Integrand size = 19, antiderivative size = 68 \[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=\frac {\sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3+m}{2},\frac {5+m}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (a \sin (e+f x))^{3+m}}{a^3 f (3+m)} \] Output:
(cos(f*x+e)^2)^(1/2)*hypergeom([3/2, 3/2+1/2*m],[5/2+1/2*m],sin(f*x+e)^2)* sec(f*x+e)*(a*sin(f*x+e))^(3+m)/a^3/f/(3+m)
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04 \[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=\frac {\sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3+m}{2},\frac {5+m}{2},\sin ^2(e+f x)\right ) \sin ^2(e+f x) (a \sin (e+f x))^m \tan (e+f x)}{f (3+m)} \] Input:
Integrate[(a*Sin[e + f*x])^m*Tan[e + f*x]^2,x]
Output:
(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(a*Sin[e + f*x])^m*Tan[e + f*x])/(f*(3 + m))
Time = 0.30 (sec) , antiderivative size = 68, 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, 3080, 3042, 3057}
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.
\(\displaystyle \int \tan ^2(e+f x) (a \sin (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (e+f x)^2 (a \sin (e+f x))^mdx\) |
\(\Big \downarrow \) 3080 |
\(\displaystyle \frac {\int \sec ^2(e+f x) (a \sin (e+f x))^{m+2}dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(a \sin (e+f x))^{m+2}}{\cos (e+f x)^2}dx}{a^2}\) |
\(\Big \downarrow \) 3057 |
\(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {m+3}{2},\frac {m+5}{2},\sin ^2(e+f x)\right )}{a^3 f (m+3)}\) |
Input:
Int[(a*Sin[e + f*x])^m*Tan[e + f*x]^2,x]
Output:
(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m))
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(n_), x_Sy mbol] :> Simp[1/a^n Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] / ; FreeQ[{a, e, f, m}, x] && IntegerQ[n] && !IntegerQ[m]
\[\int \left (a \sin \left (f x +e \right )\right )^{m} \tan \left (f x +e \right )^{2}d x\]
Input:
int((a*sin(f*x+e))^m*tan(f*x+e)^2,x)
Output:
int((a*sin(f*x+e))^m*tan(f*x+e)^2,x)
\[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2} \,d x } \] Input:
integrate((a*sin(f*x+e))^m*tan(f*x+e)^2,x, algorithm="fricas")
Output:
integral((a*sin(f*x + e))^m*tan(f*x + e)^2, x)
\[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=\int \left (a \sin {\left (e + f x \right )}\right )^{m} \tan ^{2}{\left (e + f x \right )}\, dx \] Input:
integrate((a*sin(f*x+e))**m*tan(f*x+e)**2,x)
Output:
Integral((a*sin(e + f*x))**m*tan(e + f*x)**2, x)
\[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2} \,d x } \] Input:
integrate((a*sin(f*x+e))^m*tan(f*x+e)^2,x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e))^m*tan(f*x + e)^2, x)
\[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2} \,d x } \] Input:
integrate((a*sin(f*x+e))^m*tan(f*x+e)^2,x, algorithm="giac")
Output:
integrate((a*sin(f*x + e))^m*tan(f*x + e)^2, x)
Timed out. \[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:
int(tan(e + f*x)^2*(a*sin(e + f*x))^m,x)
Output:
int(tan(e + f*x)^2*(a*sin(e + f*x))^m, x)
\[ \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx=a^{m} \left (\int \sin \left (f x +e \right )^{m} \tan \left (f x +e \right )^{2}d x \right ) \] Input:
int((a*sin(f*x+e))^m*tan(f*x+e)^2,x)
Output:
a**m*int(sin(e + f*x)**m*tan(e + f*x)**2,x)