Integrand size = 27, antiderivative size = 328 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=-\frac {4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x)}{3465 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d f}-\frac {4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}-\frac {2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f} \] Output:
-4/3465*a^3*(c+d)*(15*c^2+10*c*d+7*d^2)*(3*c^2-38*c*d+355*d^2)*cos(f*x+e)/ d^2/f/(a+a*sin(f*x+e))^(1/2)-8/3465*a^2*(5*c-d)*(c+d)*(3*c^2-38*c*d+355*d^ 2)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d/f-4/1155*a*(c+d)*(3*c^2-38*c*d+355* d^2)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f-2/693*a^3*(3*c^2-38*c*d+355*d^2)* cos(f*x+e)*(c+d*sin(f*x+e))^3/d^2/f/(a+a*sin(f*x+e))^(1/2)+2/99*a^3*(3*c-2 3*d)*cos(f*x+e)*(c+d*sin(f*x+e))^4/d^2/f/(a+a*sin(f*x+e))^(1/2)-2/11*a^2*c os(f*x+e)*(a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^4/d/f
Time = 6.52 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.75 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (164472 c^3+411840 c^2 d+373098 c d^2+114640 d^3-8 \left (693 c^3+5940 c^2 d+8382 c d^2+3250 d^3\right ) \cos (2 (e+f x))+70 d^2 (33 c+32 d) \cos (4 (e+f x))+51744 c^3 \sin (e+f x)+199980 c^2 d \sin (e+f x)+205656 c d^2 \sin (e+f x)+69890 d^3 \sin (e+f x)-5940 c^2 d \sin (3 (e+f x))-17160 c d^2 \sin (3 (e+f x))-8675 d^3 \sin (3 (e+f x))+315 d^3 \sin (5 (e+f x))\right )}{27720 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3,x]
Output:
-1/27720*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f* x])]*(164472*c^3 + 411840*c^2*d + 373098*c*d^2 + 114640*d^3 - 8*(693*c^3 + 5940*c^2*d + 8382*c*d^2 + 3250*d^3)*Cos[2*(e + f*x)] + 70*d^2*(33*c + 32* d)*Cos[4*(e + f*x)] + 51744*c^3*Sin[e + f*x] + 199980*c^2*d*Sin[e + f*x] + 205656*c*d^2*Sin[e + f*x] + 69890*d^3*Sin[e + f*x] - 5940*c^2*d*Sin[3*(e + f*x)] - 17160*c*d^2*Sin[3*(e + f*x)] - 8675*d^3*Sin[3*(e + f*x)] + 315*d ^3*Sin[5*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))
Time = 1.39 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3242, 27, 3042, 3460, 3042, 3249, 3042, 3240, 27, 3042, 3230, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} \left (a^2 (c+19 d)-a^2 (3 c-23 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3dx}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {\sin (e+f x) a+a} \left (a^2 (c+19 d)-a^2 (3 c-23 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3dx}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\sin (e+f x) a+a} \left (a^2 (c+19 d)-a^2 (3 c-23 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3dx}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3dx}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3dx}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3240 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \left (\frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 \left (3 c^2-38 c d+355 d^2\right ) \left (\frac {6}{7} (c+d) \left (\frac {-\frac {2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}}{11 d}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}\) |
Input:
Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3,x]
Output:
(-2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^4)/(11* d*f) + ((2*a^3*(3*c - 23*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(9*d*f*Sq rt[a + a*Sin[e + f*x]]) + (a^2*(3*c^2 - 38*c*d + 355*d^2)*((-2*a*Cos[e + f *x]*(c + d*Sin[e + f*x])^3)/(7*f*Sqrt[a + a*Sin[e + f*x]]) + (6*(c + d)*(( -2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*a*f) + ((-2*a^2*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a*(5* c - d)*d*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f))/(5*a)))/7))/(9*d))/ (11*d)
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[(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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
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] + Simp[2*n*((b*c + a*d)/(b*( 2*n + 1))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -1]
