3.175 \(\int (g \cos (e+f x))^{5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx\)
Optimal. Leaf size=203 \[ -\frac {8 a^3 (a \sin (e+f x)+a)^{m-3} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+3) (-m+n+4) (-m+n+5)}-\frac {4 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+4) (-m+n+5)}-\frac {a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+5)} \]
[Out]
-8*a^3*(g*cos(f*x+e))^(6-2*m)*(a+a*sin(f*x+e))^(-3+m)*(c-c*sin(f*x+e))^n/f/g/(3-m+n)/(4-m+n)/(5-m+n)-4*a^2*(g*
cos(f*x+e))^(6-2*m)*(a+a*sin(f*x+e))^(-2+m)*(c-c*sin(f*x+e))^n/f/g/(4-m+n)/(5-m+n)-a*(g*cos(f*x+e))^(6-2*m)*(a
+a*sin(f*x+e))^(-1+m)*(c-c*sin(f*x+e))^n/f/g/(5-m+n)
________________________________________________________________________________________
Rubi [A] time = 0.68, antiderivative size = 203, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used =
{2846, 2844} \[ -\frac {8 a^3 (a \sin (e+f x)+a)^{m-3} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+3) (-m+n+4) (-m+n+5)}-\frac {4 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+4) (-m+n+5)}-\frac {a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+5)} \]
Antiderivative was successfully verified.
[In]
Int[(g*Cos[e + f*x])^(5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]
[Out]
(-8*a^3*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-3 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(3 - m + n)*(4 -
m + n)*(5 - m + n)) - (4*a^2*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-2 + m)*(c - c*Sin[e + f*x])^n)
/(f*g*(4 - m + n)*(5 - m + n)) - (a*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*
x])^n)/(f*g*(5 - m + n))
Rule 2844
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 - 1)*(c + d*Sin[e +
f*x])^n)/(f*g*(m - n - 1)), 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[2*m + p - 1, 0] && NeQ[m - n - 1, 0]
Rule 2846
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 - 1)*(c + d*Sin[e +
f*x])^n)/(f*g*(m + n + p)), x] + Dist[(a*(2*m + p - 1))/(m + n + p), 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, n, p}, x] && EqQ[b*c + a*d, 0] && Eq
Q[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] && !LtQ[n, -1] && !(IGtQ[Simplify[n + p/2 - 1/2], 0] &&
GtQ[m - n, 0]) && !(ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + (3*p)/2 + 1], 0])
Rubi steps
\begin {align*} \int (g \cos (e+f x))^{5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=-\frac {a (g \cos (e+f x))^{6-2 m} (a+a \sin (e+f x))^{-1+m} (c-c \sin (e+f x))^n}{f g (5-m+n)}+\frac {(4 a) \int (g \cos (e+f x))^{5-2 m} (a+a \sin (e+f x))^{-1+m} (c-c \sin (e+f x))^n \, dx}{5-m+n}\\ &=-\frac {4 a^2 (g \cos (e+f x))^{6-2 m} (a+a \sin (e+f x))^{-2+m} (c-c \sin (e+f x))^n}{f g (4-m+n) (5-m+n)}-\frac {a (g \cos (e+f x))^{6-2 m} (a+a \sin (e+f x))^{-1+m} (c-c \sin (e+f x))^n}{f g (5-m+n)}+\frac {\left (8 a^2\right ) \int (g \cos (e+f x))^{5-2 m} (a+a \sin (e+f x))^{-2+m} (c-c \sin (e+f x))^n \, dx}{(4-m+n) (5-m+n)}\\ &=-\frac {8 a^3 (g \cos (e+f x))^{6-2 m} (a+a \sin (e+f x))^{-3+m} (c-c \sin (e+f x))^n}{f g (3-m+n) (4-m+n) (5-m+n)}-\frac {4 a^2 (g \cos (e+f x))^{6-2 m} (a+a \sin (e+f x))^{-2+m} (c-c \sin (e+f x))^n}{f g (4-m+n) (5-m+n)}-\frac {a (g \cos (e+f x))^{6-2 m} (a+a \sin (e+f x))^{-1+m} (c-c \sin (e+f x))^n}{f g (5-m+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.81, size = 1513, normalized size = 7.45 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
