Integrand size = 27, antiderivative size = 202 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=-\frac {64 a^3 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{315 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}-\frac {2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac {4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f} \] Output:
-64/315*a^3*(21*c^2+30*c*d+13*d^2)*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-16/ 315*a^2*(21*c^2+30*c*d+13*d^2)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f-2/105*a *(21*c^2+30*c*d+13*d^2)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f-4/63*(9*c-d)*d *cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f-2/9*d^2*cos(f*x+e)*(a+a*sin(f*x+e))^( 7/2)/a/f
Time = 4.41 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.89 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (7476 c^2+12480 c d+5653 d^2-4 \left (63 c^2+360 c d+254 d^2\right ) \cos (2 (e+f x))+35 d^2 \cos (4 (e+f x))+2352 c^2 \sin (e+f x)+6060 c d \sin (e+f x)+3116 d^2 \sin (e+f x)-180 c d \sin (3 (e+f x))-260 d^2 \sin (3 (e+f x))\right )}{1260 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2,x]
Output:
-1/1260*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x ])]*(7476*c^2 + 12480*c*d + 5653*d^2 - 4*(63*c^2 + 360*c*d + 254*d^2)*Cos[ 2*(e + f*x)] + 35*d^2*Cos[4*(e + f*x)] + 2352*c^2*Sin[e + f*x] + 6060*c*d* Sin[e + f*x] + 3116*d^2*Sin[e + f*x] - 180*c*d*Sin[3*(e + f*x)] - 260*d^2* Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))
Time = 0.80 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3240, 27, 3042, 3230, 3042, 3126, 3042, 3126, 3042, 3125}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2dx\) |
| \(\Big \downarrow \) 3240 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (\sin (e+f x) a+a)^{5/2} \left (a \left (9 c^2+7 d^2\right )+2 a (9 c-d) d \sin (e+f x)\right )dx}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^{5/2} \left (a \left (9 c^2+7 d^2\right )+2 a (9 c-d) d \sin (e+f x)\right )dx}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^{5/2} \left (a \left (9 c^2+7 d^2\right )+2 a (9 c-d) d \sin (e+f x)\right )dx}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {3}{7} a \left (21 c^2+30 c d+13 d^2\right ) \int (\sin (e+f x) a+a)^{5/2}dx-\frac {4 a d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a \left (21 c^2+30 c d+13 d^2\right ) \int (\sin (e+f x) a+a)^{5/2}dx-\frac {4 a d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {\frac {3}{7} a \left (21 c^2+30 c d+13 d^2\right ) \left (\frac {8}{5} a \int (\sin (e+f x) a+a)^{3/2}dx-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\right )-\frac {4 a d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a \left (21 c^2+30 c d+13 d^2\right ) \left (\frac {8}{5} a \int (\sin (e+f x) a+a)^{3/2}dx-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\right )-\frac {4 a d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {\frac {3}{7} a \left (21 c^2+30 c d+13 d^2\right ) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\right )-\frac {4 a d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a \left (21 c^2+30 c d+13 d^2\right ) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\right )-\frac {4 a d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {3}{7} a \left (21 c^2+30 c d+13 d^2\right ) \left (\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\right )-\frac {4 a d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}}{9 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}\) |
Input:
Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2,x]
Output:
(-2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(9*a*f) + ((-4*a*(9*c - d )*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(7*f) + (3*a*(21*c^2 + 30*c*d + 13*d^2)*((-2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) + (8*a*(( -8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sq rt[a + a*Sin[e + f*x]])/(3*f)))/5))/7)/(9*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^2, x_Symbol] :> Simp[(-d^2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^ (m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^ m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && !LtQ[m, -1]
Time = 7.51 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4} d^{2}+90 \sin \left (f x +e \right )^{3} c d +130 \sin \left (f x +e \right )^{3} d^{2}+63 \sin \left (f x +e \right )^{2} c^{2}+360 \sin \left (f x +e \right )^{2} c d +219 \sin \left (f x +e \right )^{2} d^{2}+294 c^{2} \sin \left (f x +e \right )+690 d \sin \left (f x +e \right ) c +292 d^{2} \sin \left (f x +e \right )+903 c^{2}+1380 c d +584 d^{2}\right )}{315 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(168\) |
| parts | \(\frac {2 c^{2} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{2}+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {2 d^{2} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4}+130 \sin \left (f x +e \right )^{3}+219 \sin \left (f x +e \right )^{2}+292 \sin \left (f x +e \right )+584\right )}{315 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {4 c d \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{3}+12 \sin \left (f x +e \right )^{2}+23 \sin \left (f x +e \right )+46\right )}{21 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(232\) |
