Integrand size = 30, antiderivative size = 246 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx=\frac {35}{8} \left (b^2+c^2\right )^2 x-\frac {35 c \left (b^2+c^2\right )^{3/2} \cos (d+e x)}{8 e}+\frac {35 b \left (b^2+c^2\right )^{3/2} \sin (d+e x)}{8 e}-\frac {35 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{24 e}-\frac {7 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{12 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e} \] Output:
35/8*(b^2+c^2)^2*x-35/8*c*(b^2+c^2)^(3/2)*cos(e*x+d)/e+35/8*b*(b^2+c^2)^(3 /2)*sin(e*x+d)/e-35/24*(b^2+c^2)*(c*cos(e*x+d)-b*sin(e*x+d))*((b^2+c^2)^(1 /2)+b*cos(e*x+d)+c*sin(e*x+d))/e-7/12*(b^2+c^2)^(1/2)*(c*cos(e*x+d)-b*sin( e*x+d))*((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^2/e-1/4*(c*cos(e*x+d)- b*sin(e*x+d))*((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^3/e
Result contains complex when optimal does not.
Time = 2.10 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.97 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx=\frac {420 \left (b^2+c^2\right )^2 (d+e x)-672 (b-i c) (b+i c) c \sqrt {b^2+c^2} \cos (d+e x)-336 b c \left (b^2+c^2\right ) \cos (2 (d+e x))+32 c \left (-3 b^2+c^2\right ) \sqrt {b^2+c^2} \cos (3 (d+e x))-12 b c \left (b^2-c^2\right ) \cos (4 (d+e x))+672 b (b-i c) (b+i c) \sqrt {b^2+c^2} \sin (d+e x)+168 \left (b^4-c^4\right ) \sin (2 (d+e x))+32 b \left (b^2-3 c^2\right ) \sqrt {b^2+c^2} \sin (3 (d+e x))+3 \left (b^4-6 b^2 c^2+c^4\right ) \sin (4 (d+e x))}{96 e} \] Input:
Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4,x]
Output:
(420*(b^2 + c^2)^2*(d + e*x) - 672*(b - I*c)*(b + I*c)*c*Sqrt[b^2 + c^2]*C os[d + e*x] - 336*b*c*(b^2 + c^2)*Cos[2*(d + e*x)] + 32*c*(-3*b^2 + c^2)*S qrt[b^2 + c^2]*Cos[3*(d + e*x)] - 12*b*c*(b^2 - c^2)*Cos[4*(d + e*x)] + 67 2*b*(b - I*c)*(b + I*c)*Sqrt[b^2 + c^2]*Sin[d + e*x] + 168*(b^4 - c^4)*Sin [2*(d + e*x)] + 32*b*(b^2 - 3*c^2)*Sqrt[b^2 + 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 = 0.64 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3042, 3592, 3042, 3592, 3042, 3592, 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.
| \(\displaystyle \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4dx\) |
| \(\Big \downarrow \) 3592 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^3dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^3dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\) |
| \(\Big \downarrow \) 3592 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2+c^2} \left (\frac {5}{3} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^2dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2+c^2} \left (\frac {5}{3} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )^2dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\) |
| \(\Big \downarrow \) 3592 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2+c^2} \left (\frac {5}{3} \sqrt {b^2+c^2} \left (\frac {3}{2} \sqrt {b^2+c^2} \int \left (b \cos (d+e x)+c \sin (d+e x)+\sqrt {b^2+c^2}\right )dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{2 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7}{4} \sqrt {b^2+c^2} \left (\frac {5}{3} \sqrt {b^2+c^2} \left (\frac {3}{2} \sqrt {b^2+c^2} \left (x \sqrt {b^2+c^2}+\frac {b \sin (d+e x)}{e}-\frac {c \cos (d+e x)}{e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{2 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{3 e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^3}{4 e}\) |
Input:
Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4,x]
Output:
-1/4*((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3)/e + (7*Sqrt[b^2 + c^2]*(-1/3*((c*Cos[d + e*x] - b*Sin [d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/e + (5*S qrt[b^2 + c^2]*(-1/2*((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/e + (3*Sqrt[b^2 + c^2]*(Sqrt[b^2 + c^2] *x - (c*Cos[d + e*x])/e + (b*Sin[d + e*x])/e))/2))/3))/4
