Integrand size = 19, antiderivative size = 69 \[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=-\frac {a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1+m),\frac {1+m}{2},\sin ^2(e+f x)\right ) (a \sin (e+f x))^{-1+m}}{f (1-m) \sqrt {\cos ^2(e+f x)}} \] Output:
-a*cos(f*x+e)*hypergeom([-1/2, -1/2+1/2*m],[1/2+1/2*m],sin(f*x+e)^2)*(a*si n(f*x+e))^(-1+m)/f/(1-m)/(cos(f*x+e)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=\frac {a \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1+m),\frac {1+m}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (a \sin (e+f x))^{-1+m}}{f (-1+m)} \] Input:
Integrate[Cot[e + f*x]^2*(a*Sin[e + f*x])^m,x]
Output:
(a*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/2, (-1 + m)/2, (1 + m)/2, Sin [e + f*x]^2]*Sec[e + f*x]*(a*Sin[e + f*x])^(-1 + m))/(f*(-1 + m))
Time = 0.30 (sec) , antiderivative size = 69, 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.
\(\displaystyle \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x))^m}{\tan (e+f x)^2}dx\) |
\(\Big \downarrow \) 3080 |
\(\displaystyle a^2 \int \cos ^2(e+f x) (a \sin (e+f x))^{m-2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^2 \int \cos (e+f x)^2 (a \sin (e+f x))^{m-2}dx\) |
\(\Big \downarrow \) 3057 |
\(\displaystyle -\frac {a \cos (e+f x) (a \sin (e+f x))^{m-1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {m-1}{2},\frac {m+1}{2},\sin ^2(e+f x)\right )}{f (1-m) \sqrt {\cos ^2(e+f x)}}\) |
Input:
Int[Cot[e + f*x]^2*(a*Sin[e + f*x])^m,x]
Output:
-((a*Cos[e + f*x]*Hypergeometric2F1[-1/2, (-1 + m)/2, (1 + m)/2, Sin[e + f *x]^2]*(a*Sin[e + f*x])^(-1 + m))/(f*(1 - m)*Sqrt[Cos[e + f*x]^2]))
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]
\[\int \cot \left (f x +e \right )^{2} \left (a \sin \left (f x +e \right )\right )^{m}d x\]
Input:
int(cot(f*x+e)^2*(a*sin(f*x+e))^m,x)
Output:
int(cot(f*x+e)^2*(a*sin(f*x+e))^m,x)
\[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2} \,d x } \] Input:
integrate(cot(f*x+e)^2*(a*sin(f*x+e))^m,x, algorithm="fricas")
Output:
integral((a*sin(f*x + e))^m*cot(f*x + e)^2, x)
\[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=\int \left (a \sin {\left (e + f x \right )}\right )^{m} \cot ^{2}{\left (e + f x \right )}\, dx \] Input:
integrate(cot(f*x+e)**2*(a*sin(f*x+e))**m,x)
Output:
Integral((a*sin(e + f*x))**m*cot(e + f*x)**2, x)
\[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2} \,d x } \] Input:
integrate(cot(f*x+e)^2*(a*sin(f*x+e))^m,x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e))^m*cot(f*x + e)^2, x)
\[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2} \,d x } \] Input:
integrate(cot(f*x+e)^2*(a*sin(f*x+e))^m,x, algorithm="giac")
Output:
integrate((a*sin(f*x + e))^m*cot(f*x + e)^2, x)
Timed out. \[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:
int(cot(e + f*x)^2*(a*sin(e + f*x))^m,x)
Output:
int(cot(e + f*x)^2*(a*sin(e + f*x))^m, x)
\[ \int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx=a^{m} \left (\int \sin \left (f x +e \right )^{m} \cot \left (f x +e \right )^{2}d x \right ) \] Input:
int(cot(f*x+e)^2*(a*sin(f*x+e))^m,x)
Output:
a**m*int(sin(e + f*x)**m*cot(e + f*x)**2,x)