Integrand size = 38, antiderivative size = 167 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {64 a^2 (11 A-B) c^5 \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 (11 A-B) c^4 \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \] Output:
64/3465*a^2*(11*A-B)*c^5*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(5/2)+16/693*a^2* (11*A-B)*c^4*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(3/2)+2/99*a^2*(11*A-B)*c^3*c os(f*x+e)^5/f/(c-c*sin(f*x+e))^(1/2)-2/11*a^2*B*c^2*cos(f*x+e)^5*(c-c*sin( f*x+e))^(1/2)/f
Time = 12.77 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {a^2 c^2 \cos ^4(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (-5478 A+3648 B+70 (11 A-28 B) \cos (2 (e+f x))+5 (968 A-1033 B) \sin (e+f x)+315 B \sin (3 (e+f x)))}{6930 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \] Input:
Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]) ^(5/2),x]
Output:
-1/6930*(a^2*c^2*Cos[e + f*x]^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt [c - c*Sin[e + f*x]]*(-5478*A + 3648*B + 70*(11*A - 28*B)*Cos[2*(e + f*x)] + 5*(968*A - 1033*B)*Sin[e + f*x] + 315*B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)
Time = 0.91 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3042, 3446, 3042, 3335, 3042, 3153, 3042, 3153, 3042, 3152}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^{5/2} (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^2 c^2 \int \cos ^4(e+f x) (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \cos (e+f x)^4 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3335 |
| \(\displaystyle a^2 c^2 \left (\frac {1}{11} (11 A-B) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)}dx-\frac {2 B \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {1}{11} (11 A-B) \int \cos (e+f x)^4 \sqrt {c-c \sin (e+f x)}dx-\frac {2 B \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\right )\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^2 c^2 \left (\frac {1}{11} (11 A-B) \left (\frac {8}{9} c \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 B \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {1}{11} (11 A-B) \left (\frac {8}{9} c \int \frac {\cos (e+f x)^4}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 B \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\right )\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^2 c^2 \left (\frac {1}{11} (11 A-B) \left (\frac {8}{9} c \left (\frac {4}{7} c \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}}dx+\frac {2 c \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 B \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {1}{11} (11 A-B) \left (\frac {8}{9} c \left (\frac {4}{7} c \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^{3/2}}dx+\frac {2 c \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 B \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\right )\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle a^2 c^2 \left (\frac {1}{11} (11 A-B) \left (\frac {8}{9} c \left (\frac {8 c^2 \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}+\frac {2 c \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}\right )+\frac {2 c \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 B \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2) ,x]
Output:
a^2*c^2*((-2*B*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(11*f) + ((11*A - B)*((2*c*Cos[e + f*x]^5)/(9*f*Sqrt[c - c*Sin[e + f*x]]) + (8*c*((8*c^2*Cos [e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^(5/2)) + (2*c*Cos[e + f*x]^5)/(7*f *(c - c*Sin[e + f*x])^(3/2))))/9))/11)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 37.69 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.63
| method | result | size |
| default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (-315 B \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )+\left (-385 A +980 B \right ) \cos \left (f x +e \right )^{2}+\left (-1210 A +1370 B \right ) \sin \left (f x +e \right )+1562 A -1402 B \right )}{3465 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(105\) |
| parts | \(-\frac {2 a^{2} A \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{2}-14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{2} B \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (63 \sin \left (f x +e \right )^{5}-224 \sin \left (f x +e \right )^{4}+355 \sin \left (f x +e \right )^{3}-426 \sin \left (f x +e \right )^{2}+568 \sin \left (f x +e \right )-1136\right )}{693 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{2} \left (A +2 B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4}-130 \sin \left (f x +e \right )^{3}+219 \sin \left (f x +e \right )^{2}-292 \sin \left (f x +e \right )+584\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{2} \left (2 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{3}-12 \sin \left (f x +e \right )^{2}+23 \sin \left (f x +e \right )-46\right )}{21 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(346\) |
Input:
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x,method=_R ETURNVERBOSE)
Output:
-2/3465*(sin(f*x+e)-1)*c^3*(1+sin(f*x+e))^3*a^2*(-315*B*cos(f*x+e)^2*sin(f *x+e)+(-385*A+980*B)*cos(f*x+e)^2+(-1210*A+1370*B)*sin(f*x+e)+1562*A-1402* B)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (151) = 302\).
