Integrand size = 21, antiderivative size = 171 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {1}{16} b \left (18 a^2+5 b^2\right ) x-\frac {a \left (a^2+3 b^2\right ) \cos (e+f x)}{f}+\frac {a \left (a^2+6 b^2\right ) \cos ^3(e+f x)}{3 f}-\frac {3 a b^2 \cos ^5(e+f x)}{5 f}-\frac {b \left (18 a^2+5 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 f}-\frac {b \left (18 a^2+5 b^2\right ) \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac {b^3 \cos (e+f x) \sin ^5(e+f x)}{6 f} \] Output:
1/16*b*(18*a^2+5*b^2)*x-a*(a^2+3*b^2)*cos(f*x+e)/f+1/3*a*(a^2+6*b^2)*cos(f *x+e)^3/f-3/5*a*b^2*cos(f*x+e)^5/f-1/16*b*(18*a^2+5*b^2)*cos(f*x+e)*sin(f* x+e)/f-1/24*b*(18*a^2+5*b^2)*cos(f*x+e)*sin(f*x+e)^3/f-1/6*b^3*cos(f*x+e)* sin(f*x+e)^5/f
Time = 2.65 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.86 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {-360 a \left (2 a^2+5 b^2\right ) \cos (e+f x)+20 \left (4 a^3+15 a b^2\right ) \cos (3 (e+f x))+b \left (-36 a b \cos (5 (e+f x))+5 \left (216 a^2 e+60 b^2 e+216 a^2 f x+60 b^2 f x-9 \left (16 a^2+5 b^2\right ) \sin (2 (e+f x))+9 \left (2 a^2+b^2\right ) \sin (4 (e+f x))-b^2 \sin (6 (e+f x))\right )\right )}{960 f} \] Input:
Integrate[Sin[e + f*x]^3*(a + b*Sin[e + f*x])^3,x]
Output:
(-360*a*(2*a^2 + 5*b^2)*Cos[e + f*x] + 20*(4*a^3 + 15*a*b^2)*Cos[3*(e + f* x)] + b*(-36*a*b*Cos[5*(e + f*x)] + 5*(216*a^2*e + 60*b^2*e + 216*a^2*f*x + 60*b^2*f*x - 9*(16*a^2 + 5*b^2)*Sin[2*(e + f*x)] + 9*(2*a^2 + b^2)*Sin[4 *(e + f*x)] - b^2*Sin[6*(e + f*x)])))/(960*f)
Time = 0.85 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3272, 3042, 3502, 3042, 3227, 3042, 3113, 2009, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (e+f x)^3 (a+b \sin (e+f x))^3dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {1}{6} \int \sin ^3(e+f x) \left (13 a b^2 \sin ^2(e+f x)+b \left (18 a^2+5 b^2\right ) \sin (e+f x)+2 a \left (3 a^2+2 b^2\right )\right )dx-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \sin (e+f x)^3 \left (13 a b^2 \sin (e+f x)^2+b \left (18 a^2+5 b^2\right ) \sin (e+f x)+2 a \left (3 a^2+2 b^2\right )\right )dx-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \sin ^3(e+f x) \left (6 a \left (5 a^2+12 b^2\right )+5 b \left (18 a^2+5 b^2\right ) \sin (e+f x)\right )dx-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \sin (e+f x)^3 \left (6 a \left (5 a^2+12 b^2\right )+5 b \left (18 a^2+5 b^2\right ) \sin (e+f x)\right )dx-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \int \sin ^4(e+f x)dx+6 a \left (5 a^2+12 b^2\right ) \int \sin ^3(e+f x)dx\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (6 a \left (5 a^2+12 b^2\right ) \int \sin (e+f x)^3dx+5 b \left (18 a^2+5 b^2\right ) \int \sin (e+f x)^4dx\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \int \sin (e+f x)^4dx-\frac {6 a \left (5 a^2+12 b^2\right ) \int \left (1-\cos ^2(e+f x)\right )d\cos (e+f x)}{f}\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \int \sin (e+f x)^4dx-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\cos (e+f x)-\frac {1}{3} \cos ^3(e+f x)\right )}{f}\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {3}{4} \int \sin ^2(e+f x)dx-\frac {\sin ^3(e+f x) \cos (e+f x)}{4 f}\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\cos (e+f x)-\frac {1}{3} \cos ^3(e+f x)\right )}{f}\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {3}{4} \int \sin (e+f x)^2dx-\frac {\sin ^3(e+f x) \cos (e+f x)}{4 f}\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\cos (e+f x)-\frac {1}{3} \cos ^3(e+f x)\right )}{f}\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {\sin (e+f x) \cos (e+f x)}{2 f}\right )-\frac {\sin ^3(e+f x) \cos (e+f x)}{4 f}\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\cos (e+f x)-\frac {1}{3} \cos ^3(e+f x)\right )}{f}\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {3}{4} \left (\frac {x}{2}-\frac {\sin (e+f x) \cos (e+f x)}{2 f}\right )-\frac {\sin ^3(e+f x) \cos (e+f x)}{4 f}\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\cos (e+f x)-\frac {1}{3} \cos ^3(e+f x)\right )}{f}\right )-\frac {13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{5 f}\right )-\frac {b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}\) |
