Integrand size = 38, antiderivative size = 124 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a^3 (11 A+3 B) c^5 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 (11 A+3 B) c^4 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}} \] Output:
8/693*a^3*(11*A+3*B)*c^5*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)+2/99*a^3*(1 1*A+3*B)*c^4*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(5/2)-2/11*a^3*B*c^3*cos(f*x+ e)^7/f/(c-c*sin(f*x+e))^(3/2)
Leaf count is larger than twice the leaf count of optimal. \(1157\) vs. \(2(124)=248\).
Time = 13.37 (sec) , antiderivative size = 1157, normalized size of antiderivative = 9.33 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]) ^(3/2),x]
Output:
((6*A + B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3 /2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[ (e + f*x)/2])^6) - ((8*A + 3*B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^ 3*(c - c*Sin[e + f*x])^(3/2))/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^ 3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(16*f*(Cos[(e + f*x)/2] - Si n[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((6*A + B)*Co s[(7*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(112 *f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f* x)/2])^6) - ((2*A + 3*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (B*Cos[(11*(e + f*x))/2]*(a + a*Sin [e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((6*A + B)*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((8*A + 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(3*(e + f*x))/2])/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/ 2] + Sin[(e + f*x)/2])^6) - (B*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x]) ^(3/2)*Sin[(5*(e + f*x))/2])/(16*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]...
Time = 0.82 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3446, 3042, 3335, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{3/2} (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{3/2} (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3335 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{11} (11 A+3 B) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}}dx-\frac {2 B \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{11} (11 A+3 B) \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{3/2}}dx-\frac {2 B \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{11} (11 A+3 B) \left (\frac {4}{9} c \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{11} (11 A+3 B) \left (\frac {4}{9} c \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{5/2}}dx+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle a^3 c^3 \left (\frac {1}{11} (11 A+3 B) \left (\frac {8 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 c \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2) ,x]
Output:
a^3*c^3*((-2*B*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^(3/2)) + ((11*A + 3*B)*((8*c^2*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) + (2*c*Co s[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^(5/2))))/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 = 2.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (-63 B \cos \left (f x +e \right )^{2}+\sin \left (f x +e \right ) \left (77 A -105 B \right )-121 A +93 B \right )}{693 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(83\) |
| parts | \(\frac {2 a^{3} A \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-5\right )}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} B \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (15 \sin \left (f x +e \right )^{5}-35 \sin \left (f x +e \right )^{4}+40 \sin \left (f x +e \right )^{3}-48 \sin \left (f x +e \right )^{2}+64 \sin \left (f x +e \right )-128\right )}{165 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (A +3 B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4}-85 \sin \left (f x +e \right )^{3}+102 \sin \left (f x +e \right )^{2}-136 \sin \left (f x +e \right )+272\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (3 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )^{2}-3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (15 \sin \left (f x +e \right )^{3}-39 \sin \left (f x +e \right )^{2}+52 \sin \left (f x +e \right )-104\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(403\) |
Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x,method=_R ETURNVERBOSE)
Output:
2/693*(sin(f*x+e)-1)*c^2*(1+sin(f*x+e))^4*a^3*(-63*B*cos(f*x+e)^2+sin(f*x+ e)*(77*A-105*B)-121*A+93*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. 287 vs. \(2 (112) = 224\).
