\(\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx\) [188]

Optimal result

Integrand size = 45, antiderivative size = 290 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\frac {(g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n}}{f g (6+m-n)}+\frac {3 (g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n}}{c f g (4+m-n) (6+m-n)}+\frac {6 (g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{c^2 f g (2+m-n) (4+m-n) (6+m-n)}+\frac {6 (g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{c^3 f g (m-n) (2+m-n) (4+m-n) (6+m-n)} \] Output:

(g*cos(f*x+e))^(-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3+n)/f/g/(6+m- 
n)+3*(g*cos(f*x+e))^(-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2+n)/c/f/ 
g/(4+m-n)/(6+m-n)+6*(g*cos(f*x+e))^(-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+ 
e))^(-1+n)/c^2/f/g/(2+m-n)/(4+m-n)/(6+m-n)+6*(g*cos(f*x+e))^(-m-n)*(a+a*si 
n(f*x+e))^m*(c-c*sin(f*x+e))^n/c^3/f/g/(m-n)/(2+m-n)/(4+m-n)/(6+m-n)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 23.20 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.67 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\frac {(g \cos (e+f x))^{-m-n} (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^n \left (-30-46 m-18 m^2-2 m^3+46 n+36 m n+6 m^2 n-18 n^2-6 m n^2+2 n^3+6 (3+m-n) \cos (2 (e+f x))+3 \left (15+2 m^2-4 m (-3+n)-12 n+2 n^2\right ) \sin (e+f x)-3 \sin (3 (e+f x))\right )}{2 c^3 f g (m-n) (2+m-n) (4+m-n) (6+m-n) (-1+\sin (e+f x))^3} \] Input:

Integrate[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[ 
e + f*x])^(-3 + n),x]
 

Output:

((g*Cos[e + f*x])^(-m - n)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n 
*(-30 - 46*m - 18*m^2 - 2*m^3 + 46*n + 36*m*n + 6*m^2*n - 18*n^2 - 6*m*n^2 
 + 2*n^3 + 6*(3 + m - n)*Cos[2*(e + f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 
 12*n + 2*n^2)*Sin[e + f*x] - 3*Sin[3*(e + f*x)]))/(2*c^3*f*g*(m - n)*(2 + 
 m - n)*(4 + m - n)*(6 + m - n)*(-1 + Sin[e + f*x])^3)
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {3042, 3328, 3042, 3328, 3042, 3328, 3042, 3327}

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.

\(\displaystyle \int (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n-1} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n-1}dx\)

\(\Big \downarrow \) 3328

\(\displaystyle \frac {3 \int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{n-2}dx}{c (m-n+6)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{n-2}dx}{c (m-n+6)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}\)

\(\Big \downarrow \) 3328

\(\displaystyle \frac {3 \left (\frac {2 \int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{n-1}dx}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\right )}{c (m-n+6)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (\frac {2 \int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{n-1}dx}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\right )}{c (m-n+6)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}\)

\(\Big \downarrow \) 3328

\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^ndx}{c (m-n+2)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}\right )}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\right )}{c (m-n+6)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^ndx}{c (m-n+2)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}\right )}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\right )}{c (m-n+6)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}\)

\(\Big \downarrow \) 3327

\(\displaystyle \frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}+\frac {3 \left (\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}+\frac {2 \left (\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c f g (m-n) (m-n+2)}\right )}{c (m-n+4)}\right )}{c (m-n+6)}\)

Input:

Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f* 
x])^(-3 + n),x]
 

Output:

((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 
 + n))/(f*g*(6 + m - n)) + (3*(((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f 
*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(f*g*(4 + m - n)) + (2*(((g*Cos[e + 
f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*g* 
(2 + m - n)) + ((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Si 
n[e + f*x])^n)/(c*f*g*(m - n)*(2 + m - n))))/(c*(4 + m - n))))/(c*(6 + m - 
 n))
 

Defintions of rubi rules used

rule 3042

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

rule 3327

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b* 
(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(a* 
f*g*(m - n))), 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] && EqQ[m + n + p + 1, 0] && NeQ[m, n]
 

rule 3328

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b* 
(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(a* 
f*g*(2*m + p + 1))), x] + Simp[(m + n + p + 1)/(a*(2*m + p + 1))   Int[(g*C 
os[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, 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] && ILtQ[Simplify[m + n + p + 1], 0] && NeQ[2*m + p + 1, 0] && (S 
umSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

Maple [F]

\[\int \left (g \cos \left (f x +e \right )\right )^{-1-m -n} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-3+n}d x\]

Input:

int((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3+n),x)
 