Time = 38.36 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (315 \sin \left (f x +e \right )^{5} d^{3}+1155 \sin \left (f x +e \right )^{4} c \,d^{2}+1120 \sin \left (f x +e \right )^{4} d^{3}+1485 \sin \left (f x +e \right )^{3} c^{2} d +4290 \sin \left (f x +e \right )^{3} c \,d^{2}+1775 d^{3} \sin \left (f x +e \right )^{3}+693 \sin \left (f x +e \right )^{2} c^{3}+5940 \sin \left (f x +e \right )^{2} c^{2} d +7227 c \,d^{2} \sin \left (f x +e \right )^{2}+2130 d^{3} \sin \left (f x +e \right )^{2}+3234 c^{3} \sin \left (f x +e \right )+11385 \sin \left (f x +e \right ) c^{2} d +9636 \sin \left (f x +e \right ) c \,d^{2}+2840 d^{3} \sin \left (f x +e \right )+9933 c^{3}+22770 c^{2} d +19272 c \,d^{2}+5680 d^{3}\right )}{3465 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(249\) |
| parts | \(\frac {2 c^{3} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{2}+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {2 d^{3} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (63 \sin \left (f x +e \right )^{5}+224 \sin \left (f x +e \right )^{4}+355 \sin \left (f x +e \right )^{3}+426 \sin \left (f x +e \right )^{2}+568 \sin \left (f x +e \right )+1136\right )}{693 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {2 c \,d^{2} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4}+130 \sin \left (f x +e \right )^{3}+219 \sin \left (f x +e \right )^{2}+292 \sin \left (f x +e \right )+584\right )}{105 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {2 c^{2} d \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{3}+12 \sin \left (f x +e \right )^{2}+23 \sin \left (f x +e \right )+46\right )}{7 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(332\) |
Input:
int((a+sin(f*x+e)*a)^(5/2)*(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
2/3465*(1+sin(f*x+e))*a^3*(-1+sin(f*x+e))*(315*sin(f*x+e)^5*d^3+1155*sin(f *x+e)^4*c*d^2+1120*sin(f*x+e)^4*d^3+1485*sin(f*x+e)^3*c^2*d+4290*sin(f*x+e )^3*c*d^2+1775*d^3*sin(f*x+e)^3+693*sin(f*x+e)^2*c^3+5940*sin(f*x+e)^2*c^2 *d+7227*c*d^2*sin(f*x+e)^2+2130*d^3*sin(f*x+e)^2+3234*c^3*sin(f*x+e)+11385 *sin(f*x+e)*c^2*d+9636*sin(f*x+e)*c*d^2+2840*d^3*sin(f*x+e)+9933*c^3+22770 *c^2*d+19272*c*d^2+5680*d^3)/cos(f*x+e)/(a+sin(f*x+e)*a)^(1/2)/f
Time = 0.10 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.50 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=-\frac {2 \, {\left (315 \, a^{2} d^{3} \cos \left (f x + e\right )^{6} + 35 \, {\left (33 \, a^{2} c d^{2} + 32 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} + 7392 \, a^{2} c^{3} + 15840 \, a^{2} c^{2} d + 13728 \, a^{2} c d^{2} + 4000 \, a^{2} d^{3} - 5 \, {\left (297 \, a^{2} c^{2} d + 627 \, a^{2} c d^{2} + 320 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} - {\left (693 \, a^{2} c^{3} + 5940 \, a^{2} c^{2} d + 9537 \, a^{2} c d^{2} + 4370 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (2541 \, a^{2} c^{3} + 8415 \, a^{2} c^{2} d + 8679 \, a^{2} c d^{2} + 2965 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5313 \, a^{2} c^{3} + 14355 \, a^{2} c^{2} d + 13827 \, a^{2} c d^{2} + 4465 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) + {\left (315 \, a^{2} d^{3} \cos \left (f x + e\right )^{5} - 7392 \, a^{2} c^{3} - 15840 \, a^{2} c^{2} d - 13728 \, a^{2} c d^{2} - 4000 \, a^{2} d^{3} - 35 \, {\left (33 \, a^{2} c d^{2} + 23 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \, {\left (297 \, a^{2} c^{2} d + 858 \, a^{2} c d^{2} + 481 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (231 \, a^{2} c^{3} + 1485 \, a^{2} c^{2} d + 1749 \, a^{2} c d^{2} + 655 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (1617 \, a^{2} c^{3} + 6435 \, a^{2} c^{2} d + 6963 \, a^{2} c d^{2} + 2465 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3465 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-2/3465*(315*a^2*d^3*cos(f*x + e)^6 + 35*(33*a^2*c*d^2 + 32*a^2*d^3)*cos(f *x + e)^5 + 7392*a^2*c^3 + 15840*a^2*c^2*d + 13728*a^2*c*d^2 + 4000*a^2*d^ 3 - 5*(297*a^2*c^2*d + 627*a^2*c*d^2 + 320*a^2*d^3)*cos(f*x + e)^4 - (693* a^2*c^3 + 5940*a^2*c^2*d + 9537*a^2*c*d^2 + 4370*a^2*d^3)*cos(f*x + e)^3 + (2541*a^2*c^3 + 8415*a^2*c^2*d + 8679*a^2*c*d^2 + 2965*a^2*d^3)*cos(f*x + e)^2 + 2*(5313*a^2*c^3 + 14355*a^2*c^2*d + 13827*a^2*c*d^2 + 4465*a^2*d^3 )*cos(f*x + e) + (315*a^2*d^3*cos(f*x + e)^5 - 7392*a^2*c^3 - 15840*a^2*c^ 2*d - 13728*a^2*c*d^2 - 4000*a^2*d^3 - 35*(33*a^2*c*d^2 + 23*a^2*d^3)*cos( f*x + e)^4 - 5*(297*a^2*c^2*d + 858*a^2*c*d^2 + 481*a^2*d^3)*cos(f*x + e)^ 3 + 3*(231*a^2*c^3 + 1485*a^2*c^2*d + 1749*a^2*c*d^2 + 655*a^2*d^3)*cos(f* x + e)^2 + 2*(1617*a^2*c^3 + 6435*a^2*c^2*d + 6963*a^2*c*d^2 + 2465*a^2*d^ 3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)
