Integrand size = 25, antiderivative size = 215 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {1}{16} a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) x-\frac {4 a^3 (c+d)^3 \cos (e+f x)}{f}+\frac {a^3 (c+d)^2 (c+7 d) \cos ^3(e+f x)}{3 f}-\frac {3 a^3 d^2 (c+d) \cos ^5(e+f x)}{5 f}-\frac {a^3 \left (24 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \cos (e+f x) \sin (e+f x)}{16 f}-\frac {a^3 d \left (18 c^2+54 c d+23 d^2\right ) \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac {a^3 d^3 \cos (e+f x) \sin ^5(e+f x)}{6 f} \] Output:
1/16*a^3*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*x-4*a^3*(c+d)^3*cos(f*x+e)/f+1/ 3*a^3*(c+d)^2*(c+7*d)*cos(f*x+e)^3/f-3/5*a^3*d^2*(c+d)*cos(f*x+e)^5/f-1/16 *a^3*(24*c^3+90*c^2*d+78*c*d^2+23*d^3)*cos(f*x+e)*sin(f*x+e)/f-1/24*a^3*d* (18*c^2+54*c*d+23*d^2)*cos(f*x+e)*sin(f*x+e)^3/f-1/6*a^3*d^3*cos(f*x+e)*si n(f*x+e)^5/f
Time = 1.83 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.08 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {a^3 \cos (e+f x) \left (30 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (16 \left (55 c^3+135 c^2 d+114 c d^2+34 d^3\right )+15 \left (24 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \sin (e+f x)+16 \left (5 c^3+45 c^2 d+57 c d^2+17 d^3\right ) \sin ^2(e+f x)+10 d \left (18 c^2+54 c d+23 d^2\right ) \sin ^3(e+f x)+144 d^2 (c+d) \sin ^4(e+f x)+40 d^3 \sin ^5(e+f x)\right )\right )}{240 f \sqrt {\cos ^2(e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]
Output:
-1/240*(a^3*Cos[e + f*x]*(30*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*ArcSi n[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(16*(55*c^3 + 135 *c^2*d + 114*c*d^2 + 34*d^3) + 15*(24*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)* Sin[e + f*x] + 16*(5*c^3 + 45*c^2*d + 57*c*d^2 + 17*d^3)*Sin[e + f*x]^2 + 10*d*(18*c^2 + 54*c*d + 23*d^2)*Sin[e + f*x]^3 + 144*d^2*(c + d)*Sin[e + f *x]^4 + 40*d^3*Sin[e + f*x]^5)))/(f*Sqrt[Cos[e + f*x]^2])
Time = 1.34 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.59, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3242, 3042, 3447, 3042, 3502, 25, 3042, 3232, 27, 3042, 3232, 3042, 3213}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^3dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a) \left (a^2 (c+10 d)-a^2 (2 c-13 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3dx}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a) \left (a^2 (c+10 d)-a^2 (2 c-13 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3dx}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^3 \left (-\left ((2 c-13 d) \sin ^2(e+f x) a^3\right )+(c+10 d) a^3+\left (a^3 (c+10 d)-a^3 (2 c-13 d)\right ) \sin (e+f x)\right )dx}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^3 \left (-\left ((2 c-13 d) \sin (e+f x)^2 a^3\right )+(c+10 d) a^3+\left (a^3 (c+10 d)-a^3 (2 c-13 d)\right ) \sin (e+f x)\right )dx}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\int -(c+d \sin (e+f x))^3 \left (3 a^3 (c-34 d) d-a^3 \left (2 c^2-18 d c+115 d^2\right ) \sin (e+f x)\right )dx}{5 d}+\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\int (c+d \sin (e+f x))^3 \left (3 a^3 (c-34 d) d-a^3 \left (2 c^2-18 d c+115 d^2\right ) \sin (e+f x)\right )dx}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\int (c+d \sin (e+f x))^3 \left (3 a^3 (c-34 d) d-a^3 \left (2 c^2-18 d c+115 d^2\right ) \sin (e+f x)\right )dx}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {1}{4} \int 3 (c+d \sin (e+f x))^2 \left (a^3 d \left (2 c^2-118 d c-115 d^2\right )-a^3 \left (2 c^3-18 d c^2+111 d^2 c+136 d^3\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (a^3 d \left (2 c^2-118 d c-115 d^2\right )-a^3 \left (2 c^3-18 d c^2+111 d^2 c+136 d^3\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (a^3 d \left (2 c^2-118 d c-115 d^2\right )-a^3 \left (2 c^3-18 d c^2+111 d^2 c+136 d^3\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^3 d \left (2 c^3-318 d c^2-567 d^2 c-272 d^3\right )-a^3 \left (4 c^4-36 d c^3+216 d^2 c^2+626 d^3 c+345 d^4\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )+\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (a^3 d \left (2 c^3-318 d c^2-567 d^2 c-272 d^3\right )-a^3 \left (4 c^4-36 d c^3+216 d^2 c^2+626 d^3 c+345 d^4\right ) \sin (e+f x)\right )dx+\frac {a^3 \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )+\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\frac {a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {a^3 \left (2 c^2-18 c d+115 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}+\frac {3}{4} \left (\frac {a^3 \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}+\frac {1}{3} \left (-\frac {15}{2} a^3 d^2 x \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )+\frac {a^3 d \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right ) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {2 a^3 \left (2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right ) \cos (e+f x)}{f}\right )\right )}{5 d}}{6 d}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{6 d f}\) |
