Integrand size = 36, antiderivative size = 244 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\frac {768 c^3 \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (7+2 m) (9+2 m) \left (15+16 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {192 c^2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt {c-c \sin (e+f x)}}{a f (9+2 m) \left (35+24 m+4 m^2\right )}+\frac {24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)} \] Output:
768*c^3*cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)/a/f/(7+2*m)/(9+2*m)/(4*m^2+16*m+ 15)/(c-c*sin(f*x+e))^(1/2)+192*c^2*cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)*(c-c* sin(f*x+e))^(1/2)/a/f/(9+2*m)/(4*m^2+24*m+35)+24*c*cos(f*x+e)*(a+a*sin(f*x +e))^(1+m)*(c-c*sin(f*x+e))^(3/2)/a/f/(4*m^2+32*m+63)+2*cos(f*x+e)*(a+a*si n(f*x+e))^(1+m)*(c-c*sin(f*x+e))^(5/2)/a/f/(9+2*m)
Result contains complex when optimal does not.
Time = 13.38 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.85 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2) ,x]
Output:
((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((2205 + 590*m + 108 *m^2 + 8*m^3)*((3/8 + (3*I)/8)*Cos[(e + f*x)/2] + (3/8 - (3*I)/8)*Sin[(e + f*x)/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((2205 + 590*m + 10 8*m^2 + 8*m^3)*((3/8 - (3*I)/8)*Cos[(e + f*x)/2] + (3/8 + (3*I)/8)*Sin[(e + f*x)/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((191*m + 48*m^2 + 4*m^3)*((1 - I)*Cos[(3*(e + f*x))/2] - (1 + I)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((191*m + 48*m^2 + 4*m^3)*((1 + I )*Cos[(3*(e + f*x))/2] - (1 - I)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2* m)*(7 + 2*m)*(9 + 2*m)) + ((21 + 2*m)*((3/2 + (3*I)/2)*Cos[(5*(e + f*x))/2 ] + (3/2 - (3*I)/2)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((21 + 2*m)*((3/2 - (3*I)/2)*Cos[(5*(e + f*x))/2] + (3/2 + (3*I)/2)*Sin [(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((3/16 - (3*I)/16)*Cos[(7*(e + f*x))/2] - (3/16 + (3*I)/16)*Sin[(7*(e + f*x))/2])) /((7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((3/16 + (3*I)/16)*Cos[(7*(e + f*x))/ 2] - (3/16 - (3*I)/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((-1 /16 + I/16)*Cos[(9*(e + f*x))/2] - (1/16 + I/16)*Sin[(9*(e + f*x))/2])/(9 + 2*m) + ((-1/16 - I/16)*Cos[(9*(e + f*x))/2] - (1/16 - I/16)*Sin[(9*(e + f*x))/2])/(9 + 2*m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)
Time = 1.17 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3042, 3320, 3042, 3219, 3042, 3219, 3042, 3219, 3042, 3217}
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.
| \(\displaystyle \int \cos ^2(e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (e+f x)^2 (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^mdx\) |
| \(\Big \downarrow \) 3320 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^{m+1} (c-c \sin (e+f x))^{7/2}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^{m+1} (c-c \sin (e+f x))^{7/2}dx}{a c}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {12 c \int (\sin (e+f x) a+a)^{m+1} (c-c \sin (e+f x))^{5/2}dx}{2 m+9}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+9)}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {12 c \int (\sin (e+f x) a+a)^{m+1} (c-c \sin (e+f x))^{5/2}dx}{2 m+9}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+9)}}{a c}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {12 c \left (\frac {8 c \int (\sin (e+f x) a+a)^{m+1} (c-c \sin (e+f x))^{3/2}dx}{2 m+7}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+7)}\right )}{2 m+9}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+9)}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {12 c \left (\frac {8 c \int (\sin (e+f x) a+a)^{m+1} (c-c \sin (e+f x))^{3/2}dx}{2 m+7}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+7)}\right )}{2 m+9}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+9)}}{a c}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {12 c \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^{m+1} \sqrt {c-c \sin (e+f x)}dx}{2 m+5}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{f (2 m+5)}\right )}{2 m+7}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+7)}\right )}{2 m+9}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+9)}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {12 c \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^{m+1} \sqrt {c-c \sin (e+f x)}dx}{2 m+5}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{f (2 m+5)}\right )}{2 m+7}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+7)}\right )}{2 m+9}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+9)}}{a c}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle \frac {\frac {12 c \left (\frac {8 c \left (\frac {8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{f (2 m+3) (2 m+5) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{f (2 m+5)}\right )}{2 m+7}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+7)}\right )}{2 m+9}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{f (2 m+9)}}{a c}\) |
Input:
Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]
Output:
((2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(5/2) )/(f*(9 + 2*m)) + (12*c*((2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(f*(7 + 2*m)) + (8*c*((8*c^2*Cos[e + f*x]*(a + a *Sin[e + f*x])^(1 + m))/(f*(3 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/ (f*(5 + 2*m))))/(7 + 2*m)))/(9 + 2*m))/(a*c)
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(a^(p/ 2)*c^(p/2)) Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2]