Integrate[(g*Cos[e + f*x])^(5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]
[Out]
(Cos[e + f*x]^(-5 + 2*n)*(g*Cos[e + f*x])^(5 - 2*m)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(n - (n*(Log
[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))/Log[c - c*Sin[e + f*x]])*((E^(n*(-2*Log[Cos[e + f*x]] + Log
[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*(256 - 41*m + 3*m^2 + 41*n - 6*m*n + 3*n^2))/(8*(-5 + m - n
)*(-4 + m - n)*(-3 + m - n)) + ((300 - 23*m + m^2 + 23*n - 2*m*n + n^2)*((-1/16*I)*E^(n*(-2*Log[Cos[e + f*x]]
+ Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[e + f*x] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1
+ Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[e + f*x])/16))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((3
00 - 23*m + m^2 + 23*n - 2*m*n + n^2)*((I/16)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c -
c*Sin[e + f*x]]))*Cos[e + f*x] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e +
f*x]]))*Sin[e + f*x])/16))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((-11*m + m^2 + 11*n - 2*m*n + n^2)*((E^
(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[2*(e + f*x)])/4 - (I/4)*E
^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[2*(e + f*x)]))/((-5 + m
- n)*(-4 + m - n)*(-3 + m - n)) + ((-11*m + m^2 + 11*n - 2*m*n + n^2)*((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1
+ Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[2*(e + f*x)])/4 + (I/4)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1
+ Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[2*(e + f*x)]))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((
100 - 53*m + 3*m^2 + 53*n - 6*m*n + 3*n^2)*((-1/32*I)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] +
Log[c - c*Sin[e + f*x]]))*Cos[3*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c
- c*Sin[e + f*x]]))*Sin[3*(e + f*x)])/32))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((100 - 53*m + 3*m^2 + 5
3*n - 6*m*n + 3*n^2)*((I/32)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]])
)*Cos[3*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[3
*(e + f*x)])/32))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((m - n)*((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1
+ Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[4*(e + f*x)])/16 - (I/16)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*
(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[4*(e + f*x)]))/((-5 + m - n)*(-4 + m - n)) + ((m - n)*((E^
(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[4*(e + f*x)])/16 + (I/16)
*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[4*(e + f*x)]))/((-5 +
m - n)*(-4 + m - n)) + ((-1/32*I)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f
*x]]))*Cos[5*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*
Sin[5*(e + f*x)])/32)/(-5 + m - n) + ((I/32)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c -
