frac {3}{2},\frac {13}{6};\frac {5}{2};\sin ^2(e+f x)\right ) (d \sec (e+f x))^{4/3} \tan ^3(e+f x)}{3 f}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 80, normalized size = 1.40 \begin {gather*} \frac {3 d \sqrt [3]{d \sec (e+f x)} \left (-3 \sin (e+f x)+2 \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) \tan (e+f x)\right )}{7 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^2,x]

[Out]

(3*d*(d*Sec[e + f*x])^(1/3)*(-3*Sin[e + f*x] + 2*(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]*Tan[e + f*x]))/(7*f)

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (d \sec \left (f x +e \right )\right )^{\frac {4}{3}} \left (\tan ^{2}\left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

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

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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)^2,x, algorithm="fricas")

[Out]

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

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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 ^{2}{\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)**2,x)

[Out]

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

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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)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\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)^2*(d/cos(e + f*x))^(4/3),x)

[Out]

int(tan(e + f*x)^2*(d/cos(e + f*x))^(4/3), x)

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