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[a*((2*n - 1)/n) Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[a^2 - b^2 - c^2, 0] && GtQ[n, 0]
Time = 17.73 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.35
| method | result | size |
| risch | \(\frac {35 b^{4} x}{8}+\frac {35 x \,b^{2} c^{2}}{4}+\frac {35 x \,c^{4}}{8}-\frac {7 c \left (b^{2}+c^{2}\right )^{\frac {3}{2}} \cos \left (e x +d \right )}{e}+\frac {7 b \left (b^{2}+c^{2}\right )^{\frac {3}{2}} \sin \left (e x +d \right )}{e}-\frac {b^{3} c \cos \left (4 e x +4 d \right )}{8 e}+\frac {b \,c^{3} \cos \left (4 e x +4 d \right )}{8 e}+\frac {\sin \left (4 e x +4 d \right ) b^{4}}{32 e}-\frac {3 \sin \left (4 e x +4 d \right ) b^{2} c^{2}}{16 e}+\frac {\sin \left (4 e x +4 d \right ) c^{4}}{32 e}-\frac {\sqrt {b^{2}+c^{2}}\, c \cos \left (3 e x +3 d \right ) b^{2}}{e}+\frac {\sqrt {b^{2}+c^{2}}\, c^{3} \cos \left (3 e x +3 d \right )}{3 e}+\frac {\sqrt {b^{2}+c^{2}}\, b^{3} \sin \left (3 e x +3 d \right )}{3 e}-\frac {\sqrt {b^{2}+c^{2}}\, b \sin \left (3 e x +3 d \right ) c^{2}}{e}-\frac {7 b^{3} c \cos \left (2 e x +2 d \right )}{2 e}-\frac {7 b \,c^{3} \cos \left (2 e x +2 d \right )}{2 e}+\frac {7 \sin \left (2 e x +2 d \right ) b^{4}}{4 e}-\frac {7 \sin \left (2 e x +2 d \right ) c^{4}}{4 e}\) | \(331\) |
| parts | \(\left (b^{2}+c^{2}\right )^{2} x -\frac {4 b^{3} \left (\frac {c \sin \left (e x +d \right )^{4}}{4}+\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d \right )^{3}}{3}-\frac {c \sin \left (e x +d \right )^{2}}{2}-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d \right )\right )}{e}+\frac {6 b^{2} c^{2} \left (-\frac {\cos \left (e x +d \right )^{3} \sin \left (e x +d \right )}{4}+\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \cos \left (e x +d \right )^{3}+6 b^{4} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+6 b^{2} c^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {4 b \left (\frac {\sin \left (e x +d \right )^{4} c^{3}}{4}+\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d \right )^{3} c^{2}+\frac {3 \sin \left (e x +d \right )^{2} b^{2} c}{2}+\frac {3 \sin \left (e x +d \right )^{2} c^{3}}{2}+\sin \left (e x +d \right ) \left (b^{2}+c^{2}\right )^{\frac {3}{2}}\right )}{e}+\frac {b^{4} \left (\frac {\left (\cos \left (e x +d \right )^{3}+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )}{e}+\frac {c^{4} \left (-\frac {\left (\sin \left (e x +d \right )^{3}+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )}{e}+\frac {6 \left (b^{2}+c^{2}\right ) c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}-\frac {4 \sqrt {b^{2}+c^{2}}\, c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3 e}-\frac {4 c \left (b^{2}+c^{2}\right )^{\frac {3}{2}} \cos \left (e x +d \right )}{e}\) | \(464\) |
| derivativedivides | \(\frac {c^{4} \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \cos \left (e x +d \right )^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \sin \left (e x +d \right )^{3}+b^{4} \left (e x +d \right )+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \sin \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \cos \left (e x +d \right )-6 b^{3} c \cos \left (e x +d \right )^{2}-6 c^{3} b \cos \left (e x +d \right )^{2}-b^{3} c \cos \left (e x +d \right )^{4}+6 b^{2} c^{2} \left (-\frac {\cos \left (e x +d \right )^{3} \sin \left (e x +d \right )}{4}+\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )+c^{3} b \sin \left (e x +d \right )^{4}+2 b^{2} c^{2} \left (e x +d \right )+6 b^{4} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+6 c^{4} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+b^{4} \left (\frac {\left (\cos \left (e x +d \right )^{3}+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+c^{4} \left (-\frac {\left (\sin \left (e x +d \right )^{3}+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+4 \sqrt {b^{2}+c^{2}}\, b^{3} \sin \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, c^{3} \cos \left (e x +d \right )+6 b^{2} c^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+\frac {4 \sqrt {b^{2}+c^{2}}\, b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-\frac {4 \sqrt {b^{2}+c^{2}}\, c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}}{e}\) | \(514\) |
