Integrand size = 27, antiderivative size = 231 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\frac {4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt {a+a \sin (e+f x)}}+\frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}+\frac {4 (c-17 d) d (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}} \] Output:
4/315*a^2*(c-17*d)*(c+d)*(15*c^2+10*c*d+7*d^2)*cos(f*x+e)/d/f/(a+a*sin(f*x +e))^(1/2)+8/315*a*(c-17*d)*(5*c-d)*(c+d)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2 )/f+4/105*(c-17*d)*d*(c+d)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f+2/63*a^2*(c -17*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/d/f/(a+a*sin(f*x+e))^(1/2)-2/9*a^2*co s(f*x+e)*(c+d*sin(f*x+e))^4/d/f/(a+a*sin(f*x+e))^(1/2)
Time = 2.84 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (4200 c^3+9828 c^2 d+8892 c d^2+2689 d^3-4 d \left (189 c^2+351 c d+137 d^2\right ) \cos (2 (e+f x))+35 d^3 \cos (4 (e+f x))+840 c^3 \sin (e+f x)+4536 c^2 d \sin (e+f x)+4554 c d^2 \sin (e+f x)+1598 d^3 \sin (e+f x)-270 c d^2 \sin (3 (e+f x))-170 d^3 \sin (3 (e+f x))\right )}{1260 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3,x]
Output:
-1/1260*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x]) ]*(4200*c^3 + 9828*c^2*d + 8892*c*d^2 + 2689*d^3 - 4*d*(189*c^2 + 351*c*d + 137*d^2)*Cos[2*(e + f*x)] + 35*d^3*Cos[4*(e + f*x)] + 840*c^3*Sin[e + f* x] + 4536*c^2*d*Sin[e + f*x] + 4554*c*d^2*Sin[e + f*x] + 1598*d^3*Sin[e + f*x] - 270*c*d^2*Sin[3*(e + f*x)] - 170*d^3*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))
Time = 1.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3242, 27, 2011, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int -\frac {\left ((c-17 d) a^2+(c-17 d) \sin (e+f x) a^2\right ) (c+d \sin (e+f x))^3}{2 \sqrt {\sin (e+f x) a+a}}dx}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\left ((c-17 d) a^2+(c-17 d) \sin (e+f x) a^2\right ) (c+d \sin (e+f x))^3}{\sqrt {\sin (e+f x) a+a}}dx}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 2011 |
| \(\displaystyle -\frac {a (c-17 d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3dx}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a (c-17 d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3dx}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3240 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \left (\frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle -\frac {a (c-17 d) \left (\frac {6}{7} (c+d) \left (\frac {-\frac {2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}\) |
Input:
Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3,x]
Output:
(-2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(9*d*f*Sqrt[a + a*Sin[e + f*x ]]) - (a*(c - 17*d)*((-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)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + 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]
Time = 1.54 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4} d^{3}+135 \sin \left (f x +e \right )^{3} c \,d^{2}+85 d^{3} \sin \left (f x +e \right )^{3}+189 \sin \left (f x +e \right )^{2} c^{2} d +351 c \,d^{2} \sin \left (f x +e \right )^{2}+102 d^{3} \sin \left (f x +e \right )^{2}+105 c^{3} \sin \left (f x +e \right )+567 \sin \left (f x +e \right ) c^{2} d +468 \sin \left (f x +e \right ) c \,d^{2}+136 d^{3} \sin \left (f x +e \right )+525 c^{3}+1134 c^{2} d +936 c \,d^{2}+272 d^{3}\right )}{315 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(195\) |
| parts | \(\frac {2 c^{3} \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {2 d^{3} \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (35 \sin \left (f x +e \right )^{4}+85 \sin \left (f x +e \right )^{3}+102 \sin \left (f x +e \right )^{2}+136 \sin \left (f x +e \right )+272\right )}{315 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {2 c \,d^{2} \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (15 \sin \left (f x +e \right )^{3}+39 \sin \left (f x +e \right )^{2}+52 \sin \left (f x +e \right )+104\right )}{35 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}+\frac {6 c^{2} d \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (-1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(288\) |
Input:
int((a+sin(f*x+e)*a)^(3/2)*(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
