Integrand size = 20, antiderivative size = 260 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx=\frac {1}{8} \left (8 a^4+24 a^2 \left (b^2+c^2\right )+3 \left (b^2+c^2\right )^2\right ) x-\frac {5 a c \left (10 a^2+11 \left (b^2+c^2\right )\right ) \cos (d+e x)}{24 e}+\frac {5 a b \left (10 a^2+11 \left (b^2+c^2\right )\right ) \sin (d+e x)}{24 e}-\frac {7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}-\frac {(a+b \cos (d+e x)+c \sin (d+e x)) \left (c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{24 e} \]
1/8*(8*a^4+24*a^2*(b^2+c^2)+3*(b^2+c^2)^2)*x-5/24*a*c*(10*a^2+11*b^2+11*c^ 2)*cos(e*x+d)/e+5/24*a*b*(10*a^2+11*b^2+11*c^2)*sin(e*x+d)/e-7/12*(a*c*cos (e*x+d)-a*b*sin(e*x+d))*(a+b*cos(e*x+d)+c*sin(e*x+d))^2/e-1/4*(c*cos(e*x+d )-b*sin(e*x+d))*(a+b*cos(e*x+d)+c*sin(e*x+d))^3/e-1/24*(a+b*cos(e*x+d)+c*s in(e*x+d))*(c*(26*a^2+9*b^2+9*c^2)*cos(e*x+d)-b*(26*a^2+9*b^2+9*c^2)*sin(e *x+d))/e
Time = 1.99 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.91 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx=\frac {12 \left (8 a^4+24 a^2 \left (b^2+c^2\right )+3 \left (b^2+c^2\right )^2\right ) (d+e x)-96 a c \left (4 a^2+3 \left (b^2+c^2\right )\right ) \cos (d+e x)-48 b c \left (6 a^2+b^2+c^2\right ) \cos (2 (d+e x))+32 a c \left (-3 b^2+c^2\right ) \cos (3 (d+e x))-12 b c \left (b^2-c^2\right ) \cos (4 (d+e x))+96 a b \left (4 a^2+3 \left (b^2+c^2\right )\right ) \sin (d+e x)+24 \left (b^2-c^2\right ) \left (6 a^2+b^2+c^2\right ) \sin (2 (d+e x))+32 a b \left (b^2-3 c^2\right ) \sin (3 (d+e x))+3 \left (b^4-6 b^2 c^2+c^4\right ) \sin (4 (d+e x))}{96 e} \]
(12*(8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*(d + e*x) - 96*a*c*(4*a ^2 + 3*(b^2 + c^2))*Cos[d + e*x] - 48*b*c*(6*a^2 + b^2 + c^2)*Cos[2*(d + e *x)] + 32*a*c*(-3*b^2 + c^2)*Cos[3*(d + e*x)] - 12*b*c*(b^2 - c^2)*Cos[4*( d + e*x)] + 96*a*b*(4*a^2 + 3*(b^2 + c^2))*Sin[d + e*x] + 24*(b^2 - c^2)*( 6*a^2 + b^2 + c^2)*Sin[2*(d + e*x)] + 32*a*b*(b^2 - 3*c^2)*Sin[3*(d + e*x) ] + 3*(b^4 - 6*b^2*c^2 + c^4)*Sin[4*(d + e*x)])/(96*e)
Time = 0.82 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3042, 3599, 3042, 3625, 3042, 3625, 2009}
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.