c*Sin[e + f*x]]))*Cos[5*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e
+ f*x]]))*Sin[5*(e + f*x)])/32)/(-5 + m - n)))/f
________________________________________________________________________________________
fricas [B] time = 0.55, size = 651, normalized size = 3.21 \[ -\frac {{\left ({\left (m^{2} - {\left (2 \, m - 7\right )} n + n^{2} - 7 \, m + 12\right )} \cos \left (f x + e\right )^{3} - {\left (m^{2} - {\left (2 \, m - 11\right )} n + n^{2} - 11 \, m + 24\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (m^{2} - {\left (2 \, m - 9\right )} n + n^{2} - 9 \, m + 22\right )} \cos \left (f x + e\right ) - {\left ({\left (m^{2} - {\left (2 \, m - 7\right )} n + n^{2} - 7 \, m + 12\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (m^{2} - {\left (2 \, m - 9\right )} n + n^{2} - 9 \, m + 18\right )} \cos \left (f x + e\right ) - 8\right )} \sin \left (f x + e\right ) - 8\right )} \left (g \cos \left (f x + e\right )\right )^{-2 \, m + 5} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} e^{\left (2 \, n \log \left (g \cos \left (f x + e\right )\right ) - n \log \left (a \sin \left (f x + e\right ) + a\right ) + n \log \left (\frac {a c}{g^{2}}\right )\right )}}{4 \, f m^{3} - 4 \, f n^{3} - {\left (f m^{3} - f n^{3} - 12 \, f m^{2} + 3 \, {\left (f m - 4 \, f\right )} n^{2} + 47 \, f m - {\left (3 \, f m^{2} - 24 \, f m + 47 \, f\right )} n - 60 \, f\right )} \cos \left (f x + e\right )^{3} - 48 \, f m^{2} + 12 \, {\left (f m - 4 \, f\right )} n^{2} - 3 \, {\left (f m^{3} - f n^{3} - 12 \, f m^{2} + 3 \, {\left (f m - 4 \, f\right )} n^{2} + 47 \, f m - {\left (3 \, f m^{2} - 24 \, f m + 47 \, f\right )} n - 60 \, f\right )} \cos \left (f x + e\right )^{2} + 188 \, f m - 4 \, {\left (3 \, f m^{2} - 24 \, f m + 47 \, f\right )} n + 2 \, {\left (f m^{3} - f n^{3} - 12 \, f m^{2} + 3 \, {\left (f m - 4 \, f\right )} n^{2} + 47 \, f m - {\left (3 \, f m^{2} - 24 \, f m + 47 \, f\right )} n - 60 \, f\right )} \cos \left (f x + e\right ) + {\left (4 \, f m^{3} - 4 \, f n^{3} - 48 \, f m^{2} + 12 \, {\left (f m - 4 \, f\right )} n^{2} - {\left (f m^{3} - f n^{3} - 12 \, f m^{2} + 3 \, {\left (f m - 4 \, f\right )} n^{2} + 47 \, f m - {\left (3 \, f m^{2} - 24 \, f m + 47 \, f\right )} n - 60 \, f\right )} \cos \left (f x + e\right )^{2} + 188 \, f m - 4 \, {\left (3 \, f m^{2} - 24 \, f m + 47 \, f\right )} n + 2 \, {\left (f m^{3} - f n^{3} - 12 \, f m^{2} + 3 \, {\left (f m - 4 \, f\right )} n^{2} + 47 \, f m - {\left (3 \, f m^{2} - 24 \, f m + 47 \, f\right )} n - 60 \, f\right )} \cos \left (f x + e\right ) - 240 \, f\right )} \sin \left (f x + e\right ) - 240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(5-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="fricas")
[Out]
-((m^2 - (2*m - 7)*n + n^2 - 7*m + 12)*cos(f*x + e)^3 - (m^2 - (2*m - 11)*n + n^2 - 11*m + 24)*cos(f*x + e)^2
- 2*(m^2 - (2*m - 9)*n + n^2 - 9*m + 22)*cos(f*x + e) - ((m^2 - (2*m - 7)*n + n^2 - 7*m + 12)*cos(f*x + e)^2 +
2*(m^2 - (2*m - 9)*n + n^2 - 9*m + 18)*cos(f*x + e) - 8)*sin(f*x + e) - 8)*(g*cos(f*x + e))^(-2*m + 5)*(a*sin
