Integrand size = 19, antiderivative size = 53 \[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{3},\frac {2}{3},\cos ^2(e+f x)\right ) \sec ^{\frac {2}{3}}(e+f x) \sin (e+f x)}{2 f \sqrt {\sin ^2(e+f x)}} \] Output:
3/2*hypergeom([-1/2, -1/3],[2/3],cos(f*x+e)^2)*sec(f*x+e)^(2/3)*sin(f*x+e) /f/(sin(f*x+e)^2)^(1/2)
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=-\frac {3 \left (-1+\sqrt [3]{\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\sin ^2(e+f x)\right )\right ) \sec ^{\frac {2}{3}}(e+f x) \sin (e+f x)}{2 f} \] Input:
Integrate[Sec[e + f*x]^(5/3)*Sin[e + f*x]^2,x]
Output:
(-3*(-1 + (Cos[e + f*x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, Sin[e + f*x]^2])*Sec[e + f*x]^(2/3)*Sin[e + f*x])/(2*f)
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3112, 3042, 3056}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^2(e+f x) \sec ^{\frac {5}{3}}(e+f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (e+f x)^{5/3}}{\csc (e+f x)^2}dx\) |
\(\Big \downarrow \) 3112 |
\(\displaystyle \cos ^{\frac {2}{3}}(e+f x) \sec ^{\frac {2}{3}}(e+f x) \int \frac {\sin ^2(e+f x)}{\cos ^{\frac {5}{3}}(e+f x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cos ^{\frac {2}{3}}(e+f x) \sec ^{\frac {2}{3}}(e+f x) \int \frac {\sin (e+f x)^2}{\cos (e+f x)^{5/3}}dx\) |
\(\Big \downarrow \) 3056 |
\(\displaystyle \frac {3 \sin (e+f x) \sec ^{\frac {2}{3}}(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{3},\frac {2}{3},\cos ^2(e+f x)\right )}{2 f \sqrt {\sin ^2(e+f x)}}\) |
Input:
Int[Sec[e + f*x]^(5/3)*Sin[e + f*x]^2,x]
Output:
(3*Hypergeometric2F1[-1/2, -1/3, 2/3, Cos[e + f*x]^2]*Sec[e + f*x]^(2/3)*S in[e + f*x])/(2*f*Sqrt[Sin[e + f*x]^2])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b^(2*IntPart[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*F racPart[(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, C os[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(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*Cos[e + f*x])^m*(b*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x]
\[\int \sec \left (f x +e \right )^{\frac {5}{3}} \sin \left (f x +e \right )^{2}d x\]
Input:
int(sec(f*x+e)^(5/3)*sin(f*x+e)^2,x)
Output:
int(sec(f*x+e)^(5/3)*sin(f*x+e)^2,x)
\[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {5}{3}} \sin \left (f x + e\right )^{2} \,d x } \] Input:
integrate(sec(f*x+e)^(5/3)*sin(f*x+e)^2,x, algorithm="fricas")
Output:
integral(-(cos(f*x + e)^2 - 1)*sec(f*x + e)^(5/3), x)
Timed out. \[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=\text {Timed out} \] Input:
integrate(sec(f*x+e)**(5/3)*sin(f*x+e)**2,x)
Output:
Timed out
\[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {5}{3}} \sin \left (f x + e\right )^{2} \,d x } \] Input:
integrate(sec(f*x+e)^(5/3)*sin(f*x+e)^2,x, algorithm="maxima")
Output:
integrate(sec(f*x + e)^(5/3)*sin(f*x + e)^2, x)
\[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int { \sec \left (f x + e\right )^{\frac {5}{3}} \sin \left (f x + e\right )^{2} \,d x } \] Input:
integrate(sec(f*x+e)^(5/3)*sin(f*x+e)^2,x, algorithm="giac")
Output:
integrate(sec(f*x + e)^(5/3)*sin(f*x + e)^2, x)
Timed out. \[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^2\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{5/3} \,d x \] Input:
int(sin(e + f*x)^2*(1/cos(e + f*x))^(5/3),x)
Output:
int(sin(e + f*x)^2*(1/cos(e + f*x))^(5/3), x)
\[ \int \sec ^{\frac {5}{3}}(e+f x) \sin ^2(e+f x) \, dx=\int \sec \left (f x +e \right )^{\frac {5}{3}} \sin \left (f x +e \right )^{2}d x \] Input:
int(sec(f*x+e)^(5/3)*sin(f*x+e)^2,x)
Output:
int(sec(e + f*x)**(2/3)*sec(e + f*x)*sin(e + f*x)**2,x)