Output:

int((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3+n),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.91 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=-\frac {{\left (6 \, {\left (m - n + 3\right )} \cos \left (f x + e\right )^{3} - {\left (m^{3} + 3 \, {\left (m + 3\right )} n^{2} - n^{3} + 9 \, m^{2} - {\left (3 \, m^{2} + 18 \, m + 26\right )} n + 26 \, m + 24\right )} \cos \left (f x + e\right ) - 3 \, {\left (2 \, \cos \left (f x + e\right )^{3} - {\left (m^{2} - 2 \, {\left (m + 3\right )} n + n^{2} + 6 \, m + 8\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} e^{\left (2 \, {\left (n - 3\right )} \log \left (g \cos \left (f x + e\right )\right ) - {\left (n - 3\right )} \log \left (a \sin \left (f x + e\right ) + a\right ) + {\left (n - 3\right )} \log \left (\frac {a c}{g^{2}}\right )\right )}}{f m^{4} + f n^{4} + 12 \, f m^{3} - 4 \, {\left (f m + 3 \, f\right )} n^{3} + 44 \, f m^{2} + 2 \, {\left (3 \, f m^{2} + 18 \, f m + 22 \, f\right )} n^{2} + 48 \, f m - 4 \, {\left (f m^{3} + 9 \, f m^{2} + 22 \, f m + 12 \, f\right )} n} \] Input:

integrate((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3+ 
n),x, algorithm="fricas")
 

Output:

-(6*(m - n + 3)*cos(f*x + e)^3 - (m^3 + 3*(m + 3)*n^2 - n^3 + 9*m^2 - (3*m 
^2 + 18*m + 26)*n + 26*m + 24)*cos(f*x + e) - 3*(2*cos(f*x + e)^3 - (m^2 - 
 2*(m + 3)*n + n^2 + 6*m + 8)*cos(f*x + e))*sin(f*x + e))*(g*cos(f*x + e)) 
^(-m - n - 1)*(a*sin(f*x + e) + a)^m*e^(2*(n - 3)*log(g*cos(f*x + e)) - (n 
 - 3)*log(a*sin(f*x + e) + a) + (n - 3)*log(a*c/g^2))/(f*m^4 + f*n^4 + 12* 
f*m^3 - 4*(f*m + 3*f)*n^3 + 44*f*m^2 + 2*(3*f*m^2 + 18*f*m + 22*f)*n^2 + 4 
8*f*m - 4*(f*m^3 + 9*f*m^2 + 22*f*m + 12*f)*n)
 

Sympy [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))**(-1-m-n)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**( 
-3+n),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3+ 
n),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n - 3} \,d x } \] Input:

integrate((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3+ 
n),x, algorithm="giac")
 

Output:

integrate((g*cos(f*x + e))^(-m - n - 1)*(a*sin(f*x + e) + a)^m*(-c*sin(f*x 
 + e) + c)^(n - 3), x)
 

Mupad [B] (verification not implemented)

Time = 23.90 (sec) , antiderivative size = 1623, normalized size of antiderivative = 5.60 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\text {Too large to display} \] Input:

int(((a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n - 3))/(g*cos(e + f*x)) 
^(m + n + 1),x)
 

Output:

-exp(- e*4i - f*x*4i)*(c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f* 
x*1i)*1i)/2))^(n - 3)*((3*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i 
+ f*x*1i)*1i)/2))^m)/(8*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/ 
2))^(m + n + 1)*(m*48i - n*48i - m*n*88i + m*n^2*36i - m^2*n*36i - m*n^3*4 
i - m^3*n*4i + m^2*44i + m^3*12i + m^4*1i + n^2*44i - n^3*12i + n^4*1i + m 
^2*n^2*6i)) - (3*exp(e*8i + f*x*8i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - 
(exp(e*1i + f*x*1i)*1i)/2))^m)/(8*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i 
+ f*x*1i)/2))^(m + n + 1)*(m*48i - n*48i - m*n*88i + m*n^2*36i - m^2*n*36i 
 - m*n^3*4i - m^3*n*4i + m^2*44i + m^3*12i + m^4*1i + n^2*44i - n^3*12i + 
n^4*1i + m^2*n^2*6i)) - (exp(e*2i + f*x*2i)*(a + a*((exp(- e*1i - f*x*1i)* 
1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^m*(36*m - 36*n - 12*m*n + 6*m^2 + 6*n^ 
2 + 42))/(8*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(m + n + 
 1)*(m*48i - n*48i - m*n*88i + m*n^2*36i - m^2*n*36i - m*n^3*4i - m^3*n*4i 
 + m^2*44i + m^3*12i + m^4*1i + n^2*44i - n^3*12i + n^4*1i + m^2*n^2*6i)) 
+ (exp(e*6i + f*x*6i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f* 
x*1i)*1i)/2))^m*(36*m - 36*n - 12*m*n + 6*m^2 + 6*n^2 + 42))/(8*f*(g*(exp( 
- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(m + n + 1)*(m*48i - n*48i - m 
*n*88i + m*n^2*36i - m^2*n*36i - m*n^3*4i - m^3*n*4i + m^2*44i + m^3*12i + 
 m^4*1i + n^2*44i - n^3*12i + n^4*1i + m^2*n^2*6i)) + (exp(e*1i + f*x*1i)* 
(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^m*(m*...
 

Reduce [F]

\[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=-\frac {\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (-\sin \left (f x +e \right ) c +c \right )^{n}}{\cos \left (f x +e \right )^{m +n} \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-3 \cos \left (f x +e \right )^{m +n} \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+3 \cos \left (f x +e \right )^{m +n} \cos \left (f x +e \right ) \sin \left (f x +e \right )-\cos \left (f x +e \right )^{m +n} \cos \left (f x +e \right )}d x}{g^{m +n} c^{3} g} \] Input:

int((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3+n),x)
 

Output:

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