Integrand size = 48, antiderivative size = 256 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (8 a^4 b B+24 a^2 b^3 B+3 b^5 B-8 a^5 C-8 a^3 b^2 C+9 a b^4 C\right ) x+\frac {b \left (95 a^3 b B+80 a b^3 B-83 a^4 C+32 a^2 b^2 C+16 b^4 C\right ) \sin (c+d x)}{30 d}+\frac {b^2 \left (130 a^2 b B+45 b^3 B-106 a^3 C+71 a b^2 C\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {b \left (35 a b B-23 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac {b (5 b B-a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {b C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d} \] Output:
1/8*(8*B*a^4*b+24*B*a^2*b^3+3*B*b^5-8*C*a^5-8*C*a^3*b^2+9*C*a*b^4)*x+1/30* b*(95*B*a^3*b+80*B*a*b^3-83*C*a^4+32*C*a^2*b^2+16*C*b^4)*sin(d*x+c)/d+1/12 0*b^2*(130*B*a^2*b+45*B*b^3-106*C*a^3+71*C*a*b^2)*cos(d*x+c)*sin(d*x+c)/d+ 1/60*b*(35*B*a*b-23*C*a^2+16*C*b^2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+1/20*b *(5*B*b-C*a)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d+1/5*b*C*(a+b*cos(d*x+c))^4*si n(d*x+c)/d
Time = 2.45 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.12 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {480 a^4 b B c+1440 a^2 b^3 B c+180 b^5 B c-480 a^5 c C-480 a^3 b^2 c C+540 a b^4 c C+480 a^4 b B d x+1440 a^2 b^3 B d x+180 b^5 B d x-480 a^5 C d x-480 a^3 b^2 C d x+540 a b^4 C d x+60 b \left (32 a^3 b B+24 a b^3 B-24 a^4 C+12 a^2 b^2 C+5 b^4 C\right ) \sin (c+d x)+120 b^2 \left (6 a^2 b B+b^3 B-2 a^3 C+3 a b^2 C\right ) \sin (2 (c+d x))+160 a b^4 B \sin (3 (c+d x))+80 a^2 b^3 C \sin (3 (c+d x))+50 b^5 C \sin (3 (c+d x))+15 b^5 B \sin (4 (c+d x))+45 a b^4 C \sin (4 (c+d x))+6 b^5 C \sin (5 (c+d x))}{480 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2 *C*Cos[c + d*x]^2),x]
Output:
(480*a^4*b*B*c + 1440*a^2*b^3*B*c + 180*b^5*B*c - 480*a^5*c*C - 480*a^3*b^ 2*c*C + 540*a*b^4*c*C + 480*a^4*b*B*d*x + 1440*a^2*b^3*B*d*x + 180*b^5*B*d *x - 480*a^5*C*d*x - 480*a^3*b^2*C*d*x + 540*a*b^4*C*d*x + 60*b*(32*a^3*b* B + 24*a*b^3*B - 24*a^4*C + 12*a^2*b^2*C + 5*b^4*C)*Sin[c + d*x] + 120*b^2 *(6*a^2*b*B + b^3*B - 2*a^3*C + 3*a*b^2*C)*Sin[2*(c + d*x)] + 160*a*b^4*B* Sin[3*(c + d*x)] + 80*a^2*b^3*C*Sin[3*(c + d*x)] + 50*b^5*C*Sin[3*(c + d*x )] + 15*b^5*B*Sin[4*(c + d*x)] + 45*a*b^4*C*Sin[4*(c + d*x)] + 6*b^5*C*Sin [5*(c + d*x)])/(480*d)
Time = 1.03 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 3494, 3042, 3232, 3042, 3232, 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.
