3.2.0 ∫ (g cos(e + fx))−1−m−n(a + a sin(e + fx))m(c − c sin(e + fx))−3+n dx [188]

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

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

Rubi [A] (veriﬁed)

Time = 0.65 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2928, 2927} \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\frac {6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^3 f g (m-n) (m-n+2) (m-n+4) (m-n+6)}+\frac {6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n+2) (m-n+4) (m-n+6)}+\frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}+\frac {3 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{c f g (m-n+4) (m-n+6)} \]

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]

((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))/(c*f*g*(4 + m - n)*(6 + m - n)) + (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)) + (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

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] &

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

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))^{-3+n}}{f g (6+m-n)}+\frac {3 \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}{c (6+m-n)} \\ & = \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 \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^2 (4+m-n) (6+m-n)} \\ & = \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 \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^3 (2+m-n) (4+m-n) (6+m-n)} \\ & = \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)} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 21.86 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.67 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\frac {(g \cos (e+f x))^{-m-n} (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^n \left (-30-46 m-18 m^2-2 m^3+46 n+36 m n+6 m^2 n-18 n^2-6 m n^2+2 n^3+6 (3+m-n) \cos (2 (e+f x))+3 \left (15+2 m^2-4 m (-3+n)-12 n+2 n^2\right ) \sin (e+f x)-3 \sin (3 (e+f x))\right )}{2 c^3 f g (m-n) (2+m-n) (4+m-n) (6+m-n) (-1+\sin (e+f x))^3} \]

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]

((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)*(-

Maple [F]

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

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 = 0.39 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.91 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=-\frac {{\left (6 \, {\left (m - n + 3\right )} \cos \left (f x + e\right )^{3} - {\left (m^{3} + 3 \, {\left (m + 3\right )} n^{2} - n^{3} + 9 \, m^{2} - {\left (3 \, m^{2} + 18 \, m + 26\right )} n + 26 \, m + 24\right )} \cos \left (f x + e\right ) - 3 \, {\left (2 \, \cos \left (f x + e\right )^{3} - {\left (m^{2} - 2 \, {\left (m + 3\right )} n + n^{2} + 6 \, m + 8\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} e^{\left (2 \, {\left (n - 3\right )} \log \left (g \cos \left (f x + e\right )\right ) - {\left (n - 3\right )} \log \left (a \sin \left (f x + e\right ) + a\right ) + {\left (n - 3\right )} \log \left (\frac {a c}{g^{2}}\right )\right )}}{f m^{4} + f n^{4} + 12 \, f m^{3} - 4 \, {\left (f m + 3 \, f\right )} n^{3} + 44 \, f m^{2} + 2 \, {\left (3 \, f m^{2} + 18 \, f m + 22 \, f\right )} n^{2} + 48 \, f m - 4 \, {\left (f m^{3} + 9 \, f m^{2} + 22 \, f m + 12 \, f\right )} n} \]

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")

-(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

+ 48*f*m - 4*(f*m^3 + 9*f*m^2 + 22*f*m + 12*f)*n)

Sympy [F(-1)]

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

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)

Maxima [F(-2)]

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

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")

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

Giac [F]

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

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")

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 = 19.78 (sec) , antiderivative size = 1623, normalized size of antiderivative = 5.60 \[ \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx=\text {Too large to display} \]

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)

-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*((e

xp(- 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)) - (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*1

i + 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*6i - n*6i + 18i))/(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*7i + f*x*7i)*(a + a*((exp(- e*1i - f*x*1i)*

1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^m*(m*6i - n*6i + 18i))/(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*3i + f*x*3i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2

- (exp(e*1i + f*x*1i)*1i)/2))^m*(m*86i - n*86i - m*n*72i + m*n^2*12i - m^2*n*12i + m^2*36i + m^3*4i + n^2*36i

- n^3*4i + 42i))/(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*5i + f*x*5i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^m*(m*86i - n

*86i - m*n*72i + m*n^2*12i - m^2*n*12i + m^2*36i + m^3*4i + n^2*36i - n^3*4i + 42i))/(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)))

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

Optimal result