Input:
Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]
Output:
-1/6*(Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(d*f) + ((a^3*(2*c - 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*d*f) - ((a^3* (2*c^2 - 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f) + (3 *((a^3*(2*c^3 - 18*c^2*d + 111*c*d^2 + 136*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f) + ((-15*a^3*d^2*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*x )/2 + (2*a^3*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 472*c^2*d^3 + 456*c*d^4 + 1 36*d^5)*Cos[e + f*x])/f + (a^3*d*(4*c^4 - 36*c^3*d + 216*c^2*d^2 + 626*c*d ^3 + 345*d^4)*Cos[e + f*x]*Sin[e + f*x])/(2*f))/3))/4)/(5*d))/(6*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 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[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[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[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(480\) vs. \(2(203)=406\).
Time = 1.12 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.24
\[\frac {-\frac {a^{3} c^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+3 a^{3} c^{2} d \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 a^{3} c \,d^{2} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}+a^{3} d^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+3 a^{3} c^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c^{2} d \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )+9 a^{3} c \,d^{2} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 a^{3} d^{3} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}-3 a^{3} c^{3} \cos \left (f x +e \right )+9 a^{3} c^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c \,d^{2} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )+3 a^{3} d^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a^{3} c^{3} \left (f x +e \right )-3 a^{3} c^{2} d \cos \left (f x +e \right )+3 a^{3} c \,d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{3} d^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}}{f}\]
Input:
int((a+sin(f*x+e)*a)^3*(c+d*sin(f*x+e))^3,x)
Output:
1/f*(-1/3*a^3*c^3*(2+sin(f*x+e)^2)*cos(f*x+e)+3*a^3*c^2*d*(-1/4*(sin(f*x+e )^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-3/5*a^3*c*d^2*(8/3+sin(f*x+e )^4+4/3*sin(f*x+e)^2)*cos(f*x+e)+a^3*d^3*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e )^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)+3*a^3*c^3*(-1/2*cos(f*x+e )*sin(f*x+e)+1/2*f*x+1/2*e)-3*a^3*c^2*d*(2+sin(f*x+e)^2)*cos(f*x+e)+9*a^3* c*d^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-3/5*a^ 3*d^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-3*a^3*c^3*cos(f*x+e)+ 9*a^3*c^2*d*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-3*a^3*c*d^2*(2+sin( f*x+e)^2)*cos(f*x+e)+3*a^3*d^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x +e)+3/8*f*x+3/8*e)+a^3*c^3*(f*x+e)-3*a^3*c^2*d*cos(f*x+e)+3*a^3*c*d^2*(-1/ 2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/3*a^3*d^3*(2+sin(f*x+e)^2)*cos(f* x+e))
Time = 0.10 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.21 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {144 \, {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left (a^{3} c^{3} + 9 \, a^{3} c^{2} d + 15 \, a^{3} c d^{2} + 7 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} f x + 960 \, {\left (a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, a^{3} d^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 31 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (24 \, a^{3} c^{3} + 102 \, a^{3} c^{2} d + 114 \, a^{3} c d^{2} + 41 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-1/240*(144*(a^3*c*d^2 + a^3*d^3)*cos(f*x + e)^5 - 80*(a^3*c^3 + 9*a^3*c^2 *d + 15*a^3*c*d^2 + 7*a^3*d^3)*cos(f*x + e)^3 - 15*(40*a^3*c^3 + 90*a^3*c^ 2*d + 78*a^3*c*d^2 + 23*a^3*d^3)*f*x + 960*(a^3*c^3 + 3*a^3*c^2*d + 3*a^3* c*d^2 + a^3*d^3)*cos(f*x + e) + 5*(8*a^3*d^3*cos(f*x + e)^5 - 2*(18*a^3*c^ 2*d + 54*a^3*c*d^2 + 31*a^3*d^3)*cos(f*x + e)^3 + 3*(24*a^3*c^3 + 102*a^3* c^2*d + 114*a^3*c*d^2 + 41*a^3*d^3)*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 1176 vs. \(2 (206) = 412\).