Input:
Int[Sin[e + f*x]^3*(a + b*Sin[e + f*x])^3,x]
Output:
-1/6*(b^2*Cos[e + f*x]*Sin[e + f*x]^4*(a + b*Sin[e + f*x]))/f + ((-13*a*b^ 2*Cos[e + f*x]*Sin[e + f*x]^4)/(5*f) + ((-6*a*(5*a^2 + 12*b^2)*(Cos[e + f* x] - Cos[e + f*x]^3/3))/f + 5*b*(18*a^2 + 5*b^2)*(-1/4*(Cos[e + f*x]*Sin[e + f*x]^3)/f + (3*(x/2 - (Cos[e + f*x]*Sin[e + f*x])/(2*f)))/4))/5)/6
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
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 - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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]
Time = 44.80 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+3 a^{2} b \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 a \,b^{2} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}+b^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\) | \(145\) |
| default | \(\frac {-\frac {a^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+3 a^{2} b \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 a \,b^{2} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}+b^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\) | \(145\) |
| parallelrisch | \(\frac {\left (80 a^{3}+300 a \,b^{2}\right ) \cos \left (3 f x +3 e \right )+\left (-720 a^{2} b -225 b^{3}\right ) \sin \left (2 f x +2 e \right )+\left (90 a^{2} b +45 b^{3}\right ) \sin \left (4 f x +4 e \right )-36 a \,b^{2} \cos \left (5 f x +5 e \right )-5 b^{3} \sin \left (6 f x +6 e \right )+\left (-720 a^{3}-1800 a \,b^{2}\right ) \cos \left (f x +e \right )+1080 a^{2} b f x +300 b^{3} f x -640 a^{3}-1536 a \,b^{2}}{960 f}\) | \(147\) |
| parts | \(-\frac {a^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3 f}+\frac {b^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}-\frac {3 a \,b^{2} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {3 a^{2} b \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(153\) |
| risch | \(\frac {9 a^{2} b x}{8}+\frac {5 b^{3} x}{16}-\frac {3 a^{3} \cos \left (f x +e \right )}{4 f}-\frac {15 a \,b^{2} \cos \left (f x +e \right )}{8 f}-\frac {b^{3} \sin \left (6 f x +6 e \right )}{192 f}-\frac {3 a \,b^{2} \cos \left (5 f x +5 e \right )}{80 f}+\frac {3 b \sin \left (4 f x +4 e \right ) a^{2}}{32 f}+\frac {3 b^{3} \sin \left (4 f x +4 e \right )}{64 f}+\frac {\cos \left (3 f x +3 e \right ) a^{3}}{12 f}+\frac {5 \cos \left (3 f x +3 e \right ) a \,b^{2}}{16 f}-\frac {3 b \sin \left (2 f x +2 e \right ) a^{2}}{4 f}-\frac {15 b^{3} \sin \left (2 f x +2 e \right )}{64 f}\) | \(184\) |
| norman | \(\frac {-\frac {20 a^{3}+48 a \,b^{2}}{15 f}+\frac {b \left (18 a^{2}+5 b^{2}\right ) x}{16}-\frac {4 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{f}-\frac {4 \left (10 a^{3}+24 a \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3 f}-\frac {\left (16 a^{3}+48 a \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{f}-\frac {2 \left (20 a^{3}+48 a \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{5 f}-\frac {3 b \left (14 a^{2}+11 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{4 f}+\frac {3 b \left (14 a^{2}+11 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}-\frac {b \left (18 a^{2}+5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {17 b \left (18 a^{2}+5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}+\frac {17 b \left (18 a^{2}+5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{24 f}+\frac {b \left (18 a^{2}+5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}+\frac {3 b \left (18 a^{2}+5 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{8}+\frac {15 b \left (18 a^{2}+5 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{16}+\frac {5 b \left (18 a^{2}+5 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{4}+\frac {15 b \left (18 a^{2}+5 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{16}+\frac {3 b \left (18 a^{2}+5 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{8}+\frac {b \left (18 a^{2}+5 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{16}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{6}}\) | \(475\) |
