Integrand size = 31, antiderivative size = 88 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+m)} \] Output:
hypergeom([-n, 1+m],[2+m],-d*(1+sin(f*x+e))/(c-d))*(a+a*sin(f*x+e))^(1+m)* (c+d*sin(f*x+e))^n/a/f/(1+m)/(((c+d*sin(f*x+e))/(c-d))^n)
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+m)} \] Input:
Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]
Output:
(Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*f*(1 + m)*((c + d*Si n[e + f*x])/(c - d))^n)
Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3042, 3312, 80, 79}
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.
\(\displaystyle \int \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^ndx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c+d \sin (e+f x))^nd(a \sin (e+f x))}{a f}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(c+d \sin (e+f x))^n \left (\frac {a c+a d \sin (e+f x)}{a (c-d)}\right )^{-n} \int (\sin (e+f x) a+a)^m \left (\frac {c}{c-d}+\frac {d \sin (e+f x)}{c-d}\right )^nd(a \sin (e+f x))}{a f}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac {a c+a d \sin (e+f x)}{a (c-d)}\right )^{-n} \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d (\sin (e+f x) a+a)}{a (c-d)}\right )}{a f (m+1)}\) |
Input:
Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]
Output:
(Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + a*Sin[e + f*x]))/(a*(c - d) ))]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*f*(1 + m)*((a* c + a*d*Sin[e + f*x])/(a*(c - d)))^n)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
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]
\[\int \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{n}d x\]
Input:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x)
Output:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x)
\[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right ) \,d x } \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm="f ricas")
Output:
integral((a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n*cos(f*x + e), x)
\[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (c + d \sin {\left (e + f x \right )}\right )^{n} \cos {\left (e + f x \right )}\, dx \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))**m*(c+d*sin(f*x+e))**n,x)
Output:
Integral((a*(sin(e + f*x) + 1))**m*(c + d*sin(e + f*x))**n*cos(e + f*x), x )
\[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right ) \,d x } \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm="m axima")
Output:
integrate((a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n*cos(f*x + e), x)
\[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right ) \,d x } \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm="g iac")
Output:
integrate((a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n*cos(f*x + e), x)
Timed out. \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int \cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \] Input:
int(cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n,x)
Output:
int(cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x)
\[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\text {too large to display} \] Input:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x)
Output:
((sin(e + f*x)*d + c)**n*(sin(e + f*x)*a + a)**m*sin(e + f*x)*d*m + (sin(e + f*x)*d + c)**n*(sin(e + f*x)*a + a)**m*sin(e + f*x)*d*n + (sin(e + f*x) *d + c)**n*(sin(e + f*x)*a + a)**m*c*n + (sin(e + f*x)*d + c)**n*(sin(e + f*x)*a + a)**m*d*m - int(((sin(e + f*x)*d + c)**n*(sin(e + f*x)*a + a)**m* cos(e + f*x))/(sin(e + f*x)**2*d*m**2 + 2*sin(e + f*x)**2*d*m*n + sin(e + f*x)**2*d*m + sin(e + f*x)**2*d*n**2 + sin(e + f*x)**2*d*n + sin(e + f*x)* c*m**2 + 2*sin(e + f*x)*c*m*n + sin(e + f*x)*c*m + sin(e + f*x)*c*n**2 + s in(e + f*x)*c*n + sin(e + f*x)*d*m**2 + 2*sin(e + f*x)*d*m*n + sin(e + f*x )*d*m + sin(e + f*x)*d*n**2 + sin(e + f*x)*d*n + c*m**2 + 2*c*m*n + c*m + c*n**2 + c*n),x)*c**2*f*m**3*n - 2*int(((sin(e + f*x)*d + c)**n*(sin(e + f *x)*a + a)**m*cos(e + f*x))/(sin(e + f*x)**2*d*m**2 + 2*sin(e + f*x)**2*d* m*n + sin(e + f*x)**2*d*m + sin(e + f*x)**2*d*n**2 + sin(e + f*x)**2*d*n + sin(e + f*x)*c*m**2 + 2*sin(e + f*x)*c*m*n + sin(e + f*x)*c*m + sin(e + f *x)*c*n**2 + sin(e + f*x)*c*n + sin(e + f*x)*d*m**2 + 2*sin(e + f*x)*d*m*n + sin(e + f*x)*d*m + sin(e + f*x)*d*n**2 + sin(e + f*x)*d*n + c*m**2 + 2* c*m*n + c*m + c*n**2 + c*n),x)*c**2*f*m**2*n**2 - int(((sin(e + f*x)*d + c )**n*(sin(e + f*x)*a + a)**m*cos(e + f*x))/(sin(e + f*x)**2*d*m**2 + 2*sin (e + f*x)**2*d*m*n + sin(e + f*x)**2*d*m + sin(e + f*x)**2*d*n**2 + sin(e + f*x)**2*d*n + sin(e + f*x)*c*m**2 + 2*sin(e + f*x)*c*m*n + sin(e + f*x)* c*m + sin(e + f*x)*c*n**2 + sin(e + f*x)*c*n + sin(e + f*x)*d*m**2 + 2*...