Integrand size = 38, antiderivative size = 161 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {64 a^3 (13 A+B) c^6 \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^{7/2}}+\frac {16 a^3 (13 A+B) c^5 \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^3 (13 A+B) c^4 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}} \] Output:
64/9009*a^3*(13*A+B)*c^6*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)+16/1287*a^3 *(13*A+B)*c^5*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(5/2)+2/143*a^3*(13*A+B)*c^4 *cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(3/2)-2/13*a^3*B*c^3*cos(f*x+e)^7/f/(c-c* sin(f*x+e))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1351\) vs. \(2(161)=322\).
Time = 13.54 (sec) , antiderivative size = 1351, normalized size of antiderivative = 8.39 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]) ^(5/2),x]
Output:
(5*A*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/( 8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f *x)/2])^6) - (5*(4*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(96*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A - B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((5*A + 2*B )*Cos[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/ (112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((A - 2*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(C os[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*A + B)*Cos[(11*(e + f*x))/2]* (a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(352*f*(Cos[(e + f*x)/2 ] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*Cos[ (13*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2))/(416* f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x )/2])^6) + (5*A*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x ])^(5/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (5*(4*A + B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)*Sin[(3*(e + f*x))/2])/(96*f*(Cos[(e + f*x)/2] - Sin[(e + f...
Time = 0.99 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{5/2} (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3335 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{13} (13 A+B) \int \frac {\cos ^6(e+f x)}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 B \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{13} (13 A+B) \int \frac {\cos (e+f x)^6}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 B \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{13} (13 A+B) \left (\frac {8}{11} c \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}}dx+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {2 B \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{13} (13 A+B) \left (\frac {8}{11} c \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{3/2}}dx+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {2 B \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{13} (13 A+B) \left (\frac {8}{11} c \left (\frac {4}{9} c \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {2 B \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{13} (13 A+B) \left (\frac {8}{11} c \left (\frac {4}{9} c \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{5/2}}dx+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {2 B \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{13} (13 A+B) \left (\frac {8}{11} c \left (\frac {8 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )+\frac {2 c \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {2 B \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2) ,x]
Output:
a^3*c^3*((-2*B*Cos[e + f*x]^7)/(13*f*Sqrt[c - c*Sin[e + f*x]]) + ((13*A + B)*((2*c*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^(3/2)) + (8*c*((8*c^2* Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) + (2*c*Cos[e + f*x]^7)/( 9*f*(c - c*Sin[e + f*x])^(5/2))))/11))/13)
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 = 191.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (-693 B \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )+\left (-819 A +2016 B \right ) \cos \left (f x +e \right )^{2}+\left (-2366 A +2590 B \right ) \sin \left (f x +e \right )+2782 A -2558 B \right )}{9009 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(105\) |
| parts | \(-\frac {2 a^{3} A \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{2}-14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} B \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3465 \sin \left (f x +e \right )^{6}-11970 \sin \left (f x +e \right )^{5}+18305 \sin \left (f x +e \right )^{4}-20920 \sin \left (f x +e \right )^{3}+25104 \sin \left (f x +e \right )^{2}-33472 \sin \left (f x +e \right )+66944\right )}{45045 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (A +3 B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (63 \sin \left (f x +e \right )^{5}-224 \sin \left (f x +e \right )^{4}+355 \sin \left (f x +e \right )^{3}-426 \sin \left (f x +e \right )^{2}+568 \sin \left (f x +e \right )-1136\right )}{693 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (3 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{3}-12 \sin \left (f x +e \right )^{2}+23 \sin \left (f x +e \right )-46\right )}{21 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4}-130 \sin \left (f x +e \right )^{3}+219 \sin \left (f x +e \right )^{2}-292 \sin \left (f x +e \right )+584\right )}{105 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(457\) |
Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x,method=_R ETURNVERBOSE)
Output:
-2/9009*(sin(f*x+e)-1)*c^3*(1+sin(f*x+e))^4*a^3*(-693*B*cos(f*x+e)^2*sin(f *x+e)+(-819*A+2016*B)*cos(f*x+e)^2+(-2366*A+2590*B)*sin(f*x+e)+2782*A-2558 *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. 334 vs. \(2 (145) = 290\).