| orering | \(\text {Expression too large to display}\) | \(3179\) |
Input:
int(sin(f*x+e)^3*(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
1/f*(-1/3*a^3*(2+sin(f*x+e)^2)*cos(f*x+e)+3*a^2*b*(-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*b^2*(8/3+sin(f*x+e)^4+4/3*sin( f*x+e)^2)*cos(f*x+e)+b^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))
Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.81 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {144 \, a b^{2} \cos \left (f x + e\right )^{5} - 80 \, {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} f x + 240 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, b^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, a^{2} b + 13 \, b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (30 \, a^{2} b + 11 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \] Input:
integrate(sin(f*x+e)^3*(a+b*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-1/240*(144*a*b^2*cos(f*x + e)^5 - 80*(a^3 + 6*a*b^2)*cos(f*x + e)^3 - 15* (18*a^2*b + 5*b^3)*f*x + 240*(a^3 + 3*a*b^2)*cos(f*x + e) + 5*(8*b^3*cos(f *x + e)^5 - 2*(18*a^2*b + 13*b^3)*cos(f*x + e)^3 + 3*(30*a^2*b + 11*b^3)*c os(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (155) = 310\).
Time = 0.37 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.30 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\begin {cases} - \frac {a^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {9 a^{2} b x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {9 a^{2} b x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {9 a^{2} b x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {15 a^{2} b \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {9 a^{2} b \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {3 a b^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 a b^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {8 a b^{2} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {5 b^{3} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {15 b^{3} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {15 b^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {5 b^{3} x \cos ^{6}{\left (e + f x \right )}}{16} - \frac {11 b^{3} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} - \frac {5 b^{3} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {5 b^{3} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\left (e \right )}\right )^{3} \sin ^{3}{\left (e \right )} & \text {otherwise} \end {cases} \] Input:
integrate(sin(f*x+e)**3*(a+b*sin(f*x+e))**3,x)
Output:
Piecewise((-a**3*sin(e + f*x)**2*cos(e + f*x)/f - 2*a**3*cos(e + f*x)**3/( 3*f) + 9*a**2*b*x*sin(e + f*x)**4/8 + 9*a**2*b*x*sin(e + f*x)**2*cos(e + f *x)**2/4 + 9*a**2*b*x*cos(e + f*x)**4/8 - 15*a**2*b*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 9*a**2*b*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*a*b**2*sin( e + f*x)**4*cos(e + f*x)/f - 4*a*b**2*sin(e + f*x)**2*cos(e + f*x)**3/f - 8*a*b**2*cos(e + f*x)**5/(5*f) + 5*b**3*x*sin(e + f*x)**6/16 + 15*b**3*x*s in(e + f*x)**4*cos(e + f*x)**2/16 + 15*b**3*x*sin(e + f*x)**2*cos(e + f*x) **4/16 + 5*b**3*x*cos(e + f*x)**6/16 - 11*b**3*sin(e + f*x)**5*cos(e + f*x )/(16*f) - 5*b**3*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 5*b**3*sin(e + f *x)*cos(e + f*x)**5/(16*f), Ne(f, 0)), (x*(a + b*sin(e))**3*sin(e)**3, Tru e))
Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a b^{2} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{3}}{960 \, f} \] Input:
integrate(sin(f*x+e)^3*(a+b*sin(f*x+e))^3,x, algorithm="maxima")
Output:
1/960*(320*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^3 + 90*(12*f*x + 12*e + sin (4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*b - 192*(3*cos(f*x + e)^5 - 10*cos (f*x + e)^3 + 15*cos(f*x + e))*a*b^2 + 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))*b^3)/f
Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.88 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {3 \, a b^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} - \frac {b^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} x + \frac {{\left (4 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {3 \, {\left (2 \, a^{3} + 5 \, a b^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {3 \, {\left (2 \, a^{2} b + b^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {3 \, {\left (16 \, a^{2} b + 5 \, b^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \] Input:
integrate(sin(f*x+e)^3*(a+b*sin(f*x+e))^3,x, algorithm="giac")
Output:
-3/80*a*b^2*cos(5*f*x + 5*e)/f - 1/192*b^3*sin(6*f*x + 6*e)/f + 1/16*(18*a ^2*b + 5*b^3)*x + 1/48*(4*a^3 + 15*a*b^2)*cos(3*f*x + 3*e)/f - 3/8*(2*a^3 + 5*a*b^2)*cos(f*x + e)/f + 3/64*(2*a^2*b + b^3)*sin(4*f*x + 4*e)/f - 3/64 *(16*a^2*b + 5*b^3)*sin(2*f*x + 2*e)/f
Time = 18.34 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.44 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (18\,a^2+5\,b^2\right )}{8\,\left (\frac {9\,a^2\,b}{4}+\frac {5\,b^3}{8}\right )}\right )\,\left (18\,a^2+5\,b^2\right )}{8\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {9\,a^2\,b}{4}+\frac {5\,b^3}{8}\right )+4\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+\frac {16\,a\,b^2}{5}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (16\,a^3+48\,a\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {40\,a^3}{3}+32\,a\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (8\,a^3+\frac {96\,a\,b^2}{5}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {9\,a^2\,b}{4}+\frac {5\,b^3}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {21\,a^2\,b}{2}+\frac {33\,b^3}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {21\,a^2\,b}{2}+\frac {33\,b^3}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {51\,a^2\,b}{4}+\frac {85\,b^3}{24}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {51\,a^2\,b}{4}+\frac {85\,b^3}{24}\right )+\frac {4\,a^3}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {b\,\left (18\,a^2+5\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{8\,f} \] Input:
int(sin(e + f*x)^3*(a + b*sin(e + f*x))^3,x)
Output:
(b*atan((b*tan(e/2 + (f*x)/2)*(18*a^2 + 5*b^2))/(8*((9*a^2*b)/4 + (5*b^3)/ 8)))*(18*a^2 + 5*b^2))/(8*f) - (tan(e/2 + (f*x)/2)*((9*a^2*b)/4 + (5*b^3)/ 8) + 4*a^3*tan(e/2 + (f*x)/2)^8 + (16*a*b^2)/5 + tan(e/2 + (f*x)/2)^4*(48* a*b^2 + 16*a^3) + tan(e/2 + (f*x)/2)^6*(32*a*b^2 + (40*a^3)/3) + tan(e/2 + (f*x)/2)^2*((96*a*b^2)/5 + 8*a^3) - tan(e/2 + (f*x)/2)^11*((9*a^2*b)/4 + (5*b^3)/8) + tan(e/2 + (f*x)/2)^5*((21*a^2*b)/2 + (33*b^3)/4) - tan(e/2 + (f*x)/2)^7*((21*a^2*b)/2 + (33*b^3)/4) + tan(e/2 + (f*x)/2)^3*((51*a^2*b)/ 4 + (85*b^3)/24) - tan(e/2 + (f*x)/2)^9*((51*a^2*b)/4 + (85*b^3)/24) + (4* a^3)/3)/(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)) - (b*(18*a^2 + 5*b^2)*(atan(tan(e/2 + (f*x)/2)) - (f* x)/2))/(8*f)
Time = 0.17 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.21 \[ \int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {-40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b^{3}-144 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a \,b^{2}-180 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a^{2} b -50 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b^{3}-80 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a^{3}-192 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a \,b^{2}-270 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a^{2} b -75 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b^{3}-160 \cos \left (f x +e \right ) a^{3}-384 \cos \left (f x +e \right ) a \,b^{2}+160 a^{3}+270 a^{2} b f x +384 a \,b^{2}+75 b^{3} f x}{240 f} \] Input:
int(sin(f*x+e)^3*(a+b*sin(f*x+e))^3,x)
Output:
( - 40*cos(e + f*x)*sin(e + f*x)**5*b**3 - 144*cos(e + f*x)*sin(e + f*x)** 4*a*b**2 - 180*cos(e + f*x)*sin(e + f*x)**3*a**2*b - 50*cos(e + f*x)*sin(e + f*x)**3*b**3 - 80*cos(e + f*x)*sin(e + f*x)**2*a**3 - 192*cos(e + f*x)* sin(e + f*x)**2*a*b**2 - 270*cos(e + f*x)*sin(e + f*x)*a**2*b - 75*cos(e + f*x)*sin(e + f*x)*b**3 - 160*cos(e + f*x)*a**3 - 384*cos(e + f*x)*a*b**2 + 160*a**3 + 270*a**2*b*f*x + 384*a*b**2 + 75*b**3*f*x)/(240*f)