| \(\displaystyle \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \cos (d+e x)+c \sin (d+e x))^4dx\) |
| \(\Big \downarrow \) 3599 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \left (4 a^2+7 b \cos (d+e x) a+7 c \sin (d+e x) a+3 \left (b^2+c^2\right )\right )dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \left (4 a^2+7 b \cos (d+e x) a+7 c \sin (d+e x) a+3 \left (b^2+c^2\right )\right )dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {1}{4} \left (\frac {\int (a+b \cos (d+e x)+c \sin (d+e x)) \left (\left (12 a^2+23 \left (b^2+c^2\right )\right ) a^2+b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x) a+c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x) a\right )dx}{3 a}-\frac {7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {\int (a+b \cos (d+e x)+c \sin (d+e x)) \left (\left (12 a^2+23 \left (b^2+c^2\right )\right ) a^2+b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x) a+c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x) a\right )dx}{3 a}-\frac {7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {1}{4} \left (\frac {\frac {\int \left (5 b \left (10 a^2+11 \left (b^2+c^2\right )\right ) \cos (d+e x) a^3+5 c \left (10 a^2+11 \left (b^2+c^2\right )\right ) \sin (d+e x) a^3+3 \left (8 a^4+24 \left (b^2+c^2\right ) a^2+3 \left (b^2+c^2\right )^2\right ) a^2\right )dx}{2 a}-\frac {(a+b \cos (d+e x)+c \sin (d+e x)) \left (a c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-a b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{2 e}}{3 a}-\frac {7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (\frac {\frac {3 a^2 x \left (8 a^4+24 a^2 \left (b^2+c^2\right )+3 \left (b^2+c^2\right )^2\right )+\frac {5 a^3 b \left (10 a^2+11 \left (b^2+c^2\right )\right ) \sin (d+e x)}{e}-\frac {5 a^3 c \left (10 a^2+11 \left (b^2+c^2\right )\right ) \cos (d+e x)}{e}}{2 a}-\frac {(a+b \cos (d+e x)+c \sin (d+e x)) \left (a c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-a b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{2 e}}{3 a}-\frac {7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}\) |
-1/4*((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e* x])^3)/e + ((-7*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e) + (-1/2*((a + b*Cos[d + e*x] + c*Sin[d + e*x])* (a*c*(26*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - a*b*(26*a^2 + 9*(b^2 + c^2))* Sin[d + e*x]))/e + (3*a^2*(8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*x - (5*a^3*c*(10*a^2 + 11*(b^2 + c^2))*Cos[d + e*x])/e + (5*a^3*b*(10*a^2 + 11*(b^2 + c^2))*Sin[d + e*x])/e)/(2*a))/(3*a))/4
3.4.95.3.1 Defintions of rubi rules used
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[(-(c*Cos[d + e*x] - b*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Simp[1/n Int[Simp[n*a^2 + ( n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x ], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_.)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^n/(a*e*(n + 1))), x] + Simp[1/(a*(n + 1 )) Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*(b*B + c*C)*n + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))*Cos[d + e*x] + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