Time = 0.50 (sec) , antiderivative size = 1176, normalized size of antiderivative = 5.47 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**3*(c+d*sin(f*x+e))**3,x)
Output:
Piecewise((3*a**3*c**3*x*sin(e + f*x)**2/2 + 3*a**3*c**3*x*cos(e + f*x)**2 /2 + a**3*c**3*x - a**3*c**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*a**3*c**3* sin(e + f*x)*cos(e + f*x)/(2*f) - 2*a**3*c**3*cos(e + f*x)**3/(3*f) - 3*a* *3*c**3*cos(e + f*x)/f + 9*a**3*c**2*d*x*sin(e + f*x)**4/8 + 9*a**3*c**2*d *x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 9*a**3*c**2*d*x*sin(e + f*x)**2/2 + 9*a**3*c**2*d*x*cos(e + f*x)**4/8 + 9*a**3*c**2*d*x*cos(e + f*x)**2/2 - 1 5*a**3*c**2*d*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 9*a**3*c**2*d*sin(e + f *x)**2*cos(e + f*x)/f - 9*a**3*c**2*d*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 9*a**3*c**2*d*sin(e + f*x)*cos(e + f*x)/(2*f) - 6*a**3*c**2*d*cos(e + f*x )**3/f - 3*a**3*c**2*d*cos(e + f*x)/f + 27*a**3*c*d**2*x*sin(e + f*x)**4/8 + 27*a**3*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*a**3*c*d**2*x*si n(e + f*x)**2/2 + 27*a**3*c*d**2*x*cos(e + f*x)**4/8 + 3*a**3*c*d**2*x*cos (e + f*x)**2/2 - 3*a**3*c*d**2*sin(e + f*x)**4*cos(e + f*x)/f - 45*a**3*c* d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*a**3*c*d**2*sin(e + f*x)**2*co s(e + f*x)**3/f - 9*a**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 27*a**3*c *d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*a**3*c*d**2*sin(e + f*x)*cos( e + f*x)/(2*f) - 8*a**3*c*d**2*cos(e + f*x)**5/(5*f) - 6*a**3*c*d**2*cos(e + f*x)**3/f + 5*a**3*d**3*x*sin(e + f*x)**6/16 + 15*a**3*d**3*x*sin(e + f *x)**4*cos(e + f*x)**2/16 + 9*a**3*d**3*x*sin(e + f*x)**4/8 + 15*a**3*d**3 *x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*a**3*d**3*x*sin(e + f*x)**2*c...
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (203) = 406\).
