\(\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx\) [187]
Optimal result
Integrand size = 45, antiderivative size = 204 \[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\frac {(g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n}}{f g (4+m-n)}+\frac {2 (g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{c f g (2+m-n) (4+m-n)}+\frac {2 (g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{c^2 f g (m-n) (2+m-n) (4+m-n)}
\]
[Out]
(g*cos(f*x+e))^(-n-m)*(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))^(-n-m)*(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))^(-n-m)*(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)
Rubi [A] (verified)
Time = 0.46 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number
of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2928, 2927}
\[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\frac {2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n) (m-n+2) (m-n+4)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}+\frac {2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c f g (m-n+2) (m-n+4)}
\]
[In]
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]
[Out]
((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))/(c*f*g*(2 + m - n)*(4 + m - n)) + (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))
Rule 2927
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 2928
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] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), Int[(g*Cos[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] && (SumSimplerQ[m, 1] || !SumS
implerQ[n, 1])
Rubi steps \begin{align*}
\text {integral}& = \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 \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n} \, dx}{c (4+m-n)} \\ & = \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 \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx}{c^2 (2+m-n) (4+m-n)} \\ & = \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)} \\
\end{align*}
Mathematica [A] (verified)
Time = 21.41 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.62
\[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\frac {(g \cos (e+f x))^{-m-n} (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^n \left (3+4 m+m^2-4 n-2 m n+n^2-\cos (2 (e+f x))-2 (2+m-n) \sin (e+f x)\right )}{c^2 f g (m-n) (2+m-n) (4+m-n) (-1+\sin (e+f x))^2}
\]
[In]
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]
[Out]
((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)
Maple [F]
\[\int \left (g \cos \left (f x +e \right )\right )^{-1-m -n} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-2+n}d x\]
[In]
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)
[Out]
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)
Fricas [A] (verification not implemented)
none
Time = 0.35 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.90
\[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=-\frac {{\left (2 \, \cos \left (f x + e\right )^{3} + 2 \, {\left (m - n + 2\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (m^{2} - 2 \, {\left (m + 2\right )} n + n^{2} + 4 \, m + 4\right )} \cos \left (f x + e\right )\right )} \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} e^{\left (2 \, {\left (n - 2\right )} \log \left (g \cos \left (f x + e\right )\right ) - {\left (n - 2\right )} \log \left (a \sin \left (f x + e\right ) + a\right ) + {\left (n - 2\right )} \log \left (\frac {a c}{g^{2}}\right )\right )}}{f m^{3} - f n^{3} + 6 \, f m^{2} + 3 \, {\left (f m + 2 \, f\right )} n^{2} + 8 \, f m - {\left (3 \, f m^{2} + 12 \, f m + 8 \, f\right )} n}
\]
[In]
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")
[Out]
-(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*sin(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)
Sympy [F(-1)]
Timed out. \[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\text {Timed out}
\]
[In]
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)
[Out]
Timed out
Maxima [F(-2)]
Exception generated. \[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\text {Exception raised: RuntimeError}
\]
[In]
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")
[Out]
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.
Giac [F]
\[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n - 2} \,d x }
\]
[In]
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")
[Out]
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)
Mupad [B] (verification not implemented)
Time = 16.63 (sec) , antiderivative size = 887, normalized size of antiderivative = 4.35
\[
\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx=\text {Too large to display}
\]
[In]
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)
[Out]
-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*4i + 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*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*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 + 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)))