Time = 3.20 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.13
| method | result | size |
| parallelrisch | \(\frac {24 \left (6 a^{2} b^{2}-6 a^{2} c^{2}+b^{4}-c^{4}\right ) \sin \left (2 e x +2 d \right )-288 b \left (a^{2}+\frac {b^{2}}{6}+\frac {c^{2}}{6}\right ) c \cos \left (2 e x +2 d \right )+3 \left (b^{4}-6 b^{2} c^{2}+c^{4}\right ) \sin \left (4 e x +4 d \right )+32 a \left (-3 b^{2} c +c^{3}\right ) \cos \left (3 e x +3 d \right )+12 \left (-b^{3} c +c^{3} b \right ) \cos \left (4 e x +4 d \right )+32 b a \left (b^{2}-3 c^{2}\right ) \sin \left (3 e x +3 d \right )-384 \left (a^{2}+\frac {3 b^{2}}{4}+\frac {3 c^{2}}{4}\right ) c a \cos \left (e x +d \right )+384 b \left (a^{2}+\frac {3 b^{2}}{4}+\frac {3 c^{2}}{4}\right ) a \sin \left (e x +d \right )+36 c^{4} e x +4 \left (-64 a +9 b \right ) c^{3}+288 \left (a^{2}+\frac {b^{2}}{4}\right ) e x \,c^{2}+12 \left (-32 a^{3}+24 a^{2} b -32 a \,b^{2}+5 b^{3}\right ) c +96 \left (a^{4}+3 a^{2} b^{2}+\frac {3}{8} b^{4}\right ) e x}{96 e}\) | \(293\) |
| parts | \(a^{4} x -\frac {4 b^{3} \left (\frac {c \sin \left (e x +d \right )^{4}}{4}+\frac {\sin \left (e x +d \right )^{3} a}{3}-\frac {\sin \left (e x +d \right )^{2} c}{2}-a \sin \left (e x +d \right )\right )}{e}+\frac {6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )^{3}}{4}+\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )-4 a \,b^{2} c \cos \left (e x +d \right )^{3}+6 a^{2} b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {b \left (a +c \sin \left (e x +d \right )\right )^{4}}{e c}+\frac {b^{4} \left (\frac {\left (\cos \left (e x +d \right )^{3}+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )}{e}+\frac {c^{4} \left (-\frac {\left (\sin \left (e x +d \right )^{3}+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )}{e}-\frac {4 a^{3} c \cos \left (e x +d \right )}{e}+\frac {6 a^{2} c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}-\frac {4 a \,c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3 e}\) | \(326\) |
| derivativedivides | \(\frac {b^{4} \left (\frac {\left (\cos \left (e x +d \right )^{3}+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+c^{4} \left (-\frac {\left (\sin \left (e x +d \right )^{3}+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+a^{4} \left (e x +d \right )-b^{3} c \cos \left (e x +d \right )^{4}+6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )^{3}}{4}+\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )+c^{3} b \sin \left (e x +d \right )^{4}-6 a^{2} b c \cos \left (e x +d \right )^{2}-4 a \,b^{2} c \cos \left (e x +d \right )^{3}+4 a b \,c^{2} \sin \left (e x +d \right )^{3}+4 a^{3} b \sin \left (e x +d \right )-4 a^{3} c \cos \left (e x +d \right )+6 a^{2} b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+6 a^{2} c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+\frac {4 a \,b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-\frac {4 a \,c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}}{e}\) | \(335\) |