| \(\displaystyle \int (a+b \cos (c+d x))^3 \left (a^2 (-C)+a b B+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (a^2 (-C)+a b B+b^2 B \sin \left (c+d x+\frac {\pi }{2}\right )+b^2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3494 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^4 \left (C \cos (c+d x) b^3+(b B-a C) b^2\right )dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+(b B-a C) b^2\right )dx}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{5} \int (a+b \cos (c+d x))^3 \left ((5 b B-a C) \cos (c+d x) b^3+\left (4 C b^2+5 a (b B-a C)\right ) b^2\right )dx+\frac {b^3 C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((5 b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (4 C b^2+5 a (b B-a C)\right ) b^2\right )dx+\frac {b^3 C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (\left (-23 C a^2+35 b B a+16 b^2 C\right ) \cos (c+d x) b^3+\left (-20 C a^3+20 b B a^2+13 b^2 C a+15 b^3 B\right ) b^2\right )dx+\frac {b^3 (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {b^3 C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (\left (-23 C a^2+35 b B a+16 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-20 C a^3+20 b B a^2+13 b^2 C a+15 b^3 B\right ) b^2\right )dx+\frac {b^3 (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {b^3 C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (\left (-106 C a^3+130 b B a^2+71 b^2 C a+45 b^3 B\right ) \cos (c+d x) b^3+\left (-60 C a^4+60 b B a^3-7 b^2 C a^2+115 b^3 B a+32 b^4 C\right ) b^2\right )dx+\frac {b^3 \left (-23 a^2 C+35 a b B+16 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {b^3 (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {b^3 C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (-106 C a^3+130 b B a^2+71 b^2 C a+45 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-60 C a^4+60 b B a^3-7 b^2 C a^2+115 b^3 B a+32 b^4 C\right ) b^2\right )dx+\frac {b^3 \left (-23 a^2 C+35 a b B+16 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {b^3 (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {b^3 C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {b^3 \left (-23 a^2 C+35 a b B+16 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {b^4 \left (-106 a^3 C+130 a^2 b B+71 a b^2 C+45 b^3 B\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {2 b^3 \left (-83 a^4 C+95 a^3 b B+32 a^2 b^2 C+80 a b^3 B+16 b^4 C\right ) \sin (c+d x)}{d}+\frac {15}{2} b^2 x \left (-8 a^5 C+8 a^4 b B-8 a^3 b^2 C+24 a^2 b^3 B+9 a b^4 C+3 b^5 B\right )\right )\right )+\frac {b^3 (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {b^3 C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{b^2}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos [c + d*x]^2),x]
Output:
((b^3*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((b^3*(5*b*B - a*C)*( a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + ((b^3*(35*a*b*B - 23*a^2*C + 1 6*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((15*b^2*(8*a^4*b*B + 24*a^2*b^3*B + 3*b^5*B - 8*a^5*C - 8*a^3*b^2*C + 9*a*b^4*C)*x)/2 + (2*b^ 3*(95*a^3*b*B + 80*a*b^3*B - 83*a^4*C + 32*a^2*b^2*C + 16*b^4*C)*Sin[c + d *x])/d + (b^4*(130*a^2*b*B + 45*b^3*B - 106*a^3*C + 71*a*b^2*C)*Cos[c + d* x]*Sin[c + d*x])/(2*d))/3)/4)/5)/b^2
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_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*B - a*C + b*C*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Time = 202.42 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.82
| method | result | size |
| parts | \(a^{4} \left (B b -C a \right ) x +\frac {\left (B \,b^{5}+3 C a \,b^{4}\right ) \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (4 a \,b^{4} B +2 C \,a^{2} b^{3}\right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (6 B \,a^{2} b^{3}-2 C \,a^{3} b^{2}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 a^{3} b^{2} B -3 C \,a^{4} b \right ) \sin \left (d x +c \right )}{d}+\frac {C \,b^{5} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(210\) |
