\(\int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx\) [617]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=-\frac {d \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}-\frac {2^{\frac {1}{2}+m} (c+c m+d m) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f (1+m)} \] Output:

-d*cos(f*x+e)*(a+a*sin(f*x+e))^m/f/(1+m)-2^(1/2+m)*(c*m+d*m+c)*cos(f*x+e)* 
hypergeom([1/2, 1/2-m],[3/2],1/2-1/2*sin(f*x+e))*(1+sin(f*x+e))^(-1/2-m)*( 
a+a*sin(f*x+e))^m/f/(1+m)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 0.58 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=\frac {2^m \left ((c-d) B_{\frac {1}{2} (1+\sin (e+f x))}\left (\frac {1}{2}+m,\frac {1}{2}\right )+2 d B_{\frac {1}{2} (1+\sin (e+f x))}\left (\frac {3}{2}+m,\frac {1}{2}\right )\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) (1+\sin (e+f x))^{-m} (a (1+\sin (e+f x)))^m}{f} \] Input:

Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x]),x]
 

Output:

(2^m*((c - d)*Beta[(1 + Sin[e + f*x])/2, 1/2 + m, 1/2] + 2*d*Beta[(1 + Sin 
[e + f*x])/2, 3/2 + m, 1/2])*Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(a*(1 + Sin 
[e + f*x]))^m)/(f*(1 + Sin[e + f*x])^m)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3230, 3042, 3131, 3042, 3130}

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.

\(\displaystyle \int (a \sin (e+f x)+a)^m (c+d \sin (e+f x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^m (c+d \sin (e+f x))dx\)

\(\Big \downarrow \) 3230

\(\displaystyle \frac {(c m+c+d m) \int (\sin (e+f x) a+a)^mdx}{m+1}-\frac {d \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c m+c+d m) \int (\sin (e+f x) a+a)^mdx}{m+1}-\frac {d \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}\)

\(\Big \downarrow \) 3131

\(\displaystyle \frac {(c m+c+d m) (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m \int (\sin (e+f x)+1)^mdx}{m+1}-\frac {d \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c m+c+d m) (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m \int (\sin (e+f x)+1)^mdx}{m+1}-\frac {d \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}\)

\(\Big \downarrow \) 3130

\(\displaystyle -\frac {2^{m+\frac {1}{2}} (c m+c+d m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f (m+1)}-\frac {d \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}\)

Input:

Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x]),x]
 

Output:

-((d*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*(c + 
 c*m + d*m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + 
 f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m) 
)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3130

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 
 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeome 
tric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; FreeQ[{a, 
 b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]
 

rule 3131

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPar 
t[n]*((a + b*Sin[c + d*x])^FracPart[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n] 
)   Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && 
EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]
 

rule 3230

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( 
f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1))   Int[(a + b*Sin[e 
 + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] 
&& EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]
 

Maple [F]

\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )d x\]

Input:

int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e)),x)
 

Output:

int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e)),x)
 

Fricas [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e)),x, algorithm="fricas")
 

Output:

integral((d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
 

Sympy [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (c + d \sin {\left (e + f x \right )}\right )\, dx \] Input:

integrate((a+a*sin(f*x+e))**m*(c+d*sin(f*x+e)),x)
 

Output:

Integral((a*(sin(e + f*x) + 1))**m*(c + d*sin(e + f*x)), x)
 

Maxima [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e)),x, algorithm="maxima")
 

Output:

integrate((d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
 

Giac [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e)),x, algorithm="giac")
 

Output:

integrate((d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \] Input:

int((a + a*sin(e + f*x))^m*(c + d*sin(e + f*x)),x)
 

Output:

int((a + a*sin(e + f*x))^m*(c + d*sin(e + f*x)), x)
 

Reduce [F]

\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx=\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m}d x \right ) c +\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sin \left (f x +e \right )d x \right ) d \] Input:

int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e)),x)
 

Output:

int((sin(e + f*x)*a + a)**m,x)*c + int((sin(e + f*x)*a + a)**m*sin(e + f*x 
),x)*d