Time = 0.09 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.07 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (693 \, B a^{3} c^{2} \cos \left (f x + e\right )^{7} + 63 \, {\left (13 \, A + 12 \, B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{6} - 7 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{5} + 10 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{4} - 16 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{3} + 32 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{2} - 128 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right ) - 256 \, {\left (13 \, A + B\right )} a^{3} c^{2} + {\left (693 \, B a^{3} c^{2} \cos \left (f x + e\right )^{6} - 63 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{5} - 70 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{4} - 80 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{3} - 96 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{2} - 128 \, {\left (13 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right ) - 256 \, {\left (13 \, A + B\right )} a^{3} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{9009 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="fricas")
Output:
-2/9009*(693*B*a^3*c^2*cos(f*x + e)^7 + 63*(13*A + 12*B)*a^3*c^2*cos(f*x + e)^6 - 7*(13*A + B)*a^3*c^2*cos(f*x + e)^5 + 10*(13*A + B)*a^3*c^2*cos(f* x + e)^4 - 16*(13*A + B)*a^3*c^2*cos(f*x + e)^3 + 32*(13*A + B)*a^3*c^2*co s(f*x + e)^2 - 128*(13*A + B)*a^3*c^2*cos(f*x + e) - 256*(13*A + B)*a^3*c^ 2 + (693*B*a^3*c^2*cos(f*x + e)^6 - 63*(13*A + B)*a^3*c^2*cos(f*x + e)^5 - 70*(13*A + B)*a^3*c^2*cos(f*x + e)^4 - 80*(13*A + B)*a^3*c^2*cos(f*x + e) ^3 - 96*(13*A + B)*a^3*c^2*cos(f*x + e)^2 - 128*(13*A + B)*a^3*c^2*cos(f*x + e) - 256*(13*A + B)*a^3*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f *cos(f*x + e) - f*sin(f*x + e) + f)
Timed out. \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3*(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)^3*(-c*sin(f*x + e) + c )^(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (145) = 290\).
Time = 0.40 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.31 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="giac")
Output:
-1/288288*sqrt(2)*(180180*A*a^3*c^2*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(sin (-1/4*pi + 1/2*f*x + 1/2*e)) + 693*B*a^3*c^2*cos(-13/4*pi + 13/2*f*x + 13/ 2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 15015*(4*A*a^3*c^2*sgn(sin(-1/4 *pi + 1/2*f*x + 1/2*e)) + B*a^3*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*c os(-3/4*pi + 3/2*f*x + 3/2*e) - 9009*(2*A*a^3*c^2*sgn(sin(-1/4*pi + 1/2*f* x + 1/2*e)) - B*a^3*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-5/4*pi + 5/2*f*x + 5/2*e) - 2574*(5*A*a^3*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*B*a^3*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) + 2002*(A*a^3*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*B*a^3*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) + 81 9*(2*A*a^3*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^3*c^2*sgn(sin(-1/ 4*pi + 1/2*f*x + 1/2*e)))*cos(-11/4*pi + 11/2*f*x + 11/2*e))*sqrt(c)/f
Timed out. \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(5/2) ,x)
Output:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(5/2) , x)
\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {c}\, a^{3} c^{2} \left (\left (\int \sqrt {-\sin \left (f x +e \right )+1}d x \right ) a +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{6}d x \right ) b +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{5}d x \right ) a +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{5}d x \right ) b +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) a -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) b -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) a -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) b \right ) \] Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(c)*a**3*c**2*(int(sqrt( - sin(e + f*x) + 1),x)*a + int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**6,x)*b + int(sqrt( - sin(e + f*x) + 1)*sin(e + f *x)**5,x)*a + int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**5,x)*b + int(sqr t( - sin(e + f*x) + 1)*sin(e + f*x)**4,x)*a - 2*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**4,x)*b - 2*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3 ,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)**2,x)*b + int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x),x)*a + i nt(sqrt( - sin(e + f*x) + 1)*sin(e + f*x),x)*b)