\(\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx\) [167]

Optimal result

Integrand size = 42, antiderivative size = 124 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\frac {2^{-\frac {3}{4}-m} (g \cos (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (5+4 m),\frac {1}{4} (11+4 m),\frac {1}{4} (9+4 m),\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1}{4} (7+4 m)} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f g (5+4 m)} \] Output:

2^(-3/4-m)*(g*cos(f*x+e))^(5/2)*hypergeom([5/4+m, 11/4+m],[9/4+m],1/2+1/2* 
sin(f*x+e))*(1-sin(f*x+e))^(7/4+m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3 
-m)/f/g/(5+4*m)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.30, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3332, 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.

\(\displaystyle \int (g \cos (e+f x))^{3/2} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (g \cos (e+f x))^{3/2} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}dx\)

\(\Big \downarrow \) 3332

\(\displaystyle (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m} \int (g \cos (e+f x))^{2 m+\frac {3}{2}} (c-c \sin (e+f x))^{-2 m-3}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m} \int (g \cos (e+f x))^{2 m+\frac {3}{2}} (c-c \sin (e+f x))^{-2 m-3}dx\)

\(\Big \downarrow \) 3168

\(\displaystyle \frac {c^2 (g \cos (e+f x))^{5/2} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{\frac {1}{4} (-4 m-5)+m} (c \sin (e+f x)+c)^{\frac {1}{4} (-4 m-5)} \int (c-c \sin (e+f x))^{\frac {1}{4} (-4 m-11)} (\sin (e+f x) c+c)^{\frac {1}{4} (4 m+1)}d\sin (e+f x)}{f g}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {2^{-m-\frac {11}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m+\frac {3}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{\frac {1}{4} (-4 m-5)-\frac {3}{4}} (c \sin (e+f x)+c)^{\frac {1}{4} (-4 m-5)} \int \left (\frac {1}{2}-\frac {1}{2} \sin (e+f x)\right )^{\frac {1}{4} (-4 m-11)} (\sin (e+f x) c+c)^{\frac {1}{4} (4 m+1)}d\sin (e+f x)}{f g}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {2^{-m-\frac {3}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m+\frac {3}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{\frac {1}{4} (-4 m-5)-\frac {3}{4}} (c \sin (e+f x)+c)^{\frac {1}{4} (-4 m-5)+\frac {1}{4} (4 m+5)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (4 m+5),\frac {1}{4} (4 m+11),\frac {1}{4} (4 m+9),\frac {1}{2} (\sin (e+f x)+1)\right )}{c f g (4 m+5)}\)

Input:

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

Output:

(2^(-3/4 - m)*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(5 + 4*m)/4, (11 + 
4*m)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(3/4 + m)*(a 
 + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3/4 + (-5 - 4*m)/4)*(c + c*Sin 
[e + f*x])^((-5 - 4*m)/4 + (5 + 4*m)/4))/(c*f*g*(5 + 4*m))
 

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]
 

rule 3332

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^ 
IntPart[m]*c^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]*((c + d*Sin[e + f* 
x])^FracPart[m]/(g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m])))   Int[ 
(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a 
, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] 
&& (FractionQ[m] ||  !FractionQ[n])
 

Maple [F]

\[\int \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-3-m}d x\]

Input:

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3-m),x)
 

Output:

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3-m),x)
 

Fricas [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3} \,d x } \] Input:

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3-m), 
x, algorithm="fricas")
 

Output:

integral(sqrt(g*cos(f*x + e))*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c) 
^(-m - 3)*g*cos(f*x + e), x)
 

Sympy [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**(-3- 
m),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3} \,d x } \] Input:

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3-m), 
x, algorithm="maxima")
 

Output:

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + 
 c)^(-m - 3), x)
 

Giac [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3} \,d x } \] Input:

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3-m), 
x, algorithm="giac")
 

Output:

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + 
 c)^(-m - 3), x)
 

Mupad [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+3}} \,d x \] Input:

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^( 
m + 3),x)
 

Output:

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^( 
m + 3), x)
 

Reduce [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=-\frac {\sqrt {g}\, \left (\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )}{\left (-\sin \left (f x +e \right ) c +c \right )^{m} \sin \left (f x +e \right )^{3}-3 \left (-\sin \left (f x +e \right ) c +c \right )^{m} \sin \left (f x +e \right )^{2}+3 \left (-\sin \left (f x +e \right ) c +c \right )^{m} \sin \left (f x +e \right )-\left (-\sin \left (f x +e \right ) c +c \right )^{m}}d x \right ) g}{c^{3}} \] Input:

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3-m),x)
 

Output:

( - sqrt(g)*int(((sin(e + f*x)*a + a)**m*sqrt(cos(e + f*x))*cos(e + f*x))/ 
(( - sin(e + f*x)*c + c)**m*sin(e + f*x)**3 - 3*( - sin(e + f*x)*c + c)**m 
*sin(e + f*x)**2 + 3*( - sin(e + f*x)*c + c)**m*sin(e + f*x) - ( - sin(e + 
 f*x)*c + c)**m),x)*g)/c**3