\(\int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx\) [371]

Optimal result

Integrand size = 25, antiderivative size = 94 \[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=-\frac {\sqrt {2} (e \cos (c+d x))^{1-2 m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1-2 m),\frac {1}{2} (3-2 m),\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^m}{d e (1-2 m) \sqrt {1+\sin (c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96 \[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=\frac {\sqrt {2} \cos (c+d x) (e \cos (c+d x))^{-2 m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2}-m,\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^m}{d (-1+2 m) \sqrt {1+\sin (c+d x)}} \] Input:

Integrate[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(2*m),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3168, 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.

\(\displaystyle \int (a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m}dx\)

\(\Big \downarrow \) 3168

\(\displaystyle \frac {a^2 (a-a \sin (c+d x))^{\frac {1}{2} (2 m-1)} (a \sin (c+d x)+a)^{\frac {1}{2} (2 m-1)} (e \cos (c+d x))^{1-2 m} \int \frac {(a-a \sin (c+d x))^{\frac {1}{2} (-2 m-1)}}{\sqrt {\sin (c+d x) a+a}}d\sin (c+d x)}{d e}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {a^2 2^{-m-\frac {1}{2}} (1-\sin (c+d x))^{m+\frac {1}{2}} (a-a \sin (c+d x))^{-m+\frac {1}{2} (2 m-1)-\frac {1}{2}} (a \sin (c+d x)+a)^{\frac {1}{2} (2 m-1)} (e \cos (c+d x))^{1-2 m} \int \frac {\left (\frac {1}{2}-\frac {1}{2} \sin (c+d x)\right )^{\frac {1}{2} (-2 m-1)}}{\sqrt {\sin (c+d x) a+a}}d\sin (c+d x)}{d e}\)

\(\Big \downarrow \) 79

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

Input:

Int[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(2*m),x]
 

Output:

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

Defintions of rubi rules used

rule 79

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]))
 

rule 80

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])
 

rule 3042

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

rule 3168

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.), x_Symbol] :> Simp[a^2*((g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin 
[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2)))   Subst[Int[(a + 
b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; Fre 
eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m]
 

Maple [F]

\[\int \left (a +a \sin \left (d x +c \right )\right )^{m} \left (e \cos \left (d x +c \right )\right )^{-2 m}d x\]

Input:

int((a+a*sin(d*x+c))^m/((e*cos(d*x+c))^(2*m)),x)
 

Output:

int((a+a*sin(d*x+c))^m/((e*cos(d*x+c))^(2*m)),x)
 

Fricas [F]

\[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{2 \, m}} \,d x } \] Input:

integrate((a+a*sin(d*x+c))^m/((e*cos(d*x+c))^(2*m)),x, algorithm="fricas")
 

Output:

integral((a*sin(d*x + c) + a)^m/(e*cos(d*x + c))^(2*m), x)
 

Sympy [F]

\[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{- 2 m}\, dx \] Input:

integrate((a+a*sin(d*x+c))**m/((e*cos(d*x+c))**(2*m)),x)
 

Output:

Integral((a*(sin(c + d*x) + 1))**m/(e*cos(c + d*x))**(2*m), x)
 

Maxima [F]

\[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{2 \, m}} \,d x } \] Input:

integrate((a+a*sin(d*x+c))^m/((e*cos(d*x+c))^(2*m)),x, algorithm="maxima")
 

Output:

integrate((a*sin(d*x + c) + a)^m/(e*cos(d*x + c))^(2*m), x)
 

Giac [F]

\[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{2 \, m}} \,d x } \] Input:

integrate((a+a*sin(d*x+c))^m/((e*cos(d*x+c))^(2*m)),x, algorithm="giac")
 

Output:

integrate((a*sin(d*x + c) + a)^m/(e*cos(d*x + c))^(2*m), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{2\,m}} \,d x \] Input:

int((a + a*sin(c + d*x))^m/(e*cos(c + d*x))^(2*m),x)
 

Output:

int((a + a*sin(c + d*x))^m/(e*cos(c + d*x))^(2*m), x)
 

Reduce [F]

\[ \int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx=\frac {\int \frac {\left (\sin \left (d x +c \right ) a +a \right )^{m}}{\cos \left (d x +c \right )^{2 m}}d x}{e^{2 m}} \] Input:

int((a+a*sin(d*x+c))^m/((e*cos(d*x+c))^(2*m)),x)
 

Output:

int((sin(c + d*x)*a + a)**m/cos(c + d*x)**(2*m),x)/e**(2*m)