| default | \(\frac {c^{4} \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \cos \left (e x +d \right )^{3}+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \sin \left (e x +d \right )^{3}+b^{4} \left (e x +d \right )+4 \sqrt {b^{2}+c^{2}}\, b \,c^{2} \sin \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, b^{2} c \cos \left (e x +d \right )-6 b^{3} c \cos \left (e x +d \right )^{2}-6 c^{3} b \cos \left (e x +d \right )^{2}-b^{3} c \cos \left (e x +d \right )^{4}+6 b^{2} c^{2} \left (-\frac {\cos \left (e x +d \right )^{3} \sin \left (e x +d \right )}{4}+\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{8}+\frac {e x}{8}+\frac {d}{8}\right )+c^{3} b \sin \left (e x +d \right )^{4}+2 b^{2} c^{2} \left (e x +d \right )+6 b^{4} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+6 c^{4} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+b^{4} \left (\frac {\left (\cos \left (e x +d \right )^{3}+\frac {3 \cos \left (e x +d \right )}{2}\right ) \sin \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+c^{4} \left (-\frac {\left (\sin \left (e x +d \right )^{3}+\frac {3 \sin \left (e x +d \right )}{2}\right ) \cos \left (e x +d \right )}{4}+\frac {3 e x}{8}+\frac {3 d}{8}\right )+4 \sqrt {b^{2}+c^{2}}\, b^{3} \sin \left (e x +d \right )-4 \sqrt {b^{2}+c^{2}}\, c^{3} \cos \left (e x +d \right )+6 b^{2} c^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+6 b^{2} c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+\frac {4 \sqrt {b^{2}+c^{2}}\, b^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-\frac {4 \sqrt {b^{2}+c^{2}}\, c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}}{e}\) | \(514\) |
| norman | \(\frac {\left (\frac {35}{8} b^{4}+\frac {35}{4} b^{2} c^{2}+\frac {35}{8} c^{4}\right ) x +\left (\frac {35}{2} b^{4}+35 b^{2} c^{2}+\frac {35}{2} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\left (\frac {35}{2} b^{4}+35 b^{2} c^{2}+\frac {35}{2} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}+\left (\frac {35}{8} b^{4}+\frac {35}{4} b^{2} c^{2}+\frac {35}{8} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{8}+\left (\frac {105}{4} b^{4}+\frac {105}{2} b^{2} c^{2}+\frac {105}{4} c^{4}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+\frac {-48 \sqrt {b^{2}+c^{2}}\, b^{2} c -40 \sqrt {b^{2}+c^{2}}\, c^{3}}{3 e}-\frac {7 \left (-64 \sqrt {b^{2}+c^{2}}\, b^{3}-96 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+9 b^{4}+18 b^{2} c^{2}-15 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{12 e}-\frac {\left (-64 \sqrt {b^{2}+c^{2}}\, b^{3}-32 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+29 b^{4}-6 b^{2} c^{2}-27 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{7}}{4 e}+\frac {\left (64 \sqrt {b^{2}+c^{2}}\, b^{3}+32 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+29 b^{4}-6 b^{2} c^{2}-27 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{4 e}+\frac {7 \left (64 \sqrt {b^{2}+c^{2}}\, b^{3}+96 \sqrt {b^{2}+c^{2}}\, b \,c^{2}+9 b^{4}+18 b^{2} c^{2}-15 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{12 e}+\frac {4 \left (-24 \sqrt {b^{2}+c^{2}}\, b^{2} c -34 \sqrt {b^{2}+c^{2}}\, c^{3}+24 b^{3} c +18 c^{3} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{3 e}+\frac {2 \left (-24 \sqrt {b^{2}+c^{2}}\, b^{2} c -20 \sqrt {b^{2}+c^{2}}\, c^{3}+24 b^{3} c +32 c^{3} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{e}+\frac {4 \left (-8 \sqrt {b^{2}+c^{2}}\, b^{2} c -2 \sqrt {b^{2}+c^{2}}\, c^{3}+8 b^{3} c +6 c^{3} b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{4}}\) | \(630\) |