(f*x + e) + a)^m*e^(2*n*log(g*cos(f*x + e)) - n*log(a*sin(f*x + e) + a) + n*log(a*c/g^2))/(4*f*m^3 - 4*f*n^3 -
(f*m^3 - f*n^3 - 12*f*m^2 + 3*(f*m - 4*f)*n^2 + 47*f*m - (3*f*m^2 - 24*f*m + 47*f)*n - 60*f)*cos(f*x + e)^3 -
48*f*m^2 + 12*(f*m - 4*f)*n^2 - 3*(f*m^3 - f*n^3 - 12*f*m^2 + 3*(f*m - 4*f)*n^2 + 47*f*m - (3*f*m^2 - 24*f*m
+ 47*f)*n - 60*f)*cos(f*x + e)^2 + 188*f*m - 4*(3*f*m^2 - 24*f*m + 47*f)*n + 2*(f*m^3 - f*n^3 - 12*f*m^2 + 3*(
f*m - 4*f)*n^2 + 47*f*m - (3*f*m^2 - 24*f*m + 47*f)*n - 60*f)*cos(f*x + e) + (4*f*m^3 - 4*f*n^3 - 48*f*m^2 + 1
2*(f*m - 4*f)*n^2 - (f*m^3 - f*n^3 - 12*f*m^2 + 3*(f*m - 4*f)*n^2 + 47*f*m - (3*f*m^2 - 24*f*m + 47*f)*n - 60*
f)*cos(f*x + e)^2 + 188*f*m - 4*(3*f*m^2 - 24*f*m + 47*f)*n + 2*(f*m^3 - f*n^3 - 12*f*m^2 + 3*(f*m - 4*f)*n^2
+ 47*f*m - (3*f*m^2 - 24*f*m + 47*f)*n - 60*f)*cos(f*x + e) - 240*f)*sin(f*x + e) - 240*f)
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(5-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [F] time = 66.34, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{5-2 m} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*cos(f*x+e))^(5-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)
[Out]
int((g*cos(f*x+e))^(5-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x)
________________________________________________________________________________________
maxima [B] time = 0.98, size = 978, normalized size = 4.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))^(5-2*m)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm="maxima")
[Out]
((m^2 - m*(2*n + 11) + n^2 + 11*n + 32)*a^m*c^n*g^5 - 2*(m^2 - m*(2*n + 15) + n^2 + 15*n + 60)*a^m*c^n*g^5*sin
(f*x + e)/(cos(f*x + e) + 1) - (3*m^2 - m*(6*n + 1) + 3*n^2 + n - 160)*a^m*c^n*g^5*sin(f*x + e)^2/(cos(f*x + e
) + 1)^2 + 8*(m^2 - m*(2*n + 7) + n^2 + 7*n - 20)*a^m*c^n*g^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 2*(m^2 - m
*(2*n - 5) + n^2 - 5*n + 160)*a^m*c^n*g^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 4*(3*m^2 - m*(6*n + 13) + 3*n^
2 + 13*n + 116)*a^m*c^n*g^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 2*(m^2 - m*(2*n - 5) + n^2 - 5*n + 160)*a^m*
c^n*g^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 8*(m^2 - m*(2*n + 7) + n^2 + 7*n - 20)*a^m*c^n*g^5*sin(f*x + e)^
7/(cos(f*x + e) + 1)^7 - (3*m^2 - m*(6*n + 1) + 3*n^2 + n - 160)*a^m*c^n*g^5*sin(f*x + e)^8/(cos(f*x + e) + 1)
^8 - 2*(m^2 - m*(2*n + 15) + n^2 + 15*n + 60)*a^m*c^n*g^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + (m^2 - m*(2*n
+ 11) + n^2 + 11*n + 32)*a^m*c^n*g^5*sin(f*x + e)^10/(cos(f*x + e) + 1)^10)*e^(2*n*log(sin(f*x + e)/(cos(f*x +
e) + 1) - 1) - 2*m*log(-sin(f*x + e)/(cos(f*x + e) + 1) + 1) + m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)
- n*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/(((m^3 - 3*m^2*(n + 4) - n^3 + (3*n^2 + 24*n + 47)*m - 12*n
^2 - 47*n - 60)*g^(2*m) + 5*(m^3 - 3*m^2*(n + 4) - n^3 + (3*n^2 + 24*n + 47)*m - 12*n^2 - 47*n - 60)*g^(2*m)*s
in(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*(m^3 - 3*m^2*(n + 4) - n^3 + (3*n^2 + 24*n + 47)*m - 12*n^2 - 47*n - 6
0)*g^(2*m)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 10*(m^3 - 3*m^2*(n + 4) - n^3 + (3*n^2 + 24*n + 47)*m - 12*n^