Input:
int((a+sin(f*x+e)*a)^(5/2)*(c+d*sin(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
2/315*(1+sin(f*x+e))*a^3*(-1+sin(f*x+e))*(35*sin(f*x+e)^4*d^2+90*sin(f*x+e )^3*c*d+130*sin(f*x+e)^3*d^2+63*sin(f*x+e)^2*c^2+360*sin(f*x+e)^2*c*d+219* sin(f*x+e)^2*d^2+294*c^2*sin(f*x+e)+690*d*sin(f*x+e)*c+292*d^2*sin(f*x+e)+ 903*c^2+1380*c*d+584*d^2)/cos(f*x+e)/(a+sin(f*x+e)*a)^(1/2)/f
Time = 0.09 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.68 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=-\frac {2 \, {\left (35 \, a^{2} d^{2} \cos \left (f x + e\right )^{5} - 5 \, {\left (18 \, a^{2} c d + 19 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} - {\left (63 \, a^{2} c^{2} + 360 \, a^{2} c d + 289 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (231 \, a^{2} c^{2} + 510 \, a^{2} c d + 263 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (483 \, a^{2} c^{2} + 870 \, a^{2} c d + 419 \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (35 \, a^{2} d^{2} \cos \left (f x + e\right )^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} + 10 \, {\left (9 \, a^{2} c d + 13 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (21 \, a^{2} c^{2} + 90 \, a^{2} c d + 53 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (147 \, a^{2} c^{2} + 390 \, a^{2} c d + 211 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^2,x, algorithm="fricas")
Output:
-2/315*(35*a^2*d^2*cos(f*x + e)^5 - 5*(18*a^2*c*d + 19*a^2*d^2)*cos(f*x + e)^4 + 672*a^2*c^2 + 960*a^2*c*d + 416*a^2*d^2 - (63*a^2*c^2 + 360*a^2*c*d + 289*a^2*d^2)*cos(f*x + e)^3 + (231*a^2*c^2 + 510*a^2*c*d + 263*a^2*d^2) *cos(f*x + e)^2 + 2*(483*a^2*c^2 + 870*a^2*c*d + 419*a^2*d^2)*cos(f*x + e) - (35*a^2*d^2*cos(f*x + e)^4 + 672*a^2*c^2 + 960*a^2*c*d + 416*a^2*d^2 + 10*(9*a^2*c*d + 13*a^2*d^2)*cos(f*x + e)^3 - 3*(21*a^2*c^2 + 90*a^2*c*d + 53*a^2*d^2)*cos(f*x + e)^2 - 2*(147*a^2*c^2 + 390*a^2*c*d + 211*a^2*d^2)*c os(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*si n(f*x + e) + f)
\[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{2}\, dx \] Input:
integrate((a+a*sin(f*x+e))**(5/2)*(c+d*sin(f*x+e))**2,x)
Output:
Integral((a*(sin(e + f*x) + 1))**(5/2)*(c + d*sin(e + f*x))**2, x)
\[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{2} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^2,x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^2, x)
Time = 0.19 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.64 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=\frac {\sqrt {2} {\left (35 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) + 630 \, {\left (20 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 30 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 13 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 420 \, {\left (5 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 11 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 252 \, {\left (a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 45 \, {\left (4 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right )\right )} \sqrt {a}}{2520 \, f} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^2,x, algorithm="giac")
Output:
1/2520*sqrt(2)*(35*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-9/4*pi + 9/2*f*x + 9/2*e) + 630*(20*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 13*a^2*d^2*sgn(cos(-1/4 *pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 420*(5*a^2*c^2*s gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 11*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3 /2*f*x + 3/2*e) + 252*(a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*a^2 *c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*a^2*d^2*sgn(cos(-1/4*pi + 1/2 *f*x + 1/2*e)))*sin(-5/4*pi + 5/2*f*x + 5/2*e) + 45*(4*a^2*c*d*sgn(cos(-1/ 4*pi + 1/2*f*x + 1/2*e)) + 5*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))* sin(-7/4*pi + 7/2*f*x + 7/2*e))*sqrt(a)/f
Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \] Input:
int((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2,x)
Output:
int((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2, x)
\[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {\sin \left (f x +e \right )+1}d x \right ) c^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) c d +2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) c^{2}+4 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) c d \right ) \] Input:
int((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^2,x)
Output:
sqrt(a)*a**2*(int(sqrt(sin(e + f*x) + 1),x)*c**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**4,x)*d**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3 ,x)*c*d + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3,x)*d**2 + int(sqrt( sin(e + f*x) + 1)*sin(e + f*x)**2,x)*c**2 + 4*int(sqrt(sin(e + f*x) + 1)*s in(e + f*x)**2,x)*c*d + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*d**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*c**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*c*d)