Time = 0.04 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.18 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{3} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{3} + 960 \, {\left (f x + e\right )} a^{3} c^{3} + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{2} d + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{2} d + 2160 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{2} d - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c d^{2} + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c d^{2} + 270 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d^{2} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d^{2} - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} d^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} d^{3} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{3} + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} d^{3} - 2880 \, a^{3} c^{3} \cos \left (f x + e\right ) - 2880 \, a^{3} c^{2} d \cos \left (f x + e\right )}{960 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm="maxima")
Output:
1/960*(320*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c^3 + 720*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^3 + 960*(f*x + e)*a^3*c^3 + 2880*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c^2*d + 90*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin( 2*f*x + 2*e))*a^3*c^2*d + 2160*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c^2*d - 192*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*c*d^2 + 2880*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*c*d^2 + 270*(12*f*x + 12*e + s in(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3*c*d^2 + 720*(2*f*x + 2*e - sin(2 *f*x + 2*e))*a^3*c*d^2 - 192*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*co s(f*x + e))*a^3*d^3 + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3*d^3 + 5*(4 *sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^3*d^3 + 90*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e)) *a^3*d^3 - 2880*a^3*c^3*cos(f*x + e) - 2880*a^3*c^2*d*cos(f*x + e))/f
Time = 0.15 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.69 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {a^{3} d^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {a^{3} d^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} - \frac {3 \, a^{3} c d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{16} \, {\left (24 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} x + \frac {1}{2} \, {\left (2 \, a^{3} c^{3} + 3 \, a^{3} c d^{2}\right )} x - \frac {3 \, {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, a^{3} c^{3} + 36 \, a^{3} c^{2} d + 51 \, a^{3} c d^{2} + 15 \, a^{3} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {3 \, {\left (10 \, a^{3} c^{3} + 18 \, a^{3} c^{2} d + 23 \, a^{3} c d^{2} + 5 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {3 \, {\left (4 \, a^{3} c^{2} d + a^{3} d^{3}\right )} \cos \left (f x + e\right )}{4 \, f} + \frac {3 \, {\left (2 \, a^{3} c^{2} d + 6 \, a^{3} c d^{2} + 3 \, a^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {3 \, {\left (16 \, a^{3} c^{3} + 64 \, a^{3} c^{2} d + 48 \, a^{3} c d^{2} + 21 \, a^{3} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm="giac")
Output:
1/12*a^3*d^3*cos(3*f*x + 3*e)/f - 1/192*a^3*d^3*sin(6*f*x + 6*e)/f - 3/4*a ^3*c*d^2*sin(2*f*x + 2*e)/f + 1/16*(24*a^3*c^3 + 90*a^3*c^2*d + 54*a^3*c*d ^2 + 23*a^3*d^3)*x + 1/2*(2*a^3*c^3 + 3*a^3*c*d^2)*x - 3/80*(a^3*c*d^2 + a ^3*d^3)*cos(5*f*x + 5*e)/f + 1/48*(4*a^3*c^3 + 36*a^3*c^2*d + 51*a^3*c*d^2 + 15*a^3*d^3)*cos(3*f*x + 3*e)/f - 3/8*(10*a^3*c^3 + 18*a^3*c^2*d + 23*a^ 3*c*d^2 + 5*a^3*d^3)*cos(f*x + e)/f - 3/4*(4*a^3*c^2*d + a^3*d^3)*cos(f*x + e)/f + 3/64*(2*a^3*c^2*d + 6*a^3*c*d^2 + 3*a^3*d^3)*sin(4*f*x + 4*e)/f - 3/64*(16*a^3*c^3 + 64*a^3*c^2*d + 48*a^3*c*d^2 + 21*a^3*d^3)*sin(2*f*x + 2*e)/f