2 - 47*n - 60)*g^(2*m)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 5*(m^3 - 3*m^2*(n + 4) - n^3 + (3*n^2 + 24*n + 47
)*m - 12*n^2 - 47*n - 60)*g^(2*m)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + (m^3 - 3*m^2*(n + 4) - n^3 + (3*n^2 +
24*n + 47)*m - 12*n^2 - 47*n - 60)*g^(2*m)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10)*f)
________________________________________________________________________________________
mupad [B] time = 17.21, size = 1149, normalized size = 5.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*cos(e + f*x))^(5 - 2*m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n,x)
[Out]
((c - c*sin(e + f*x))^n*((exp(e*2i + f*x*2i)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(5 - 2*m)*(a
+ a*sin(e + f*x))^m*(22*n - 22*m - 4*m*n + 2*m^2 + 2*n^2 + 80))/(f*(47*n - 47*m - 24*m*n - 3*m*n^2 + 3*m^2*n +
12*m^2 - m^3 + 12*n^2 + n^3 + 60)) - (exp(e*5i + f*x*5i)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^
(5 - 2*m)*(a + a*sin(e + f*x))^m*(n*7i - m*7i - m*n*2i + m^2*1i + n^2*1i + 12i))/(f*(47*n - 47*m - 24*m*n - 3*
m*n^2 + 3*m^2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 60)) - ((g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(5
- 2*m)*(a + a*sin(e + f*x))^m*(7*n - 7*m - 2*m*n + m^2 + n^2 + 12))/(f*(47*n - 47*m - 24*m*n - 3*m*n^2 + 3*m^
2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 60)) + (exp(e*4i + f*x*4i)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)
/2))^(5 - 2*m)*(a + a*sin(e + f*x))^m*(29*n - 29*m - 6*m*n + 3*m^2 + 3*n^2 + 60))/(f*(47*n - 47*m - 24*m*n - 3
*m*n^2 + 3*m^2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 60)) + (exp(e*1i + f*x*1i)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e
*1i + f*x*1i)/2))^(5 - 2*m)*(a + a*sin(e + f*x))^m*(n*29i - m*29i - m*n*6i + m^2*3i + n^2*3i + 60i))/(f*(47*n
- 47*m - 24*m*n - 3*m*n^2 + 3*m^2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 60)) + (exp(e*3i + f*x*3i)*(g*(exp(- e*1i
- f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(5 - 2*m)*(a + a*sin(e + f*x))^m*(n*22i - m*22i - m*n*4i + m^2*2i + n^2*2
i + 80i))/(f*(47*n - 47*m - 24*m*n - 3*m*n^2 + 3*m^2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 60))))/(5*exp(e*1i + f*
x*1i) - 10*exp(e*3i + f*x*3i) + exp(e*5i + f*x*5i) + (n*47i - m*47i - m*n*24i - m*n^2*3i + m^2*n*3i + m^2*12i
- m^3*1i + n^2*12i + n^3*1i + 60i)/(47*n - 47*m - 24*m*n - 3*m*n^2 + 3*m^2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 6
0) - (10*exp(e*2i + f*x*2i)*(n*47i - m*47i - m*n*24i - m*n^2*3i + m^2*n*3i + m^2*12i - m^3*1i + n^2*12i + n^3*
1i + 60i))/(47*n - 47*m - 24*m*n - 3*m*n^2 + 3*m^2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 60) + (5*exp(e*4i + f*x*4
i)*(n*47i - m*47i - m*n*24i - m*n^2*3i + m^2*n*3i + m^2*12i - m^3*1i + n^2*12i + n^3*1i + 60i))/(47*n - 47*m -
24*m*n - 3*m*n^2 + 3*m^2*n + 12*m^2 - m^3 + 12*n^2 + n^3 + 60))
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*cos(f*x+e))**(5-2*m)*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**n,x)
[Out]
Timed out
________________________________________________________________________________________