| parallelrisch | \(\frac {\left (720 B \,a^{2} b^{3}+120 B \,b^{5}-240 C \,a^{3} b^{2}+360 C a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (160 a \,b^{4} B +80 C \,a^{2} b^{3}+50 C \,b^{5}\right ) \sin \left (3 d x +3 c \right )+\left (15 B \,b^{5}+45 C a \,b^{4}\right ) \sin \left (4 d x +4 c \right )+6 C \,b^{5} \sin \left (5 d x +5 c \right )+\left (1920 a^{3} b^{2} B +1440 a \,b^{4} B -1440 C \,a^{4} b +720 C \,a^{2} b^{3}+300 C \,b^{5}\right ) \sin \left (d x +c \right )+480 \left (B \,a^{4} b +3 B \,a^{2} b^{3}-C \,a^{3} b^{2}+\frac {3}{8} B \,b^{5}-C \,a^{5}+\frac {9}{8} C a \,b^{4}\right ) d x}{480 d}\) | \(215\) |
| derivativedivides | \(\frac {\frac {C \,b^{5} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,b^{5} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C a \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a \,b^{4} B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {2 C \,a^{2} b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 B \,a^{2} b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-2 C \,a^{3} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b^{2} B \sin \left (d x +c \right )-3 C \,a^{4} b \sin \left (d x +c \right )+B \,a^{4} b \left (d x +c \right )-C \,a^{5} \left (d x +c \right )}{d}\) | \(276\) |
| default | \(\frac {\frac {C \,b^{5} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,b^{5} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C a \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a \,b^{4} B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {2 C \,a^{2} b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 B \,a^{2} b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-2 C \,a^{3} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b^{2} B \sin \left (d x +c \right )-3 C \,a^{4} b \sin \left (d x +c \right )+B \,a^{4} b \left (d x +c \right )-C \,a^{5} \left (d x +c \right )}{d}\) | \(276\) |
| risch | \(x B \,a^{4} b +3 x B \,a^{2} b^{3}-x C \,a^{3} b^{2}+\frac {3 x B \,b^{5}}{8}-x C \,a^{5}+\frac {9 x C a \,b^{4}}{8}+\frac {4 \sin \left (d x +c \right ) a^{3} b^{2} B}{d}+\frac {3 \sin \left (d x +c \right ) a \,b^{4} B}{d}-\frac {3 \sin \left (d x +c \right ) C \,a^{4} b}{d}+\frac {3 \sin \left (d x +c \right ) C \,a^{2} b^{3}}{2 d}+\frac {5 \sin \left (d x +c \right ) C \,b^{5}}{8 d}+\frac {C \,b^{5} \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) B \,b^{5}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C a \,b^{4}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a \,b^{4} B}{3 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{2} b^{3}}{6 d}+\frac {5 \sin \left (3 d x +3 c \right ) C \,b^{5}}{48 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b^{3}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B \,b^{5}}{4 d}-\frac {\sin \left (2 d x +2 c \right ) C \,a^{3} b^{2}}{2 d}+\frac {3 \sin \left (2 d x +2 c \right ) C a \,b^{4}}{4 d}\) | \(326\) |
| norman | \(\frac {\left (B \,a^{4} b +3 B \,a^{2} b^{3}-C \,a^{3} b^{2}+\frac {3}{8} B \,b^{5}-C \,a^{5}+\frac {9}{8} C a \,b^{4}\right ) x +\left (B \,a^{4} b +3 B \,a^{2} b^{3}-C \,a^{3} b^{2}+\frac {3}{8} B \,b^{5}-C \,a^{5}+\frac {9}{8} C a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (5 B \,a^{4} b +15 B \,a^{2} b^{3}-5 C \,a^{3} b^{2}+\frac {15}{8} B \,b^{5}-5 C \,a^{5}+\frac {45}{8} C a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (5 B \,a^{4} b +15 B \,a^{2} b^{3}-5 C \,a^{3} b^{2}+\frac {15}{8} B \,b^{5}-5 C \,a^{5}+\frac {45}{8} C a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (10 B \,a^{4} b +30 B \,a^{2} b^{3}-10 C \,a^{3} b^{2}+\frac {15}{4} B \,b^{5}-10 C \,a^{5}+\frac {45}{4} C a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (10 B \,a^{4} b +30 B \,a^{2} b^{3}-10 C \,a^{3} b^{2}+\frac {15}{4} B \,b^{5}-10 