Integrand size = 31, antiderivative size = 59 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d) f (1+m)} \] Output:
hypergeom([1, 1+m],[2+m],-d*(1+sin(f*x+e))/(c-d))*(a+a*sin(f*x+e))^(1+m)/a /(c-d)/f/(1+m)
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d) f (1+m)} \] Input:
Integrate[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x]),x]
Output:
(Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)*f*(1 + m))
Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3042, 3312, 27, 78}
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.
\(\displaystyle \int \frac {\cos (e+f x) (a \sin (e+f x)+a)^m}{c+d \sin (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (e+f x) (a \sin (e+f x)+a)^m}{c+d \sin (e+f x)}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \frac {a (\sin (e+f x) a+a)^m}{a c+a d \sin (e+f x)}d(a \sin (e+f x))}{a f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m}{a c+a d \sin (e+f x)}d(a \sin (e+f x))}{f}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {(a \sin (e+f x)+a)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (\sin (e+f x) a+a)}{a (c-d)}\right )}{a f (m+1) (c-d)}\) |
Input:
Int[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x]),x]
Output:
(Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + a*Sin[e + f*x]))/(a*(c - d)) )]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)*f*(1 + m))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
\[\int \frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{c +d \sin \left (f x +e \right )}d x\]
Input:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x)
Output:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x)
\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d \sin \left (f x + e\right ) + c} \,d x } \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm="fri cas")
Output:
integral((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c), x)
Timed out. \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))**m/(c+d*sin(f*x+e)),x)
Output:
Timed out
\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d \sin \left (f x + e\right ) + c} \,d x } \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm="max ima")
Output:
integrate((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c), x)
\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{d \sin \left (f x + e\right ) + c} \,d x } \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm="gia c")
Output:
integrate((a*sin(f*x + e) + a)^m*cos(f*x + e)/(d*sin(f*x + e) + c), x)
Timed out. \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\int \frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{c+d\,\sin \left (e+f\,x\right )} \,d x \] Input:
int((cos(e + f*x)*(a + a*sin(e + f*x))^m)/(c + d*sin(e + f*x)),x)
Output:
int((cos(e + f*x)*(a + a*sin(e + f*x))^m)/(c + d*sin(e + f*x)), x)
\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx=\frac {\left (a +a \sin \left (f x +e \right )\right )^{m}-\left (\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \cos \left (f x +e \right )}{\sin \left (f x +e \right )^{2} d +\sin \left (f x +e \right ) c +\sin \left (f x +e \right ) d +c}d x \right ) c f m +\left (\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \cos \left (f x +e \right )}{\sin \left (f x +e \right )^{2} d +\sin \left (f x +e \right ) c +\sin \left (f x +e \right ) d +c}d x \right ) d f m}{d f m} \] Input:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x)
Output:
((sin(e + f*x)*a + a)**m - int(((sin(e + f*x)*a + a)**m*cos(e + f*x))/(sin (e + f*x)**2*d + sin(e + f*x)*c + sin(e + f*x)*d + c),x)*c*f*m + int(((sin (e + f*x)*a + a)**m*cos(e + f*x))/(sin(e + f*x)**2*d + sin(e + f*x)*c + si n(e + f*x)*d + c),x)*d*f*m)/(d*f*m)