Input:
int(((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x,method=_RETURNVERBOSE)
Output:
35/8*b^4*x+35/4*x*b^2*c^2+35/8*x*c^4-7*c*(b^2+c^2)^(3/2)*cos(e*x+d)/e+7*b* (b^2+c^2)^(3/2)*sin(e*x+d)/e-1/8*b^3*c/e*cos(4*e*x+4*d)+1/8*b*c^3/e*cos(4* e*x+4*d)+1/32/e*sin(4*e*x+4*d)*b^4-3/16/e*sin(4*e*x+4*d)*b^2*c^2+1/32/e*si n(4*e*x+4*d)*c^4-(b^2+c^2)^(1/2)*c/e*cos(3*e*x+3*d)*b^2+1/3*(b^2+c^2)^(1/2 )*c^3/e*cos(3*e*x+3*d)+1/3*(b^2+c^2)^(1/2)*b^3/e*sin(3*e*x+3*d)-(b^2+c^2)^ (1/2)*b/e*sin(3*e*x+3*d)*c^2-7/2*b^3*c/e*cos(2*e*x+2*d)-7/2*b*c^3/e*cos(2* e*x+2*d)+7/4/e*sin(2*e*x+2*d)*b^4-7/4/e*sin(2*e*x+2*d)*c^4
Time = 0.08 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.90 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx=-\frac {24 \, {\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{4} - 105 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} e x + 48 \, {\left (3 \, b^{3} c + 4 \, b c^{3}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (2 \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{3} + {\left (27 \, b^{4} + 6 \, b^{2} c^{2} - 29 \, c^{4}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 32 \, {\left ({\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (b^{2} c + 2 \, c^{3}\right )} \cos \left (e x + d\right ) - {\left (5 \, b^{3} + 6 \, b c^{2} + {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}}{24 \, e} \] Input:
integrate(((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="fric as")
Output:
-1/24*(24*(b^3*c - b*c^3)*cos(e*x + d)^4 - 105*(b^4 + 2*b^2*c^2 + c^4)*e*x + 48*(3*b^3*c + 4*b*c^3)*cos(e*x + d)^2 - 3*(2*(b^4 - 6*b^2*c^2 + c^4)*co s(e*x + d)^3 + (27*b^4 + 6*b^2*c^2 - 29*c^4)*cos(e*x + d))*sin(e*x + d) + 32*((3*b^2*c - c^3)*cos(e*x + d)^3 + 3*(b^2*c + 2*c^3)*cos(e*x + d) - (5*b ^3 + 6*b*c^2 + (b^3 - 3*b*c^2)*cos(e*x + d)^2)*sin(e*x + d))*sqrt(b^2 + c^ 2))/e
Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (231) = 462\).
Time = 0.59 (sec) , antiderivative size = 857, normalized size of antiderivative = 3.48 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx =\text {Too large to display} \] Input:
integrate(((b**2+c**2)**(1/2)+b*cos(e*x+d)+c*sin(e*x+d))**4,x)
Output:
Piecewise((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*sin(d + e*x)**2 + 3*b**4*x*cos(d + e*x)**4/8 + 3*b**4* x*cos(d + e*x)**2 + b**4*x + 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) + 3*b**4*sin(d + e*x)*cos(d + e*x )/e - b**3*c*cos(d + e*x)**4/e - 6*b**3*c*cos(d + e*x)**2/e + 8*b**3*sqrt( b**2 + c**2)*sin(d + e*x)**3/(3*e) + 4*b**3*sqrt(b**2 + c**2)*sin(d + e*x) *cos(d + e*x)**2/e + 4*b**3*sqrt(b**2 + c**2)*sin(d + e*x)/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 + 6 *b**2*c**2*x*sin(d + e*x)**2 + 3*b**2*c**2*x*cos(d + e*x)**4/4 + 6*b**2*c* *2*x*cos(d + e*x)**2 + 2*b**2*c**2*x + 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) - 4*b**2*c*sq rt(b**2 + c**2)*cos(d + e*x)**3/e - 4*b**2*c*sqrt(b**2 + c**2)*cos(d + e*x )/e + b*c**3*sin(d + e*x)**4/e - 6*b*c**3*cos(d + e*x)**2/e + 4*b*c**2*sqr t(b**2 + c**2)*sin(d + e*x)**3/e + 4*b*c**2*sqrt(b**2 + c**2)*sin(d + e*x) /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*sin(d + e*x)**2 + 3*c**4*x*cos(d + e*x)**4/8 + 3*c**4*x*cos( d + e*x)**2 + c**4*x - 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) - 3*c**4*sin(d + e*x)*cos(d + e*x)/e - 4*c**3*sqrt(b**2 + c**2)*sin(d + e*x)**2*cos(d + e*x)/e - 8*c**3*sqrt(b**2 + c**2)*cos(d + e*x)**3/(3*e) - 4*c**3*sqrt(b**2 + c**2)*cos(d + e*x)/...