\[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \] Input:
integrate((a+a*sin(f*x+e))**(5/2)*(c+d*sin(f*x+e))**3,x)
Output:
Integral((a*(sin(e + f*x) + 1))**(5/2)*(c + d*sin(e + f*x))**3, x)
\[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{3} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^3,x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^3, x)
Time = 0.22 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.48 \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^3,x, algorithm="giac")
Output:
1/55440*sqrt(2)*(315*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-11/4 *pi + 11/2*f*x + 11/2*e) + 6930*(40*a^2*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/ 2*e)) + 90*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 78*a^2*c*d^2*sg n(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 23*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 2310*(20*a^2*c^3*sgn(cos(-1/4*p i + 1/2*f*x + 1/2*e)) + 66*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 60*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 19*a^2*d^3*sgn(cos(-1/ 4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 693*(8*a^2*c^3* sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 60*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f *x + 1/2*e)) + 72*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 25*a^2*d ^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-5/4*pi + 5/2*f*x + 5/2*e) + 4 95*(12*a^2*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*a^2*c*d^2*sgn(co s(-1/4*pi + 1/2*f*x + 1/2*e)) + 13*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2 *e)))*sin(-7/4*pi + 7/2*f*x + 7/2*e) + 385*(6*a^2*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*a^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-9/ 4*pi + 9/2*f*x + 9/2*e))*sqrt(a)/f
Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \] Input:
int((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3,x)
Output:
int((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3, x)
\[ \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {\sin \left (f x +e \right )+1}d x \right ) c^{3}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{5}d x \right ) d^{3}+3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) c \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) d^{3}+3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) c^{2} d +6 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) c \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) d^{3}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) c^{3}+6 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) c^{2} d +3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) c \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) c^{3}+3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) c^{2} d \right ) \] Input:
int((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^3,x)
Output:
sqrt(a)*a**2*(int(sqrt(sin(e + f*x) + 1),x)*c**3 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**5,x)*d**3 + 3*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**4 ,x)*c*d**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**4,x)*d**3 + 3*int( sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3,x)*c**2*d + 6*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**3 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*c**3 + 6*int(sq rt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*c**2*d + 3*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*c*d**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x), x)*c**3 + 3*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*c**2*d)