| default | \(\frac {b^{4} \left (\frac {\left (\cos \left (e x +d \right )^{3}+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+c^{4} \left (-\frac {\left (\sin \left (e x +d \right )^{3}+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+a^{4} \left (e x +d \right )-b^{3} c \cos \left (e x +d \right )^{4}+6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )^{3}}{4}+\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )+c^{3} b \sin \left (e x +d \right )^{4}-6 a^{2} b c \cos \left (e x +d \right )^{2}-4 a \,b^{2} c \cos \left (e x +d \right )^{3}+4 a b \,c^{2} \sin \left (e x +d \right )^{3}+4 a^{3} b \sin \left (e x +d \right )-4 a^{3} c \cos \left (e x +d \right )+6 a^{2} b^{2} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+6 a^{2} c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+\frac {4 a \,b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-\frac {4 a \,c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}}{e}\) | \(335\) |
| risch | \(a^{4} x -\frac {c \,b^{3} \cos \left (4 e x +4 d \right )}{8 e}+\frac {c^{3} b \cos \left (4 e x +4 d \right )}{8 e}-\frac {3 \sin \left (4 e x +4 d \right ) b^{2} c^{2}}{16 e}+\frac {4 a^{3} b \sin \left (e x +d \right )}{e}+\frac {3 a \,b^{3} \sin \left (e x +d \right )}{e}+\frac {a \,c^{3} \cos \left (3 e x +3 d \right )}{3 e}+\frac {a \,b^{3} \sin \left (3 e x +3 d \right )}{3 e}+\frac {3 \sin \left (2 e x +2 d \right ) a^{2} b^{2}}{2 e}-\frac {3 \sin \left (2 e x +2 d \right ) a^{2} c^{2}}{2 e}-\frac {4 a^{3} c \cos \left (e x +d \right )}{e}-\frac {3 a \,c^{3} \cos \left (e x +d \right )}{e}+\frac {3 a b \sin \left (e x +d \right ) c^{2}}{e}-\frac {a c \cos \left (3 e x +3 d \right ) b^{2}}{e}-\frac {a b \sin \left (3 e x +3 d \right ) c^{2}}{e}-\frac {3 c b \cos \left (2 e x +2 d \right ) a^{2}}{e}+\frac {3 c^{4} x}{8}-\frac {c \,b^{3} \cos \left (2 e x +2 d \right )}{2 e}-\frac {c^{3} b \cos \left (2 e x +2 d \right )}{2 e}-\frac {3 a c \cos \left (e x +d \right ) b^{2}}{e}+\frac {3 x \,b^{2} c^{2}}{4}+\frac {\sin \left (4 e x +4 d \right ) b^{4}}{32 e}+\frac {\sin \left (4 e x +4 d \right ) c^{4}}{32 e}+\frac {\sin \left (2 e x +2 d \right ) b^{4}}{4 e}-\frac {\sin \left (2 e x +2 d \right ) c^{4}}{4 e}+3 x \,a^{2} c^{2}+\frac {3 b^{4} x}{8}+3 a^{2} b^{2} x\) | \(431\) |
| norman | \(\frac {\left (a^{4}+3 a^{2} b^{2}+\frac {3}{8} b^{4}+3 a^{2} c^{2}+\frac {3}{4} b^{2} c^{2}+\frac {3}{8} c^{4}\right ) x +\left (a^{4}+3 a^{2} b^{2}+\frac {3}{8} b^{4}+3 a^{2} c^{2}+\frac {3}{4} b^{2} c^{2}+\frac {3}{8} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{8}+\left (4 a^{4}+12 a^{2} b^{2}+\frac {3}{2} b^{4}+12 a^{2} c^{2}+3 b^{2} c^{2}+\frac {3}{2} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\left (4 a^{4}+12 a^{2} b^{2}+\frac {3}{2} b^{4}+12 a^{2} c^{2}+3 b^{2} c^{2}+\frac {3}{2} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}+\left (6 a^{4}+18 a^{2} b^{2}+\frac {9}{4} b^{4}+18 a^{2} c^{2}+\frac {9}{2} b^{2} c^{2}+\frac {9}{4} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}-\frac {24 a^{3} c +24 a \,b^{2} c +16 a \,c^{3}}{3 e}-\frac {4 \left (2 a^{3} c -6 a^{2} b c +6 a \,b^{2} c -2 b^{3} c \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{e}-\frac {2 \left (12 a^{3} c -24 a^{2} b c +12 a \,b^{2} c +8 a \,c^{3}-8 c^{3} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{e}-\frac {4 \left (18 a^{3} c -18 a^{2} b c +6 a \,b^{2} c +16 a \,c^{3}-6 b^{3} c \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{3 e}+\frac {\left (32 a^{3} b -24 a^{2} b^{2}+24 a^{2} c^{2}+32 a \,b^{3}-5 b^{4}+6 b^{2} c^{2}+3 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{7}}{4 e}+\frac {\left (32 a^{3} b +24 a^{2} b^{2}-24 a^{2} c^{2}+32 a \,b^{3}+5 b^{4}-6 b^{2} c^{2}-3 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{4 e}+\frac {\left (288 a^{3} b -72 a^{2} b^{2}+72 a^{2} c^{2}+160 a \,b^{3}+384 a b \,c^{2}+9 b^{4}-126 b^{2} c^{2}+33 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{12 e}+\frac {\left (288 a^{3} b +72 a^{2} b^{2}-72 a^{2} c^{2}+160 a \,b^{3}+384 a b \,c^{2}-9 b^{4}+126 b^{2} c^{2}-33 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{12 e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{4}}\) | \(698\) |
1/96*(24*(6*a^2*b^2-6*a^2*c^2+b^4-c^4)*sin(2*e*x+2*d)-288*b*(a^2+1/6*b^2+1 /6*c^2)*c*cos(2*e*x+2*d)+3*(b^4-6*b^2*c^2+c^4)*sin(4*e*x+4*d)+32*a*(-3*b^2 *c+c^3)*cos(3*e*x+3*d)+12*(-b^3*c+b*c^3)*cos(4*e*x+4*d)+32*b*a*(b^2-3*c^2) *sin(3*e*x+3*d)-384*(a^2+3/4*b^2+3/4*c^2)*c*a*cos(e*x+d)+384*b*(a^2+3/4*b^ 2+3/4*c^2)*a*sin(e*x+d)+36*c^4*e*x+4*(-64*a+9*b)*c^3+288*(a^2+1/4*b^2)*e*x *c^2+12*(-32*a^3+24*a^2*b-32*a*b^2+5*b^3)*c+96*(a^4+3*a^2*b^2+3/8*b^4)*e*x )/e
Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.98 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx=-\frac {24 \, {\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{4} + 32 \, {\left (3 \, a b^{2} c - a c^{3}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4} + 3 \, c^{4} + 6 \, {\left (4 \, a^{2} + b^{2}\right )} c^{2}\right )} e x + 48 \, {\left (3 \, a^{2} b c + b c^{3}\right )} \cos \left (e x + d\right )^{2} + 96 \, {\left (a^{3} c + a c^{3}\right )} \cos \left (e x + d\right ) - {\left (96 \, a^{3} b + 64 \, a b^{3} + 96 \, a b c^{2} + 6 \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{3} + 32 \, {\left (a b^{3} - 3 \, a b c^{2}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (24 \, a^{2} b^{2} + 3 \, b^{4} - 5 \, c^{4} - 6 \, {\left (4 \, a^{2} - b^{2}\right )} c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{24 \, e} \]
-1/24*(24*(b^3*c - b*c^3)*cos(e*x + d)^4 + 32*(3*a*b^2*c - a*c^3)*cos(e*x + d)^3 - 3*(8*a^4 + 24*a^2*b^2 + 3*b^4 + 3*c^4 + 6*(4*a^2 + b^2)*c^2)*e*x + 48*(3*a^2*b*c + b*c^3)*cos(e*x + d)^2 + 96*(a^3*c + a*c^3)*cos(e*x + d) - (96*a^3*b + 64*a*b^3 + 96*a*b*c^2 + 6*(b^4 - 6*b^2*c^2 + c^4)*cos(e*x + d)^3 + 32*(a*b^3 - 3*a*b*c^2)*cos(e*x + d)^2 + 3*(24*a^2*b^2 + 3*b^4 - 5*c ^4 - 6*(4*a^2 - b^2)*c^2)*cos(e*x + d))*sin(e*x + d))/e
Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (253) = 506\).
Time = 0.29 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.62 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b \sin {\left (d + e x \right )}}{e} - \frac {4 a^{3} c \cos {\left (d + e x \right )}}{e} + 3 a^{2} b^{2} x \sin ^{2}{\left (d + e x \right )} + 3 a^{2} b^{2} x \cos ^{2}{\left (d + e x \right )} + \frac {3 a^{2} b^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} + \frac {6 a^{2} b c \sin ^{2}{\left (d + e x \right )}}{e} + 3 a^{2} c^{2} x \sin ^{2}{\left (d + e x \right )} + 3 a^{2} c^{2} x \cos ^{2}{\left (d + e x \right )} - \frac {3 a^{2} c^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} + \frac {8 a b^{3} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac {4 a b^{3} \sin {\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} - \frac {4 a b^{2} c \cos ^{3}{\left (d + e x \right )}}{e} + \frac {4 a b c^{2} \sin ^{3}{\left (d + e x \right )}}{e} - \frac {4 a c^{3} \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {8 a c^{3} \cos ^{3}{\left (d + e x \right )}}{3 e} + \frac {3 b^{4} x \sin ^{4}{\left (d + e x \right )}}{8} + \frac {3 b^{4} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{4} + \frac {3 b^{4} x \cos ^{4}{\left (d + e x \right )}}{8} + \frac {3 b^{4} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{8 e} + \frac {5 b^{4} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{8 e} - \frac {b^{3} c \cos ^{4}{\left (d + e x \right )}}{e} + \frac {3 b^{2} c^{2} x \sin ^{4}{\left (d + e x \right )}}{4} + \frac {3 b^{2} c^{2} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{2} + \frac {3 b^{2} c^{2} x \cos ^{4}{\left (d + e x \right )}}{4} + \frac {3 b^{2} c^{2} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{4 e} - \frac {3 b^{2} c^{2} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{4 e} + \frac {b c^{3} \sin ^{4}{\left (d + e x \right )}}{e} + \frac {3 c^{4} x \sin ^{4}{\left (d + e x \right )}}{8} + \frac {3 c^{4} x \sin ^{2}{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{4} + \frac {3 c^{4} x \cos ^{4}{\left (d + e x \right )}}{8} - \frac {5 c^{4} \sin ^{3}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{8 e} - \frac {3 c^{4} \sin {\left (d + e x \right )} \cos ^{3}{\left (d + e x \right )}}{8 e} & \text {for}\: e \neq 0 \\x \left (a + b \cos {\left (d \right )} + c \sin {\left (d \right )}\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((a**4*x + 4*a**3*b*sin(d + e*x)/e - 4*a**3*c*cos(d + e*x)/e + 3* a**2*b**2*x*sin(d + e*x)**2 + 3*a**2*b**2*x*cos(d + e*x)**2 + 3*a**2*b**2* sin(d + e*x)*cos(d + e*x)/e + 6*a**2*b*c*sin(d + e*x)**2/e + 3*a**2*c**2*x *sin(d + e*x)**2 + 3*a**2*c**2*x*cos(d + e*x)**2 - 3*a**2*c**2*sin(d + e*x )*cos(d + e*x)/e + 8*a*b**3*sin(d + e*x)**3/(3*e) + 4*a*b**3*sin(d + e*x)* cos(d + e*x)**2/e - 4*a*b**2*c*cos(d + e*x)**3/e + 4*a*b*c**2*sin(d + e*x) **3/e - 4*a*c**3*sin(d + e*x)**2*cos(d + e*x)/e - 8*a*c**3*cos(d + e*x)**3 /(3*e) + 3*b**4*x*sin(d + e*x)**4/8 + 3*b**4*x*sin(d + e*x)**2*cos(d + e*x )**2/4 + 3*b**4*x*cos(d + e*x)**4/8 + 3*b**4*sin(d + e*x)**3*cos(d + e*x)/ (8*e) + 5*b**4*sin(d + e*x)*cos(d + e*x)**3/(8*e) - b**3*c*cos(d + e*x)**4 /e + 3*b**2*c**2*x*sin(d + e*x)**4/4 + 3*b**2*c**2*x*sin(d + e*x)**2*cos(d + e*x)**2/2 + 3*b**2*c**2*x*cos(d + e*x)**4/4 + 3*b**2*c**2*sin(d + e*x)* *3*cos(d + e*x)/(4*e) - 3*b**2*c**2*sin(d + e*x)*cos(d + e*x)**3/(4*e) + b *c**3*sin(d + e*x)**4/e + 3*c**4*x*sin(d + e*x)**4/8 + 3*c**4*x*sin(d + e* x)**2*cos(d + e*x)**2/4 + 3*c**4*x*cos(d + e*x)**4/8 - 5*c**4*sin(d + e*x) **3*cos(d + e*x)/(8*e) - 3*c**4*sin(d + e*x)*cos(d + e*x)**3/(8*e), Ne(e, 0)), (x*(a + b*cos(d) + c*sin(d))**4, True))
Time = 0.23 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.27 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx=-\frac {b^{3} c \cos \left (e x + d\right )^{4}}{e} + \frac {b c^{3} \sin \left (e x + d\right )^{4}}{e} + a^{4} x + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) + 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} b^{4}}{32 \, e} + \frac {3 \, {\left (4 \, e x + 4 \, d - \sin \left (4 \, e x + 4 \, d\right )\right )} b^{2} c^{2}}{16 \, e} + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) - 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} c^{4}}{32 \, e} - 4 \, a^{3} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {b \sin \left (e x + d\right )}{e}\right )} - \frac {3}{2} \, {\left (\frac {4 \, b c \cos \left (e x + d\right )^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} a^{2} - \frac {4}{3} \, {\left (\frac {3 \, b^{2} c \cos \left (e x + d\right )^{3}}{e} - \frac {3 \, b c^{2} \sin \left (e x + d\right )^{3}}{e} + \frac {{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{e} - \frac {{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{e}\right )} a \]
-b^3*c*cos(e*x + d)^4/e + b*c^3*sin(e*x + d)^4/e + a^4*x + 1/32*(12*e*x + 12*d + sin(4*e*x + 4*d) + 8*sin(2*e*x + 2*d))*b^4/e + 3/16*(4*e*x + 4*d - sin(4*e*x + 4*d))*b^2*c^2/e + 1/32*(12*e*x + 12*d + sin(4*e*x + 4*d) - 8*s in(2*e*x + 2*d))*c^4/e - 4*a^3*(c*cos(e*x + d)/e - b*sin(e*x + d)/e) - 3/2 *(4*b*c*cos(e*x + d)^2/e - (2*e*x + 2*d + sin(2*e*x + 2*d))*b^2/e - (2*e*x + 2*d - sin(2*e*x + 2*d))*c^2/e)*a^2 - 4/3*(3*b^2*c*cos(e*x + d)^3/e - 3* b*c^2*sin(e*x + d)^3/e + (sin(e*x + d)^3 - 3*sin(e*x + d))*b^3/e - (cos(e* x + d)^3 - 3*cos(e*x + d))*c^3/e)*a
Time = 0.34 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.10 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx=\frac {1}{8} \, {\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4} + 24 \, a^{2} c^{2} + 6 \, b^{2} c^{2} + 3 \, c^{4}\right )} x - \frac {{\left (b^{3} c - b c^{3}\right )} \cos \left (4 \, e x + 4 \, d\right )}{8 \, e} - \frac {{\left (3 \, a b^{2} c - a c^{3}\right )} \cos \left (3 \, e x + 3 \, d\right )}{3 \, e} - \frac {{\left (6 \, a^{2} b c + b^{3} c + b