Time = 0.04 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.44 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx=-\frac {b^{3} c \cos \left (e x + d\right )^{4}}{e} + \frac {b c^{3} \sin \left (e x + d\right )^{4}}{e} + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) + 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} b^{4}}{32 \, e} + \frac {3 \, {\left (4 \, e x + 4 \, d - \sin \left (4 \, e x + 4 \, d\right )\right )} b^{2} c^{2}}{16 \, e} + \frac {{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) - 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} c^{4}}{32 \, e} + {\left (b^{2} + c^{2}\right )}^{2} x - 4 \, {\left (b^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {b \sin \left (e x + d\right )}{e}\right )} - \frac {3}{2} \, {\left (\frac {4 \, b c \cos \left (e x + d\right )^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} {\left (b^{2} + c^{2}\right )} - \frac {4}{3} \, {\left (\frac {3 \, b^{2} c \cos \left (e x + d\right )^{3}}{e} - \frac {3 \, b c^{2} \sin \left (e x + d\right )^{3}}{e} + \frac {{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{e} - \frac {{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{e}\right )} \sqrt {b^{2} + c^{2}} \] Input:
integrate(((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="maxi ma")
Output:
-b^3*c*cos(e*x + d)^4/e + b*c^3*sin(e*x + d)^4/e + 1/32*(12*e*x + 12*d + s in(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*sin(2*e*x + 2*d))*c^4/e + (b^2 + c^2)^2*x - 4*(b^2 + c^2)^(3/2)*(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)*(b^2 + c^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)*sqrt(b^2 + c^2)
Time = 0.20 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.17 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx=\frac {35}{8} \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )} x - \frac {{\left (b^{3} c - b c^{3}\right )} \cos \left (4 \, e x + 4 \, d\right )}{8 \, e} - \frac {{\left (3 \, \sqrt {b^{2} + c^{2}} b^{2} c - \sqrt {b^{2} + c^{2}} c^{3}\right )} \cos \left (3 \, e x + 3 \, d\right )}{3 \, e} - \frac {7 \, {\left (b^{3} c + b c^{3}\right )} \cos \left (2 \, e x + 2 \, d\right )}{2 \, e} - \frac {7 \, {\left (\sqrt {b^{2} + c^{2}} b^{2} c + \sqrt {b^{2} + c^{2}} c^{3}\right )} \cos \left (e x + d\right )}{e} + \frac {{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \sin \left (4 \, e x + 4 \, d\right )}{32 \, e} + \frac {{\left (\sqrt {b^{2} + c^{2}} b^{3} - 3 \, \sqrt {b^{2} + c^{2}} b c^{2}\right )} \sin \left (3 \, e x + 3 \, d\right )}{3 \, e} + \frac {7 \, {\left (b^{4} - c^{4}\right )} \sin \left (2 \, e x + 2 \, d\right )}{4 \, e} + \frac {7 \, {\left (\sqrt {b^{2} + c^{2}} b^{3} + \sqrt {b^{2} + c^{2}} b c^{2}\right )} \sin \left (e x + d\right )}{e} \] Input:
integrate(((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="giac ")
Output:
35/8*(b^4 + 2*b^2*c^2 + c^4)*x - 1/8*(b^3*c - b*c^3)*cos(4*e*x + 4*d)/e - 1/3*(3*sqrt(b^2 + c^2)*b^2*c - sqrt(b^2 + c^2)*c^3)*cos(3*e*x + 3*d)/e - 7 /2*(b^3*c + b*c^3)*cos(2*e*x + 2*d)/e - 7*(sqrt(b^2 + c^2)*b^2*c + sqrt(b^ 2 + c^2)*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*(sqrt(b^2 + c^2)*b^3 - 3*sqrt(b^2 + c^2)*b*c^2)*sin(3*e*x + 3*d )/e + 7/4*(b^4 - c^4)*sin(2*e*x + 2*d)/e + 7*(sqrt(b^2 + c^2)*b^3 + sqrt(b ^2 + c^2)*b*c^2)*sin(e*x + d)/e
Time = 20.34 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.12 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx=\frac {35\,\mathrm {atan}\left (\frac {35\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,{\left (b^2+c^2\right )}^2}{4\,\left (\frac {35\,b^4}{4}+\frac {35\,b^2\,c^2}{2}+\frac {35\,c^4}{4}\right )}\right )\,{\left (b^2+c^2\right )}^2}{4\,e}-\frac {35\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )-\frac {e\,x}{2}\right )\,{\left (b^2+c^2\right )}^2}{4\,e}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\left (16\,b^3+8\,b\,c^2\right )\,\sqrt {b^2+c^2}+\frac {29\,b^4}{4}-\frac {27\,c^4}{4}-\frac {3\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (24\,b\,c^3+32\,b^3\,c-\left (32\,b^2\,c+8\,c^3\right )\,\sqrt {b^2+c^2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (64\,b\,c^3+48\,b^3\,c-\left (48\,b^2\,c+40\,c^3\right )\,\sqrt {b^2+c^2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (24\,b\,c^3+32\,b^3\,c-\left (32\,b^2\,c+\frac {136\,c^3}{3}\right )\,\sqrt {b^2+c^2}\right )-\left (16\,b^2\,c+\frac {40\,c^3}{3}\right )\,\sqrt {b^2+c^2}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^7\,\left (\left (16\,b^3+8\,b\,c^2\right )\,\sqrt {b^2+c^2}-\frac {29\,b^4}{4}+\frac {27\,c^4}{4}+\frac {3\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (\left (\frac {112\,b^3}{3}+56\,b\,c^2\right )\,\sqrt {b^2+c^2}+\frac {21\,b^4}{4}-\frac {35\,c^4}{4}+\frac {21\,b^2\,c^2}{2}\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (\left (\frac {112\,b^3}{3}+56\,b\,c^2\right )\,\sqrt {b^2+c^2}-\frac {21\,b^4}{4}+\frac {35\,c^4}{4}-\frac {21\,b^2\,c^2}{2}\right )}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \] Input:
int((b*cos(d + e*x) + c*sin(d + e*x) + (b^2 + c^2)^(1/2))^4,x)
Output:
(35*atan((35*tan(d/2 + (e*x)/2)*(b^2 + c^2)^2)/(4*((35*b^4)/4 + (35*c^4)/4 + (35*b^2*c^2)/2)))*(b^2 + c^2)^2)/(4*e) - (35*(atan(tan(d/2 + (e*x)/2)) - (e*x)/2)*(b^2 + c^2)^2)/(4*e) + (tan(d/2 + (e*x)/2)*((8*b*c^2 + 16*b^3)* (b^2 + c^2)^(1/2) + (29*b^4)/4 - (27*c^4)/4 - (3*b^2*c^2)/2) + tan(d/2 + ( e*x)/2)^6*(24*b*c^3 + 32*b^3*c - (32*b^2*c + 8*c^3)*(b^2 + c^2)^(1/2)) + t an(d/2 + (e*x)/2)^4*(64*b*c^3 + 48*b^3*c - (48*b^2*c + 40*c^3)*(b^2 + c^2) ^(1/2)) + tan(d/2 + (e*x)/2)^2*(24*b*c^3 + 32*b^3*c - (32*b^2*c + (136*c^3 )/3)*(b^2 + c^2)^(1/2)) - (16*b^2*c + (40*c^3)/3)*(b^2 + c^2)^(1/2) + tan( d/2 + (e*x)/2)^7*((8*b*c^2 + 16*b^3)*(b^2 + c^2)^(1/2) - (29*b^4)/4 + (27* c^4)/4 + (3*b^2*c^2)/2) + tan(d/2 + (e*x)/2)^3*((56*b*c^2 + (112*b^3)/3)*( b^2 + c^2)^(1/2) + (21*b^4)/4 - (35*c^4)/4 + (21*b^2*c^2)/2) + tan(d/2 + ( e*x)/2)^5*((56*b*c^2 + (112*b^3)/3)*(b^2 + c^2)^(1/2) - (21*b^4)/4 + (35*c ^4)/4 - (21*b^2*c^2)/2))/(e*(4*tan(d/2 + (e*x)/2)^2 + 6*tan(d/2 + (e*x)/2) ^4 + 4*tan(d/2 + (e*x)/2)^6 + tan(d/2 + (e*x)/2)^8 + 1))
Time = 0.20 (sec) , antiderivative size = 744, normalized size of antiderivative = 3.02 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^4 \, dx =\text {Too large to display} \] Input:
int(((b^2+c^2)^(1/2)+b*cos(e*x+d)+c*sin(e*x+d))^4,x)
Output:
(9*cos(d + e*x)**4*b**4*e*x - 24*cos(d + e*x)**4*b**3*c + 18*cos(d + e*x)* *4*b**2*c**2*e*x - 24*cos(d + e*x)**4*b*c**3 + 9*cos(d + e*x)**4*c**4*e*x - 96*sqrt(b**2 + c**2)*cos(d + e*x)**3*b**2*c - 64*sqrt(b**2 + c**2)*cos(d + e*x)**3*c**3 + 15*cos(d + e*x)**3*sin(d + e*x)*b**4 - 18*cos(d + e*x)** 3*sin(d + e*x)*b**2*c**2 - 9*cos(d + e*x)**3*sin(d + e*x)*c**4 + 96*sqrt(b **2 + c**2)*cos(d + e*x)**2*sin(d + e*x)*b**3 + 18*cos(d + e*x)**2*sin(d + e*x)**2*b**4*e*x + 36*cos(d + e*x)**2*sin(d + e*x)**2*b**2*c**2*e*x - 48* cos(d + e*x)**2*sin(d + e*x)**2*b*c**3 + 18*cos(d + e*x)**2*sin(d + e*x)** 2*c**4*e*x + 72*cos(d + e*x)**2*b**4*e*x - 144*cos(d + e*x)**2*b**3*c + 14 4*cos(d + e*x)**2*b**2*c**2*e*x - 144*cos(d + e*x)**2*b*c**3 + 72*cos(d + e*x)**2*c**4*e*x - 96*sqrt(b**2 + c**2)*cos(d + e*x)*sin(d + e*x)**2*c**3 - 96*sqrt(b**2 + c**2)*cos(d + e*x)*b**2*c - 96*sqrt(b**2 + c**2)*cos(d + e*x)*c**3 + 9*cos(d + e*x)*sin(d + e*x)**3*b**4 + 18*cos(d + e*x)*sin(d + e*x)**3*b**2*c**2 - 15*cos(d + e*x)*sin(d + e*x)**3*c**4 + 72*cos(d + e*x) *sin(d + e*x)*b**4 - 72*cos(d + e*x)*sin(d + e*x)*c**4 + 64*sqrt(b**2 + c* *2)*sin(d + e*x)**3*b**3 + 96*sqrt(b**2 + c**2)*sin(d + e*x)**3*b*c**2 + 9 6*sqrt(b**2 + c**2)*sin(d + e*x)*b**3 + 96*sqrt(b**2 + c**2)*sin(d + e*x)* b*c**2 + 9*sin(d + e*x)**4*b**4*e*x + 18*sin(d + e*x)**4*b**2*c**2*e*x + 9 *sin(d + e*x)**4*c**4*e*x + 72*sin(d + e*x)**2*b**4*e*x + 144*sin(d + e*x) **2*b**2*c**2*e*x + 72*sin(d + e*x)**2*c**4*e*x + 24*b**4*e*x + 48*b**2...