2/315*(1+sin(f*x+e))*a^2*(-1+sin(f*x+e))*(35*sin(f*x+e)^4*d^3+135*sin(f*x+ e)^3*c*d^2+85*d^3*sin(f*x+e)^3+189*sin(f*x+e)^2*c^2*d+351*c*d^2*sin(f*x+e) ^2+102*d^3*sin(f*x+e)^2+105*c^3*sin(f*x+e)+567*sin(f*x+e)*c^2*d+468*sin(f* x+e)*c*d^2+136*d^3*sin(f*x+e)+525*c^3+1134*c^2*d+936*c*d^2+272*d^3)/cos(f* x+e)/(a+sin(f*x+e)*a)^(1/2)/f
Time = 0.09 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.47 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=-\frac {2 \, {\left (35 \, a d^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (27 \, a c d^{2} + 10 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} - {\left (189 \, a c^{2} d + 351 \, a c d^{2} + 172 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{3} + 378 \, a c^{2} d + 387 \, a c d^{2} + 134 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (525 \, a c^{3} + 1323 \, a c^{2} d + 1287 \, a c d^{2} + 409 \, a d^{3}\right )} \cos \left (f x + e\right ) - {\left (35 \, a d^{3} \cos \left (f x + e\right )^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} + 5 \, {\left (27 \, a c d^{2} + 17 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (63 \, a c^{2} d + 72 \, a c d^{2} + 29 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (105 \, a c^{3} + 567 \, a c^{2} d + 603 \, a c d^{2} + 221 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-2/315*(35*a*d^3*cos(f*x + e)^5 - 5*(27*a*c*d^2 + 10*a*d^3)*cos(f*x + e)^4 + 420*a*c^3 + 756*a*c^2*d + 684*a*c*d^2 + 188*a*d^3 - (189*a*c^2*d + 351* a*c*d^2 + 172*a*d^3)*cos(f*x + e)^3 + (105*a*c^3 + 378*a*c^2*d + 387*a*c*d ^2 + 134*a*d^3)*cos(f*x + e)^2 + (525*a*c^3 + 1323*a*c^2*d + 1287*a*c*d^2 + 409*a*d^3)*cos(f*x + e) - (35*a*d^3*cos(f*x + e)^4 + 420*a*c^3 + 756*a*c ^2*d + 684*a*c*d^2 + 188*a*d^3 + 5*(27*a*c*d^2 + 17*a*d^3)*cos(f*x + e)^3 - 3*(63*a*c^2*d + 72*a*c*d^2 + 29*a*d^3)*cos(f*x + e)^2 - (105*a*c^3 + 567 *a*c^2*d + 603*a*c*d^2 + 221*a*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)
\[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \] Input:
integrate((a+a*sin(f*x+e))**(3/2)*(c+d*sin(f*x+e))**3,x)
Output:
Integral((a*(sin(e + f*x) + 1))**(3/2)*(c + d*sin(e + f*x))**3, x)
\[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{3} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c)^3, x)
Time = 0.19 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.54 \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x, algorithm="giac")
Output:
1/2520*sqrt(2)*(35*a*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) + 1890*(4*a*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 8* a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 7*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*a*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) + 210*(4*a*c^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 18*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 15*a*c*d^2*sgn(cos(-1/4* pi + 1/2*f*x + 1/2*e)) + 5*a*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) + 378*(2*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/ 2*e)) + 3*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + a*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) + 135*(2*a*c*d^2*s gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + a*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))*sqrt(a)/f
Timed out. \[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \] Input:
int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^3,x)
Output:
int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^3, x)
\[ \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\sqrt {a}\, a \left (\left (\int \sqrt {\sin \left (f x +e \right )+1}d x \right ) c^{3}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) d^{3}+3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) c \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) d^{3}+3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) c^{2} d +3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) c \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) c^{3}+3 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) c^{2} d \right ) \] Input:
int((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x)
Output:
sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1),x)*c**3 + 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*d**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3,x)*d**3 + 3*int(sqrt( 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 + 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)