C \,a^{5}+\frac {45}{4} C a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {4 b \left (180 B \,a^{3} b +100 B a \,b^{3}-135 a^{4} C +50 C \,a^{2} b^{2}+29 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {b \left (32 B \,a^{3} b -24 B \,a^{2} b^{2}+32 B a \,b^{3}-5 B \,b^{4}-24 a^{4} C +8 a^{3} b C +16 C \,a^{2} b^{2}-15 C a \,b^{3}+8 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}+\frac {b \left (32 B \,a^{3} b +24 B \,a^{2} b^{2}+32 B a \,b^{3}+5 B \,b^{4}-24 a^{4} C -8 a^{3} b C +16 C \,a^{2} b^{2}+15 C a \,b^{3}+8 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {b \left (192 B \,a^{3} b -72 B \,a^{2} b^{2}+128 B a \,b^{3}-3 B \,b^{4}-144 a^{4} C +24 a^{3} b C +64 C \,a^{2} b^{2}-9 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {b \left (192 B \,a^{3} b +72 B \,a^{2} b^{2}+128 B a \,b^{3}+3 B \,b^{4}-144 a^{4} C -24 a^{3} b C +64 C \,a^{2} b^{2}+9 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(732\) |
| orering | \(\text {Expression too large to display}\) | \(11647\) |
Input:
int((a+b*cos(d*x+c))^3*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^2),x ,method=_RETURNVERBOSE)
Output:
a^4*(B*b-C*a)*x+(B*b^5+3*C*a*b^4)/d*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin (d*x+c)+3/8*d*x+3/8*c)+1/3*(4*B*a*b^4+2*C*a^2*b^3)/d*(2+cos(d*x+c)^2)*sin( d*x+c)+(6*B*a^2*b^3-2*C*a^3*b^2)/d*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2* c)+(4*B*a^3*b^2-3*C*a^4*b)/d*sin(d*x+c)+1/5*C*b^5/d*(8/3+cos(d*x+c)^4+4/3* cos(d*x+c)^2)*sin(d*x+c)
Time = 0.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.83 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=-\frac {15 \, {\left (8 \, C a^{5} - 8 \, B a^{4} b + 8 \, C a^{3} b^{2} - 24 \, B a^{2} b^{3} - 9 \, C a b^{4} - 3 \, B b^{5}\right )} d x - {\left (24 \, C b^{5} \cos \left (d x + c\right )^{4} - 360 \, C a^{4} b + 480 \, B a^{3} b^{2} + 160 \, C a^{2} b^{3} + 320 \, B a b^{4} + 64 \, C b^{5} + 30 \, {\left (3 \, C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (5 \, C a^{2} b^{3} + 10 \, B a b^{4} + 2 \, C b^{5}\right )} \cos \left (d x + c\right )^{2} - 15 \, {\left (8 \, C a^{3} b^{2} - 24 \, B a^{2} b^{3} - 9 \, C a b^{4} - 3 \, B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c )^2),x, algorithm="fricas")
Output:
-1/120*(15*(8*C*a^5 - 8*B*a^4*b + 8*C*a^3*b^2 - 24*B*a^2*b^3 - 9*C*a*b^4 - 3*B*b^5)*d*x - (24*C*b^5*cos(d*x + c)^4 - 360*C*a^4*b + 480*B*a^3*b^2 + 1 60*C*a^2*b^3 + 320*B*a*b^4 + 64*C*b^5 + 30*(3*C*a*b^4 + B*b^5)*cos(d*x + c )^3 + 16*(5*C*a^2*b^3 + 10*B*a*b^4 + 2*C*b^5)*cos(d*x + c)^2 - 15*(8*C*a^3 *b^2 - 24*B*a^2*b^3 - 9*C*a*b^4 - 3*B*b^5)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (255) = 510\).
Time = 0.31 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.42 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\begin {cases} B a^{4} b x + \frac {4 B a^{3} b^{2} \sin {\left (c + d x \right )}}{d} + 3 B a^{2} b^{3} x \sin ^{2}{\left (c + d x \right )} + 3 B a^{2} b^{3} x \cos ^{2}{\left (c + d x \right )} + \frac {3 B a^{2} b^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {8 B a b^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {4 B a b^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 B b^{5} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B b^{5} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B b^{5} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 B b^{5} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 B b^{5} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - C a^{5} x - \frac {3 C a^{4} b \sin {\left (c + d x \right )}}{d} - C a^{3} b^{2} x \sin ^{2}{\left (c + d x \right )} - C a^{3} b^{2} x \cos ^{2}{\left (c + d x \right )} - \frac {C a^{3} b^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 C a^{2} b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 C a^{2} b^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a