3.3.85 \(\int \frac {\tan ^2(e+f x)}{(d \sec (e+f x))^{2/3}} \, dx\) [285]

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

[Out]

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

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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)^{7/6} \tan ^3(e+f x) \, _2F_1\left (\frac {7}{6},\frac {3}{2};\frac {5}{2};\sin ^2(e+f x)\right )}{3 f (d \sec (e+f x))^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.14, size = 79, normalized size = 1.39 \begin {gather*} \frac {-\frac {3}{2} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right ) \sin (2 (e+f x))+3 \cos ^2(e+f x)^{5/6} \tan (e+f x)}{f \cos ^2(e+f x)^{5/6} (d \sec (e+f x))^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{2}\left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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