3.188 \(\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx\)
Optimal. Leaf size=290 \[ \frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n+2) (m-n+4) (m-n+6)}+\frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^3 f g (m-n) (m-n+2) (m-n+4) (m-n+6)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}+\frac{3 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{c f g (m-n+4) (m-n+6)} \]
[Out]
((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
)*(2 + m - n)*(4 + m - n)*(6 + m - n))
________________________________________________________________________________________
Rubi [A] time = 0.937238, antiderivative size = 290, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 2, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.044, Rules used =
{2849, 2848} \[ \frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n+2) (m-n+4) (m-n+6)}+\frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^3 f g (m-n) (m-n+2) (m-n+4) (m-n+6)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}+\frac{3 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{c f g (m-n+4) (m-n+6)} \]
Antiderivative was successfully verified.
[In]
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]
[Out]
((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
)*(2 + m - n)*(4 + m - n)*(6 + m - n))
Rule 2849
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])
Rule 2848
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]
Rubi steps
\begin{align*} \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 \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 [B] time = 37.399, size = 2681, normalized size = 9.24 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
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]
[Out]
-((2^(-4 - m + 2*n)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[e + f*x]*(g*Cos[e + f*x])^(-1 - m - n)*Csc[(-e + Pi/2 -
f*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-
e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] +
Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 + 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 + Pi/2 - f*x)] + 3*
Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x])*(Cos[Pi/4 + (e - Pi/2 +
f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(-7 + 2*n))/(f*(m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^(2*(-3 + n))*((2^(-4 - m + 2*n)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Csc[(-e + Pi/2 - f
*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e
+ Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Si
n[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(-3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Cos[e + f*x] + 12*(3 + m
- n)*Sin[2*(-e + Pi/2 - f*x)] - 9*Sin[3*(-e + Pi/2 - f*x)]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) - (
2^(-4 - m + 2*n)*m*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*Csc[(-e + Pi/2 - f*x)/2]^5*(Cos[(-e + Pi/2 - f*x)/8]*(-
Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f
*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - 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 +
Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m -
n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) - (3*2^(-4 - m + 2*n)*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*Csc[(-e + Pi
/2 - f*x)/2]^7*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Co
s[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8
] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - 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 + Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-
3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) + (2^(-3 - m + 2*n)*n*Cos[
(-e + Pi/2 - f*x)/2]^(2*m)*Csc[(-e + Pi/2 - f*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + S
in[(3*(-e + Pi/2 - f*x))/8]))^(-1 + 2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Cos[(-e + Pi/2 - f*x)/8]/8 + (3*Cos[(3*(-
e + Pi/2 - f*x))/8])/8) - (Sin[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8])
)/8)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 -
f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(-30 - 46*m - 18*m^2 - 2*m^3 + 46*n + 36*m*n + 6*m^2*n - 1
8*n^2 - 6*m*n^2 + 2*n^3 - 6*(3 + m - n)*Cos[2*(-e + Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2
- 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) + (2^(-4 - m + 2*n
)*(-m - n)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Csc[(-e + Pi/2 - f*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/
2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[
(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-1 - m - n)*(Cos[(-e
+ Pi/2 - f*x)/8]*(-Cos[(-e + Pi/2 - f*x)/8]/8 + (3*Cos[(3*(-e + Pi/2 - f*x))/8])/8 - (5*Cos[(5*(-e + Pi/2 - f
*x))/8])/8 + (7*Cos[(7*(-e + Pi/2 - f*x))/8])/8) - (Sin[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[
(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))/8)*(-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 + Pi/2 - f*x)] + 3*C
os[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m - n)*(2 + m - n)*(4
+ m - n)*(6 + m - n)))*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] + Sin[Pi/4 + (e - Pi/2 + f*x)/2])))
________________________________________________________________________________________
Maple [F] time = 0.645, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{-1-m-n} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{-3+n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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)
[Out]
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)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
Exception raised: RuntimeError
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Fricas [A] time = 2.01214, size = 664, normalized size = 2.29 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
-(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)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
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)