b^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {9 C a b^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a b^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {15 C a b^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C b^{5} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C b^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C b^{5} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{3} \left (B a b + B b^{2} \cos {\left (c \right )} - C a^{2} + C b^{2} \cos ^{2}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((a+b*cos(d*x+c))**3*(B*a*b-a**2*C+b**2*B*cos(d*x+c)+b**2*C*cos(d *x+c)**2),x)
Output:
Piecewise((B*a**4*b*x + 4*B*a**3*b**2*sin(c + d*x)/d + 3*B*a**2*b**3*x*sin (c + d*x)**2 + 3*B*a**2*b**3*x*cos(c + d*x)**2 + 3*B*a**2*b**3*sin(c + d*x )*cos(c + d*x)/d + 8*B*a*b**4*sin(c + d*x)**3/(3*d) + 4*B*a*b**4*sin(c + d *x)*cos(c + d*x)**2/d + 3*B*b**5*x*sin(c + d*x)**4/8 + 3*B*b**5*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*B*b**5*x*cos(c + d*x)**4/8 + 3*B*b**5*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5*B*b**5*sin(c + d*x)*cos(c + d*x)**3/(8*d) - C*a**5*x - 3*C*a**4*b*sin(c + d*x)/d - C*a**3*b**2*x*sin(c + d*x)**2 - C*a**3*b**2*x*cos(c + d*x)**2 - C*a**3*b**2*sin(c + d*x)*cos(c + d*x)/d + 4*C*a**2*b**3*sin(c + d*x)**3/(3*d) + 2*C*a**2*b**3*sin(c + d*x)*cos(c + d *x)**2/d + 9*C*a*b**4*x*sin(c + d*x)**4/8 + 9*C*a*b**4*x*sin(c + d*x)**2*c os(c + d*x)**2/4 + 9*C*a*b**4*x*cos(c + d*x)**4/8 + 9*C*a*b**4*sin(c + d*x )**3*cos(c + d*x)/(8*d) + 15*C*a*b**4*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 8*C*b**5*sin(c + d*x)**5/(15*d) + 4*C*b**5*sin(c + d*x)**3*cos(c + d*x)** 2/(3*d) + C*b**5*sin(c + d*x)*cos(c + d*x)**4/d, Ne(d, 0)), (x*(a + b*cos( c))**3*(B*a*b + B*b**2*cos(c) - C*a**2 + C*b**2*cos(c)**2), True))
Time = 0.04 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.03 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=-\frac {480 \, {\left (d x + c\right )} C a^{5} - 480 \, {\left (d x + c\right )} B a^{4} b + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b^{2} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{3} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{3} + 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{4} - 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{5} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{5} + 1440 \, C a^{4} b \sin \left (d x + c\right ) - 1920 \, B a^{3} b^{2} \sin \left (d x + c\right )}{480 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c )^2),x, algorithm="maxima")
Output:
-1/480*(480*(d*x + c)*C*a^5 - 480*(d*x + c)*B*a^4*b + 240*(2*d*x + 2*c + s in(2*d*x + 2*c))*C*a^3*b^2 - 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^2*b^ 3 + 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2*b^3 + 640*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a*b^4 - 45*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2 *d*x + 2*c))*C*a*b^4 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*b^5 - 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c) )*C*b^5 + 1440*C*a^4*b*sin(d*x + c) - 1920*B*a^3*b^2*sin(d*x + c))/d
Time = 0.15 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.89 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {C b^{5} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {1}{8} \, {\left (8 \, C a^{5} - 8 \, B a^{4} b + 8 \, C a^{3} b^{2} - 24 \, B a^{2} b^{3} - 9 \, C a b^{4} - 3 \, B b^{5}\right )} x + \frac {{\left (3 \, C a b^{4} + B b^{5}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (8 \, C a^{2} b^{3} + 16 \, B a b^{4} + 5 \, C b^{5}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (2 \, C a^{3} b^{2} - 6 \, B a^{2} b^{3} - 3 \, C a b^{4} - B b^{5}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac {{\left (24 \, C a^{4} b - 32 \, B a^{3} b^{2} - 12 \, C a^{2} b^{3} - 24 \, B a b^{4} - 5 \, C b^{5}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c )^2),x, algorithm="giac")
