\(\int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx\) [643]

Optimal result

Integrand size = 27, antiderivative size = 132 \[ \int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx=\frac {e (e \cos (c+d x))^{-2-m} \operatorname {Hypergeometric2F1}\left (1+m,\frac {2+m}{2},2+m,\frac {2 (a+b \sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right ) (1-\sin (c+d x)) \left (-\frac {(a-b) (1-\sin (c+d x))}{(a+b) (1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{(a+b) d (1+m)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.48 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx=-\frac {e (e \cos (c+d x))^{-2-m} \operatorname {Hypergeometric2F1}\left (1+m,\frac {2+m}{2},2+m,-\frac {2 (a+b \sin (c+d x))}{(a-b) (-1+\sin (c+d x))}\right ) (1+\sin (c+d x)) \left (\frac {(a+b) (1+\sin (c+d x))}{(a-b) (-1+\sin (c+d x))}\right )^{m/2} (a+b \sin (c+d x))^{1+m}}{(a-b) d (1+m)} \] Input:

Integrate[(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^m,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {3042, 3177}

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.

\(\displaystyle \int (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^m \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3177

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

Input:

Int[(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^m,x]
 

Output:

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

Defintions of rubi rules used

rule 3042

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

rule 3177

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*(1 - Sin[e + f*x])* 
(a + b*Sin[e + f*x])^(m + 1)*(((-(a - b))*((1 - Sin[e + f*x])/((a + b)*(1 + 
 Sin[e + f*x]))))^(m/2)/(f*(a + b)*(m + 1)))*Hypergeometric2F1[m + 1, m/2 + 
 1, m + 2, 2*((a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x])))], x] /; Fr 
eeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 1, 0]
 

Maple [F]

\[\int \left (e \cos \left (d x +c \right )\right )^{-1-m} \left (a +b \sin \left (d x +c \right )\right )^{m}d x\]

Input:

int((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x)
 

Output:

int((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x)
 

Fricas [F]

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

integrate((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x, algorithm="fricas")
 

Output:

integral((e*cos(d*x + c))^(-m - 1)*(b*sin(d*x + c) + a)^m, x)
 

Sympy [F]

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

integrate((e*cos(d*x+c))**(-1-m)*(a+b*sin(d*x+c))**m,x)
 

Output:

Integral((e*cos(c + d*x))**(-m - 1)*(a + b*sin(c + d*x))**m, x)
 

Maxima [F]

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

integrate((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x, algorithm="maxima")
 

Output:

integrate((e*cos(d*x + c))^(-m - 1)*(b*sin(d*x + c) + a)^m, x)
 

Giac [F]

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

integrate((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x, algorithm="giac")
 

Output:

integrate((e*cos(d*x + c))^(-m - 1)*(b*sin(d*x + c) + a)^m, x)
 

Mupad [F(-1)]

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

int((a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 1),x)
 

Output:

int((a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 1), x)
 

Reduce [F]

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

int((e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^m,x)
 

Output:

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