Time = 0.09 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.87 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (315 \, B a^{2} c^{2} \cos \left (f x + e\right )^{6} - 35 \, {\left (11 \, A - 10 \, B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{5} + 5 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{3} + 16 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{2} - 64 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right ) - 128 \, {\left (11 \, A - B\right )} a^{2} c^{2} - {\left (315 \, B a^{2} c^{2} \cos \left (f x + e\right )^{5} + 35 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{4} + 40 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{3} + 48 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{2} + 64 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right ) + 128 \, {\left (11 \, A - B\right )} a^{2} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3465 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="fricas")
Output:
-2/3465*(315*B*a^2*c^2*cos(f*x + e)^6 - 35*(11*A - 10*B)*a^2*c^2*cos(f*x + e)^5 + 5*(11*A - B)*a^2*c^2*cos(f*x + e)^4 - 8*(11*A - B)*a^2*c^2*cos(f*x + e)^3 + 16*(11*A - B)*a^2*c^2*cos(f*x + e)^2 - 64*(11*A - B)*a^2*c^2*cos (f*x + e) - 128*(11*A - B)*a^2*c^2 - (315*B*a^2*c^2*cos(f*x + e)^5 + 35*(1 1*A - B)*a^2*c^2*cos(f*x + e)^4 + 40*(11*A - B)*a^2*c^2*cos(f*x + e)^3 + 4 8*(11*A - B)*a^2*c^2*cos(f*x + e)^2 + 64*(11*A - B)*a^2*c^2*cos(f*x + e) + 128*(11*A - B)*a^2*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f* x + e) - f*sin(f*x + e) + f)
\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=a^{2} \left (\int A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- 2 A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\, dx + \int B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{5}{\left (e + f x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(5/2),x)
Output:
a**2*(Integral(A*c**2*sqrt(-c*sin(e + f*x) + c), x) + Integral(-2*A*c**2*s qrt(-c*sin(e + f*x) + c)*sin(e + f*x)**2, x) + Integral(A*c**2*sqrt(-c*sin (e + f*x) + c)*sin(e + f*x)**4, x) + Integral(B*c**2*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x), x) + Integral(-2*B*c**2*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**3, x) + Integral(B*c**2*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**5 , x))
\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^2*(-c*sin(f*x + e) + c )^(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (151) = 302\).
Time = 0.38 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.04 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (315 \, B a^{2} c^{2} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6930 \, {\left (6 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2310 \, {\left (4 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) - 693 \, {\left (8 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) - 495 \, {\left (2 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 385 \, {\left (2 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right )\right )} \sqrt {c}}{55440 \, f} \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="giac")
Output:
-1/55440*sqrt(2)*(315*B*a^2*c^2*cos(-11/4*pi + 11/2*f*x + 11/2*e)*sgn(sin( -1/4*pi + 1/2*f*x + 1/2*e)) + 6930*(6*A*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - B*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1 /2*f*x + 1/2*e) + 2310*(4*A*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3/4*pi + 3/2*f*x + 3/2 *e) - 693*(8*A*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*B*a^2*c^2*s gn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-5/4*pi + 5/2*f*x + 5/2*e) - 495*( 2*A*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a^2*c^2*sgn(sin(-1/4 *pi + 1/2*f*x + 1/2*e)))*cos(-7/4*pi + 7/2*f*x + 7/2*e) + 385*(2*A*a^2*c^2 *sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - B*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-9/4*pi + 9/2*f*x + 9/2*e))*sqrt(c)/f
Timed out. \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(5/2) ,x)
Output:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(5/2) , x)
\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {c}\, a^{2} c^{2} \left (\left (\int \sqrt {-\sin \left (f x +e \right )+1}d x \right ) a +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{5}d x \right ) b +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) a -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) b \right ) \] Input:
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(c)*a**2*c**2*(int(sqrt( - sin(e + f*x) + 1),x)*a + int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**5,x)*b + int(sqrt( - sin(e + f*x) + 1)*sin(e + f *x)**4,x)*a - 2*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3,x)*b - 2*int (sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2,x)*a + int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x),x)*b)