Integrand size = 45, antiderivative size = 204 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\frac {(g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n}}{f g (4+m-n)}+\frac {2 (g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{c f g (2+m-n) (4+m-n)}+\frac {2 (g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{c^2 f g (m-n) (2+m-n) (4+m-n)} \] Output:
(g*cos(f*x+e))^(-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2+n)/f/g/(4+m- n)+2*(g*cos(f*x+e))^(-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-1+n)/c/f/ g/(2+m-n)/(4+m-n)+2*(g*cos(f*x+e))^(-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+ e))^n/c^2/f/g/(m-n)/(2+m-n)/(4+m-n)
Time = 22.88 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.62 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\frac {(g \cos (e+f x))^{-m-n} (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^n \left (3+4 m+m^2-4 n-2 m n+n^2-\cos (2 (e+f x))-2 (2+m-n) \sin (e+f x)\right )}{c^2 f g (m-n) (2+m-n) (4+m-n) (-1+\sin (e+f x))^2} \] Input:
Integrate[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[ e + f*x])^(-2 + n),x]
Output:
((g*Cos[e + f*x])^(-m - n)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n *(3 + 4*m + m^2 - 4*n - 2*m*n + n^2 - Cos[2*(e + f*x)] - 2*(2 + m - n)*Sin [e + f*x]))/(c^2*f*g*(m - n)*(2 + m - n)*(4 + m - n)*(-1 + Sin[e + f*x])^2 )
Time = 1.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
| \(\displaystyle \int (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n-1} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n-1}dx\) |
| \(\Big \downarrow \) 3328 |
| \(\displaystyle \frac {2 \int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{n-1}dx}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{n-1}dx}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\) |
| \(\Big \downarrow \) 3328 |
| \(\displaystyle \frac {2 \left (\frac {\int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^ndx}{c (m-n+2)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}\right )}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {\int (g \cos (e+f x))^{-m-n-1} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^ndx}{c (m-n+2)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}\right )}{c (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}\) |
| \(\Big \downarrow \) 3327 |
| \(\displaystyle \frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}+\frac {2 \left (\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c f g (m-n) (m-n+2)}\right )}{c (m-n+4)}\) |
Input:
Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f* x])^(-2 + n),x]
Output:
((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*Sin[e + f*x])^n)/(c*f*g*(m - n)*( 2 + m - n))))/(c*(4 + m - n))
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]
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])
\[\int \left (g \cos \left (f x +e \right )\right )^{-1-m -n} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-2+n}d x\]
Input:
int((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2+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))^(-2+n),x)
Time = 0.10 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.90 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=-\frac {{\left (2 \, \cos \left (f x + e\right )^{3} + 2 \, {\left (m - n + 2\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (m^{2} - 2 \, {\left (m + 2\right )} n + n^{2} + 4 \, m + 4\right )} \cos \left (f x + e\right )\right )} \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} e^{\left (2 \, {\left (n - 2\right )} \log \left (g \cos \left (f x + e\right )\right ) - {\left (n - 2\right )} \log \left (a \sin \left (f x + e\right ) + a\right ) + {\left (n - 2\right )} \log \left (\frac {a c}{g^{2}}\right )\right )}}{f m^{3} - f n^{3} + 6 \, f m^{2} + 3 \, {\left (f m + 2 \, f\right )} n^{2} + 8 \, f m - {\left (3 \, f m^{2} + 12 \, f m + 8 \, f\right )} n} \] Input:
integrate((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2+ n),x, algorithm="fricas")
Output:
-(2*cos(f*x + e)^3 + 2*(m - n + 2)*cos(f*x + e)*sin(f*x + e) - (m^2 - 2*(m + 2)*n + n^2 + 4*m + 4)*cos(f*x + e))*(g*cos(f*x + e))^(-m - n - 1)*(a*si n(f*x + e) + a)^m*e^(2*(n - 2)*log(g*cos(f*x + e)) - (n - 2)*log(a*sin(f*x + e) + a) + (n - 2)*log(a*c/g^2))/(f*m^3 - f*n^3 + 6*f*m^2 + 3*(f*m + 2*f )*n^2 + 8*f*m - (3*f*m^2 + 12*f*m + 8*f)*n)
Timed out. \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(-1-m-n)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**( -2+n),x)
Output:
Timed out
Exception generated. \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2+ n),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n - 2} \,d x } \] Input:
integrate((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2+ 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 - 2), x)
Time = 23.02 (sec) , antiderivative size = 887, normalized size of antiderivative = 4.35 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\text {Too large to display} \] Input:
int(((a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(n - 2))/(g*cos(e + f*x)) ^(m + n + 1),x)
Output:
-exp(- e*3i - f*x*3i)*(c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f* x*1i)*1i)/2))^(n - 2)*((a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f *x*1i)*1i)/2))^m/(4*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^ (m + n + 1)*(8*m - 8*n - 12*m*n + 3*m*n^2 - 3*m^2*n + 6*m^2 + m^3 + 6*n^2 - n^3)) + (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)/(4*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x* 1i)/2))^(m + n + 1)*(8*m - 8*n - 12*m*n + 3*m*n^2 - 3*m^2*n + 6*m^2 + m^3 + 6*n^2 - n^3)) - (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*(8*m - 8*n - 4*m*n + 2*m^2 + 2*n^2 + 5))/( 4*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(m + n + 1)*(8*m - 8*n - 12*m*n + 3*m*n^2 - 3*m^2*n + 6*m^2 + m^3 + 6*n^2 - n^3)) - (exp(e*4 i + f*x*4i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/ 2))^m*(8*m - 8*n - 4*m*n + 2*m^2 + 2*n^2 + 5))/(4*f*(g*(exp(- e*1i - f*x*1 i)/2 + exp(e*1i + f*x*1i)/2))^(m + n + 1)*(8*m - 8*n - 12*m*n + 3*m*n^2 - 3*m^2*n + 6*m^2 + m^3 + 6*n^2 - n^3)) + (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*2i - n*2i + 4i))/ (4*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(m + n + 1)*(8*m - 8*n - 12*m*n + 3*m*n^2 - 3*m^2*n + 6*m^2 + m^3 + 6*n^2 - n^3)) - (exp(e* 5i + f*x*5i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i) /2))^m*(m*2i - n*2i + 4i))/(4*f*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i +...
\[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\frac {\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (-\sin \left (f x +e \right ) c +c \right )^{n}}{\cos \left (f x +e \right )^{m +n} \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-2 \cos \left (f x +e \right )^{m +n} \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )^{m +n} \cos \left (f x +e \right )}d x}{g^{m +n} c^{2} g} \] Input:
int((g*cos(f*x+e))^(-1-m-n)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2+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)**2 - 2*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**2*g)