Time = 17.59 (sec) , antiderivative size = 773, normalized size of antiderivative = 3.60 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx =\text {Too large to display} \] Input:
int((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^3,x)
Output:
(a^3*atan((a^3*tan(e/2 + (f*x)/2)*(78*c*d^2 + 90*c^2*d + 40*c^3 + 23*d^3)) /(8*(5*a^3*c^3 + (23*a^3*d^3)/8 + (39*a^3*c*d^2)/4 + (45*a^3*c^2*d)/4)))*( 78*c*d^2 + 90*c^2*d + 40*c^3 + 23*d^3))/(8*f) - (a^3*(atan(tan(e/2 + (f*x) /2)) - (f*x)/2)*(78*c*d^2 + 90*c^2*d + 40*c^3 + 23*d^3))/(8*f) - (tan(e/2 + (f*x)/2)^10*(6*a^3*c^3 + 6*a^3*c^2*d) - tan(e/2 + (f*x)/2)^11*(3*a^3*c^3 + (23*a^3*d^3)/8 + (39*a^3*c*d^2)/4 + (45*a^3*c^2*d)/4) + tan(e/2 + (f*x) /2)^8*(34*a^3*c^3 + 4*a^3*d^3 + 36*a^3*c*d^2 + 66*a^3*c^2*d) + tan(e/2 + ( f*x)/2)^5*(6*a^3*c^3 + (75*a^3*d^3)/4 + (75*a^3*c*d^2)/2 + (57*a^3*c^2*d)/ 2) - tan(e/2 + (f*x)/2)^7*(6*a^3*c^3 + (75*a^3*d^3)/4 + (75*a^3*c*d^2)/2 + (57*a^3*c^2*d)/2) + tan(e/2 + (f*x)/2)^4*(76*a^3*c^3 + 64*a^3*d^3 + 192*a ^3*c*d^2 + 204*a^3*c^2*d) + tan(e/2 + (f*x)/2)^6*((220*a^3*c^3)/3 + (136*a ^3*d^3)/3 + 152*a^3*c*d^2 + 180*a^3*c^2*d) + tan(e/2 + (f*x)/2)^2*(38*a^3* c^3 + (136*a^3*d^3)/5 + (456*a^3*c*d^2)/5 + 102*a^3*c^2*d) + tan(e/2 + (f* x)/2)^3*(9*a^3*c^3 + (391*a^3*d^3)/24 + (189*a^3*c*d^2)/4 + (159*a^3*c^2*d )/4) - tan(e/2 + (f*x)/2)^9*(9*a^3*c^3 + (391*a^3*d^3)/24 + (189*a^3*c*d^2 )/4 + (159*a^3*c^2*d)/4) + (22*a^3*c^3)/3 + (68*a^3*d^3)/15 + tan(e/2 + (f *x)/2)*(3*a^3*c^3 + (23*a^3*d^3)/8 + (39*a^3*c*d^2)/4 + (45*a^3*c^2*d)/4) + (76*a^3*c*d^2)/5 + 18*a^3*c^2*d)/(f*(6*tan(e/2 + (f*x)/2)^2 + 15*tan(e/2 + (f*x)/2)^4 + 20*tan(e/2 + (f*x)/2)^6 + 15*tan(e/2 + (f*x)/2)^8 + 6*tan( e/2 + (f*x)/2)^10 + tan(e/2 + (f*x)/2)^12 + 1))
Time = 0.19 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.73 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {a^{3} \left (-880 \cos \left (f x +e \right ) c^{3}-544 \cos \left (f x +e \right ) d^{3}+2160 c^{2} d -360 \cos \left (f x +e \right ) \sin \left (f x +e \right ) c^{3}-144 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} d^{3}-144 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} c \,d^{2}-180 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} c^{2} d +544 d^{3}-912 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} c \,d^{2}-1350 \cos \left (f x +e \right ) \sin \left (f x +e \right ) c^{2} d -1170 \cos \left (f x +e \right ) \sin \left (f x +e \right ) c \,d^{2}+1350 c^{2} d f x +1170 c \,d^{2} f x +880 c^{3}-230 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} d^{3}-272 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} d^{3}-345 \cos \left (f x +e \right ) \sin \left (f x +e \right ) d^{3}-2160 \cos \left (f x +e \right ) c^{2} d -1824 \cos \left (f x +e \right ) c \,d^{2}+600 c^{3} f x +345 d^{3} f x -540 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} c \,d^{2}-720 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} c^{2} d -40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} d^{3}-80 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} c^{3}+1824 c \,d^{2}\right )}{240 f} \] Input:
int((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x)
Output:
(a**3*( - 40*cos(e + f*x)*sin(e + f*x)**5*d**3 - 144*cos(e + f*x)*sin(e + f*x)**4*c*d**2 - 144*cos(e + f*x)*sin(e + f*x)**4*d**3 - 180*cos(e + f*x)* sin(e + f*x)**3*c**2*d - 540*cos(e + f*x)*sin(e + f*x)**3*c*d**2 - 230*cos (e + f*x)*sin(e + f*x)**3*d**3 - 80*cos(e + f*x)*sin(e + f*x)**2*c**3 - 72 0*cos(e + f*x)*sin(e + f*x)**2*c**2*d - 912*cos(e + f*x)*sin(e + f*x)**2*c *d**2 - 272*cos(e + f*x)*sin(e + f*x)**2*d**3 - 360*cos(e + f*x)*sin(e + f *x)*c**3 - 1350*cos(e + f*x)*sin(e + f*x)*c**2*d - 1170*cos(e + f*x)*sin(e + f*x)*c*d**2 - 345*cos(e + f*x)*sin(e + f*x)*d**3 - 880*cos(e + f*x)*c** 3 - 2160*cos(e + f*x)*c**2*d - 1824*cos(e + f*x)*c*d**2 - 544*cos(e + f*x) *d**3 + 600*c**3*f*x + 880*c**3 + 1350*c**2*d*f*x + 2160*c**2*d + 1170*c*d **2*f*x + 1824*c*d**2 + 345*d**3*f*x + 544*d**3))/(240*f)