Time = 0.09 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.31 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {2 \, {\left (63 \, B a^{3} c \cos \left (f x + e\right )^{6} - 7 \, {\left (11 \, A + 12 \, B\right )} a^{3} c \cos \left (f x + e\right )^{5} - {\left (187 \, A + 177 \, B\right )} a^{3} c \cos \left (f x + e\right )^{4} + 2 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{3} - 4 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{2} + 16 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right ) + 32 \, {\left (11 \, A + 3 \, B\right )} a^{3} c - {\left (63 \, B a^{3} c \cos \left (f x + e\right )^{5} + 7 \, {\left (11 \, A + 21 \, B\right )} a^{3} c \cos \left (f x + e\right )^{4} - 10 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{3} - 12 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{2} - 16 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right ) - 32 \, {\left (11 \, A + 3 \, B\right )} a^{3} c\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{693 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x, al gorithm="fricas")
Output:
2/693*(63*B*a^3*c*cos(f*x + e)^6 - 7*(11*A + 12*B)*a^3*c*cos(f*x + e)^5 - (187*A + 177*B)*a^3*c*cos(f*x + e)^4 + 2*(11*A + 3*B)*a^3*c*cos(f*x + e)^3 - 4*(11*A + 3*B)*a^3*c*cos(f*x + e)^2 + 16*(11*A + 3*B)*a^3*c*cos(f*x + e ) + 32*(11*A + 3*B)*a^3*c - (63*B*a^3*c*cos(f*x + e)^5 + 7*(11*A + 21*B)*a ^3*c*cos(f*x + e)^4 - 10*(11*A + 3*B)*a^3*c*cos(f*x + e)^3 - 12*(11*A + 3* B)*a^3*c*cos(f*x + e)^2 - 16*(11*A + 3*B)*a^3*c*cos(f*x + e) - 32*(11*A + 3*B)*a^3*c)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*si n(f*x + e) + f)
\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=a^{3} \left (\int A c \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int 2 A c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 A c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- A c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\right )\, dx + \int B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int 2 B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\right )\, dx + \int \left (- B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{5}{\left (e + f x \right )}\right )\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(3/2),x)
Output:
a**3*(Integral(A*c*sqrt(-c*sin(e + f*x) + c), x) + Integral(2*A*c*sqrt(-c* sin(e + f*x) + c)*sin(e + f*x), x) + Integral(-2*A*c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**3, x) + Integral(-A*c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**4, x) + Integral(B*c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x), x) + I ntegral(2*B*c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**2, x) + Integral(-2* B*c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**4, x) + Integral(-B*c*sqrt(-c* sin(e + f*x) + c)*sin(e + f*x)**5, x))
\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),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 )^(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (112) = 224\).
Time = 0.37 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.37 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (693 \, B a^{3} c \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 63 \, B a^{3} c \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 ) + 1386 \, {\left (6 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{3} c \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 ) + 462 \, {\left (8 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{3} c \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 ) - 99 \, {\left (6 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{3} c \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 ) - 77 \, {\left (2 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{3} c \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}}{11088 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x, al gorithm="giac")
Output:
-1/11088*sqrt(2)*(693*B*a^3*c*cos(-5/4*pi + 5/2*f*x + 5/2*e)*sgn(sin(-1/4* pi + 1/2*f*x + 1/2*e)) - 63*B*a^3*c*cos(-11/4*pi + 11/2*f*x + 11/2*e)*sgn( sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 1386*(6*A*a^3*c*sgn(sin(-1/4*pi + 1/2*f* x + 1/2*e)) + B*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1 /2*f*x + 1/2*e) + 462*(8*A*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B *a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3/4*pi + 3/2*f*x + 3/2*e) - 99*(6*A*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^3*c*sgn(sin(-1/ 4*pi + 1/2*f*x + 1/2*e)))*cos(-7/4*pi + 7/2*f*x + 7/2*e) - 77*(2*A*a^3*c*s gn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a^3*c*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. \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \] Input:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(3/2) ,x)
Output:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(3/2) , x)
\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\sqrt {c}\, a^{3} c \left (\left (\int \sqrt {-\sin \left (f x +e \right )+1}d x \right ) a -\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{5}d x \right ) b -\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) a -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) b -2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) a +2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b +2 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a +\left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) b \right ) \] Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x)
Output:
sqrt(c)*a**3*c*(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)**4,x)*b - 2*int(sq rt( - sin(e + f*x) + 1)*sin(e + f*x)**3,x)*a + 2*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2,x)*b + 2*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x),x )*a + int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x),x)*b)