\[\int \cos \left (f x +e \right )^{2} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x)
Output:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x)
Time = 0.11 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.62 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (8 \, c^{2} m^{3} + 108 \, c^{2} m^{2} + 334 \, c^{2} m + 285 \, c^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 334 \, c^{2} m + 339 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \, {\left (2 \, c^{2} m - c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2} + {\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 238 \, c^{2} m + 195 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \, {\left (2 \, c^{2} m + 3 \, c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + {\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \sin \left (f x + e\right ) + 945 \, f} \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x, algori thm="fricas")
Output:
-2*((8*c^2*m^3 + 60*c^2*m^2 + 142*c^2*m + 105*c^2)*cos(f*x + e)^5 - (8*c^2 *m^3 + 108*c^2*m^2 + 334*c^2*m + 285*c^2)*cos(f*x + e)^4 - 2*(8*c^2*m^3 + 84*c^2*m^2 + 334*c^2*m + 339*c^2)*cos(f*x + e)^3 - 384*c^2*cos(f*x + e) - 96*(2*c^2*m - c^2)*cos(f*x + e)^2 - 768*c^2 + ((8*c^2*m^3 + 60*c^2*m^2 + 1 42*c^2*m + 105*c^2)*cos(f*x + e)^4 + 2*(8*c^2*m^3 + 84*c^2*m^2 + 238*c^2*m + 195*c^2)*cos(f*x + e)^3 - 384*c^2*cos(f*x + e) - 96*(2*c^2*m + 3*c^2)*c os(f*x + e)^2 - 768*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)*(a*sin(f* x + e) + a)^m/(16*f*m^4 + 192*f*m^3 + 824*f*m^2 + 1488*f*m + (16*f*m^4 + 1 92*f*m^3 + 824*f*m^2 + 1488*f*m + 945*f)*cos(f*x + e) - (16*f*m^4 + 192*f* m^3 + 824*f*m^2 + 1488*f*m + 945*f)*sin(f*x + e) + 945*f)
Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**(5/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (236) = 472\).
Time = 0.20 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.29 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x, algori thm="maxima")
Output:
-2*((8*m^3 + 108*m^2 + 526*m + 957)*a^m*c^(5/2) - 3*(8*m^3 + 76*m^2 + 142* m - 315)*a^m*c^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) - 24*(4*m^2 + 16*m - 81)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 16*(4*m^3 + 36*m^2 + 95*m + 315)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6*(8*m^3 + 60*m^2 + 206*m - 567)*a^m*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 6* (8*m^3 + 60*m^2 + 206*m - 567)*a^m*c^(5/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 16*(4*m^3 + 36*m^2 + 95*m + 315)*a^m*c^(5/2)*sin(f*x + e)^6/(cos(f* x + e) + 1)^6 - 24*(4*m^2 + 16*m - 81)*a^m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3*(8*m^3 + 76*m^2 + 142*m - 315)*a^m*c^(5/2)*sin(f*x + e)^8 /(cos(f*x + e) + 1)^8 + (8*m^3 + 108*m^2 + 526*m + 957)*a^m*c^(5/2)*sin(f* x + e)^9/(cos(f*x + e) + 1)^9)*e^(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))/((16*m^4 + 192*m^3 + 824*m^2 + 1488*m + 2*(16*m^4 + 192*m^3 + 824*m^2 + 1488*m + 945)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + (16*m^4 + 192*m^3 + 824*m^2 + 1488*m + 945) *sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 945)*f*(sin(f*x + e)^2/(cos(f*x + e ) + 1)^2 + 1)^(5/2))
\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x, algori thm="giac")
Output:
integrate((-c*sin(f*x + e) + c)^(5/2)*(a*sin(f*x + e) + a)^m*cos(f*x + e)^ 2, x)
Time = 24.09 (sec) , antiderivative size = 1060, normalized size of antiderivative = 4.34 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:
int(cos(e + f*x)^2*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(5/2),x)
Output:
((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*( (3*c^2*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*(48*m + 4*m^2 + 63))/(f*(m*1488i + m^2*824i + m^3*192i + m^4*16i + 945i)) - (c^2*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^m*(m*142i + m^2*60i + m^3*8i + 105i))/(8*f*(m*1488i + m ^2*824i + m^3*192i + m^4*16i + 945i)) + (3*c^2*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*(m*48i + m^2*4 i + 63i))/(f*(m*1488i + m^2*824i + m^3*192i + m^4*16i + 945i)) - (c^2*exp( e*9i + f*x*9i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1 i)/2))^m*(142*m + 60*m^2 + 8*m^3 + 105))/(8*f*(m*1488i + m^2*824i + m^3*19 2i + m^4*16i + 945i)) + (3*c^2*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*(270*m + 92*m^2 + 8*m^3 + 225) )/(8*f*(m*1488i + m^2*824i + m^3*192i + m^4*16i + 945i)) + (3*c^2*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*(m*270i + m^2*92i + m^3*8i + 225i))/(8*f*(m*1488i + m^2*824i + m^3*19 2i + m^4*16i + 945i)) + (3*c^2*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*(590*m + 108*m^2 + 8*m^3 + 220 5))/(4*f*(m*1488i + m^2*824i + m^3*192i + m^4*16i + 945i)) + (3*c^2*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*(m*590i + m^2*108i + m^3*8i + 2205i))/(4*f*(m*1488i + m^2*824i +...
\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {c}\, c^{2} \left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{2}d x -2 \left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )d x \right )+\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2}d x \right ) \] Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(c)*c**2*(int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x)**2,x) - 2*int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x),x) + int((sin(e + f*x)*a + a)**m *sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2,x))