\(\int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx\) [273]

Optimal result

Integrand size = 19, antiderivative size = 53 \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},\cos ^2(e+f x)\right ) \sec ^{\frac {4}{3}}(e+f x) \sin (e+f x)}{4 f \sqrt {\sin ^2(e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2712, 2656} \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\frac {3 \sin (e+f x) \sec ^{\frac {4}{3}}(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},\cos ^2(e+f x)\right )}{4 f \sqrt {\sin ^2(e+f x)}} \]

[In]

Int[Sec[e + f*x]^(7/3)*Sin[e + f*x]^2,x]

[Out]

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

Rule 2656

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^(2*IntPar
t[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*FracPart[(n - 1)/2])*((a*Cos[e + f*x])^(m + 1)/(a*f*(m + 1)*(Sin[e + f*
x]^2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2], x] /; FreeQ[{a
, b, e, f, m, n}, x] && SimplerQ[n, m]

Rule 2712

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a^2/b^2)*(a*
Sec[e + f*x])^(m - 1)*(b*Csc[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1), Int[1/((a*Co
s[e + f*x])^m*(b*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt [3]{\cos (e+f x)} \sqrt [3]{\sec (e+f x)}\right ) \int \frac {\sin ^2(e+f x)}{\cos ^{\frac {7}{3}}(e+f x)} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},\cos ^2(e+f x)\right ) \sec ^{\frac {4}{3}}(e+f x) \sin (e+f x)}{4 f \sqrt {\sin ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=-\frac {3 \left (-1+\cos ^2(e+f x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\sin ^2(e+f x)\right )\right ) \sec ^{\frac {4}{3}}(e+f x) \sin (e+f x)}{4 f} \]

[In]

Integrate[Sec[e + f*x]^(7/3)*Sin[e + f*x]^2,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int \tan ^{2}{\left (e + f x \right )} \sqrt [3]{\sec {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{1/3} \,d x \]

[In]

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

[Out]

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