Output:
1/80*C*b^5*sin(5*d*x + 5*c)/d - 1/8*(8*C*a^5 - 8*B*a^4*b + 8*C*a^3*b^2 - 2 4*B*a^2*b^3 - 9*C*a*b^4 - 3*B*b^5)*x + 1/32*(3*C*a*b^4 + B*b^5)*sin(4*d*x + 4*c)/d + 1/48*(8*C*a^2*b^3 + 16*B*a*b^4 + 5*C*b^5)*sin(3*d*x + 3*c)/d - 1/4*(2*C*a^3*b^2 - 6*B*a^2*b^3 - 3*C*a*b^4 - B*b^5)*sin(2*d*x + 2*c)/d - 1 /8*(24*C*a^4*b - 32*B*a^3*b^2 - 12*C*a^2*b^3 - 24*B*a*b^4 - 5*C*b^5)*sin(d *x + c)/d
Time = 1.08 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.27 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {3\,B\,b^5\,x}{8}-C\,a^5\,x+B\,a^4\,b\,x+\frac {9\,C\,a\,b^4\,x}{8}+\frac {5\,C\,b^5\,\sin \left (c+d\,x\right )}{8\,d}+3\,B\,a^2\,b^3\,x-C\,a^3\,b^2\,x+\frac {B\,b^5\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,b^5\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,C\,b^5\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {C\,b^5\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {B\,a\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {4\,B\,a^3\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,C\,a\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a^2\,b^3\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}-\frac {C\,a^3\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {3\,B\,a\,b^4\,\sin \left (c+d\,x\right )}{d}-\frac {3\,C\,a^4\,b\,\sin \left (c+d\,x\right )}{d} \] Input:
int((a + b*cos(c + d*x))^3*(C*b^2*cos(c + d*x)^2 - C*a^2 + B*a*b + B*b^2*c os(c + d*x)),x)
Output:
(3*B*b^5*x)/8 - C*a^5*x + B*a^4*b*x + (9*C*a*b^4*x)/8 + (5*C*b^5*sin(c + d *x))/(8*d) + 3*B*a^2*b^3*x - C*a^3*b^2*x + (B*b^5*sin(2*c + 2*d*x))/(4*d) + (B*b^5*sin(4*c + 4*d*x))/(32*d) + (5*C*b^5*sin(3*c + 3*d*x))/(48*d) + (C *b^5*sin(5*c + 5*d*x))/(80*d) + (B*a*b^4*sin(3*c + 3*d*x))/(3*d) + (4*B*a^ 3*b^2*sin(c + d*x))/d + (3*C*a*b^4*sin(2*c + 2*d*x))/(4*d) + (3*C*a*b^4*si n(4*c + 4*d*x))/(32*d) + (3*C*a^2*b^3*sin(c + d*x))/(2*d) + (3*B*a^2*b^3*s in(2*c + 2*d*x))/(2*d) - (C*a^3*b^2*sin(2*c + 2*d*x))/(2*d) + (C*a^2*b^3*s in(3*c + 3*d*x))/(6*d) + (3*B*a*b^4*sin(c + d*x))/d - (3*C*a^4*b*sin(c + d *x))/d
Time = 0.18 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.18 \[ \int (a+b \cos (c+d x))^3 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {-90 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{4} c -30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{6}-120 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b^{2} c +360 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{4}+225 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{4} c +75 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{6}+24 \sin \left (d x +c \right )^{5} b^{5} c -80 \sin \left (d x +c \right )^{3} a^{2} b^{3} c -160 \sin \left (d x +c \right )^{3} a \,b^{5}-80 \sin \left (d x +c \right )^{3} b^{5} c -360 \sin \left (d x +c \right ) a^{4} b c +480 \sin \left (d x +c \right ) a^{3} b^{3}+240 \sin \left (d x +c \right ) a^{2} b^{3} c +480 \sin \left (d x +c \right ) a \,b^{5}+120 \sin \left (d x +c \right ) b^{5} c -120 a^{5} c d x +120 a^{4} b^{2} d x -120 a^{3} b^{2} c d x +360 a^{2} b^{4} d x +135 a \,b^{4} c d x +45 b^{6} d x}{120 d} \] Input:
int((a+b*cos(d*x+c))^3*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^2),x )
Output:
( - 90*cos(c + d*x)*sin(c + d*x)**3*a*b**4*c - 30*cos(c + d*x)*sin(c + d*x )**3*b**6 - 120*cos(c + d*x)*sin(c + d*x)*a**3*b**2*c + 360*cos(c + d*x)*s in(c + d*x)*a**2*b**4 + 225*cos(c + d*x)*sin(c + d*x)*a*b**4*c + 75*cos(c + d*x)*sin(c + d*x)*b**6 + 24*sin(c + d*x)**5*b**5*c - 80*sin(c + d*x)**3* a**2*b**3*c - 160*sin(c + d*x)**3*a*b**5 - 80*sin(c + d*x)**3*b**5*c - 360 *sin(c + d*x)*a**4*b*c + 480*sin(c + d*x)*a**3*b**3 + 240*sin(c + d*x)*a** 2*b**3*c + 480*sin(c + d*x)*a*b**5 + 120*sin(c + d*x)*b**5*c - 120*a**5*c* d*x + 120*a**4*b**2*d*x - 120*a**3*b**2*c*d*x + 360*a**2*b**4*d*x + 135*a* b**4*c*d*x + 45*b**6*d*x)/(120*d)