c^{3}\right )} \cos \left (2 \, e x + 2 \, d\right )}{2 \, e} - \frac {{\left (4 \, a^{3} c + 3 \, a b^{2} c + 3 \, a c^{3}\right )} \cos \left (e x + d\right )}{e} + \frac {{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \sin \left (4 \, e x + 4 \, d\right )}{32 \, e} + \frac {{\left (a b^{3} - 3 \, a b c^{2}\right )} \sin \left (3 \, e x + 3 \, d\right )}{3 \, e} + \frac {{\left (6 \, a^{2} b^{2} + b^{4} - 6 \, a^{2} c^{2} - c^{4}\right )} \sin \left (2 \, e x + 2 \, d\right )}{4 \, e} + \frac {{\left (4 \, a^{3} b + 3 \, a b^{3} + 3 \, a b c^{2}\right )} \sin \left (e x + d\right )}{e} \]
1/8*(8*a^4 + 24*a^2*b^2 + 3*b^4 + 24*a^2*c^2 + 6*b^2*c^2 + 3*c^4)*x - 1/8* (b^3*c - b*c^3)*cos(4*e*x + 4*d)/e - 1/3*(3*a*b^2*c - a*c^3)*cos(3*e*x + 3 *d)/e - 1/2*(6*a^2*b*c + b^3*c + b*c^3)*cos(2*e*x + 2*d)/e - (4*a^3*c + 3* a*b^2*c + 3*a*c^3)*cos(e*x + d)/e + 1/32*(b^4 - 6*b^2*c^2 + c^4)*sin(4*e*x + 4*d)/e + 1/3*(a*b^3 - 3*a*b*c^2)*sin(3*e*x + 3*d)/e + 1/4*(6*a^2*b^2 + b^4 - 6*a^2*c^2 - c^4)*sin(2*e*x + 2*d)/e + (4*a^3*b + 3*a*b^3 + 3*a*b*c^2 )*sin(e*x + d)/e
Time = 28.62 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.45 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx=\frac {6\,b^4\,\sin \left (2\,d+2\,e\,x\right )+\frac {3\,b^4\,\sin \left (4\,d+4\,e\,x\right )}{4}-6\,c^4\,\sin \left (2\,d+2\,e\,x\right )+\frac {3\,c^4\,\sin \left (4\,d+4\,e\,x\right )}{4}+8\,a\,c^3\,\cos \left (3\,d+3\,e\,x\right )-12\,b\,c^3\,\cos \left (2\,d+2\,e\,x\right )-12\,b^3\,c\,\cos \left (2\,d+2\,e\,x\right )+3\,b\,c^3\,\cos \left (4\,d+4\,e\,x\right )-3\,b^3\,c\,\cos \left (4\,d+4\,e\,x\right )+8\,a\,b^3\,\sin \left (3\,d+3\,e\,x\right )+36\,a^2\,b^2\,\sin \left (2\,d+2\,e\,x\right )-36\,a^2\,c^2\,\sin \left (2\,d+2\,e\,x\right )-\frac {9\,b^2\,c^2\,\sin \left (4\,d+4\,e\,x\right )}{2}-72\,a\,c^3\,\cos \left (d+e\,x\right )-96\,a^3\,c\,\cos \left (d+e\,x\right )+72\,a\,b^3\,\sin \left (d+e\,x\right )+96\,a^3\,b\,\sin \left (d+e\,x\right )+24\,a^4\,e\,x+9\,b^4\,e\,x+9\,c^4\,e\,x-72\,a\,b^2\,c\,\cos \left (d+e\,x\right )+72\,a\,b\,c^2\,\sin \left (d+e\,x\right )-72\,a^2\,b\,c\,\cos \left (2\,d+2\,e\,x\right )-24\,a\,b^2\,c\,\cos \left (3\,d+3\,e\,x\right )-24\,a\,b\,c^2\,\sin \left (3\,d+3\,e\,x\right )+72\,a^2\,b^2\,e\,x+72\,a^2\,c^2\,e\,x+18\,b^2\,c^2\,e\,x}{24\,e} \]
(6*b^4*sin(2*d + 2*e*x) + (3*b^4*sin(4*d + 4*e*x))/4 - 6*c^4*sin(2*d + 2*e *x) + (3*c^4*sin(4*d + 4*e*x))/4 + 8*a*c^3*cos(3*d + 3*e*x) - 12*b*c^3*cos (2*d + 2*e*x) - 12*b^3*c*cos(2*d + 2*e*x) + 3*b*c^3*cos(4*d + 4*e*x) - 3*b ^3*c*cos(4*d + 4*e*x) + 8*a*b^3*sin(3*d + 3*e*x) + 36*a^2*b^2*sin(2*d + 2* e*x) - 36*a^2*c^2*sin(2*d + 2*e*x) - (9*b^2*c^2*sin(4*d + 4*e*x))/2 - 72*a *c^3*cos(d + e*x) - 96*a^3*c*cos(d + e*x) + 72*a*b^3*sin(d + e*x) + 96*a^3 *b*sin(d + e*x) + 24*a^4*e*x + 9*b^4*e*x + 9*c^4*e*x - 72*a*b^2*c*cos(d + e*x) + 72*a*b*c^2*sin(d + e*x) - 72*a^2*b*c*cos(2*d + 2*e*x) - 24*a*b^2*c* cos(3*d + 3*e*x) - 24*a*b*c^2*sin(3*d + 3*e*x) + 72*a^2*b^2*e*x + 72*a^2*c ^2*e*x + 18*b^2*c^2*e*x)/(24*e)