Integrand size = 24, antiderivative size = 81 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=2 \left (3 a^2+c^2\right ) x-\frac {6 a c \cos (d+e x)}{e}+\frac {6 a^2 \sin (d+e x)}{e}-\frac {2 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))}{e} \] Output:
2*(3*a^2+c^2)*x-6*a*c*cos(e*x+d)/e+6*a^2*sin(e*x+d)/e-2*(c*cos(e*x+d)-a*si n(e*x+d))*(a+a*cos(e*x+d)+c*sin(e*x+d))/e
Time = 0.47 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=4 \left (\frac {\left (3 a^2+c^2\right ) (d+e x)}{2 e}-\frac {2 a c \cos (d+e x)}{e}-\frac {a c \cos (2 (d+e x))}{2 e}+\frac {2 a^2 \sin (d+e x)}{e}+\frac {\left (a^2-c^2\right ) \sin (2 (d+e x))}{4 e}\right ) \] Input:
Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2,x]
Output:
4*(((3*a^2 + c^2)*(d + e*x))/(2*e) - (2*a*c*Cos[d + e*x])/e - (a*c*Cos[2*( d + e*x)])/(2*e) + (2*a^2*Sin[d + e*x])/e + ((a^2 - c^2)*Sin[2*(d + e*x)]) /(4*e))
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3042, 3599, 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.
\(\displaystyle \int (2 a \cos (d+e x)+2 a+2 c \sin (d+e x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (2 a \cos (d+e x)+2 a+2 c \sin (d+e x))^2dx\) |
\(\Big \downarrow \) 3599 |
\(\displaystyle \frac {1}{2} \int \left (12 \cos (d+e x) a^2+12 c \sin (d+e x) a+4 \left (3 a^2+c^2\right )\right )dx-\frac {2 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (4 x \left (3 a^2+c^2\right )+\frac {12 a^2 \sin (d+e x)}{e}-\frac {12 a c \cos (d+e x)}{e}\right )-\frac {2 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))}{e}\) |
Input:
Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2,x]
Output:
(-2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x] ))/e + (4*(3*a^2 + c^2)*x - (12*a*c*Cos[d + e*x])/e + (12*a^2*Sin[d + e*x] )/e)/2
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]
Time = 1.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95
method | result | size |
parallelrisch | \(\frac {\left (a^{2}-c^{2}\right ) \sin \left (2 e x +2 d \right )+6 a^{2} e x +2 c^{2} e x +8 a^{2} \sin \left (e x +d \right )-8 a c \cos \left (e x +d \right )-2 a c \cos \left (2 e x +2 d \right )-6 a c}{e}\) | \(77\) |
risch | \(6 a^{2} x +2 x \,c^{2}-\frac {8 a c \cos \left (e x +d \right )}{e}+\frac {8 a^{2} \sin \left (e x +d \right )}{e}-\frac {2 a c \cos \left (2 e x +2 d \right )}{e}+\frac {\sin \left (2 e x +2 d \right ) a^{2}}{e}-\frac {\sin \left (2 e x +2 d \right ) c^{2}}{e}\) | \(90\) |
derivativedivides | \(\frac {4 a^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-4 a c \cos \left (e x +d \right )^{2}+4 c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a^{2} \sin \left (e x +d \right )-8 a c \cos \left (e x +d \right )+4 a^{2} \left (e x +d \right )}{e}\) | \(101\) |
default | \(\frac {4 a^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-4 a c \cos \left (e x +d \right )^{2}+4 c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a^{2} \sin \left (e x +d \right )-8 a c \cos \left (e x +d \right )+4 a^{2} \left (e x +d \right )}{e}\) | \(101\) |
parts | \(\frac {8 a \left (\frac {c \sin \left (e x +d \right )^{2}}{2}+a \sin \left (e x +d \right )\right )}{e}+4 a^{2} x +\frac {4 a^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {4 c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}-\frac {8 a c \cos \left (e x +d \right )}{e}\) | \(107\) |
norman | \(\frac {\left (6 a^{2}+2 c^{2}\right ) x +\left (6 a^{2}+2 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+\left (12 a^{2}+4 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-\frac {16 a c}{e}+\frac {4 \left (3 a^{2}+c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{e}+\frac {4 \left (5 a^{2}-c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{2}}\) | \(136\) |
orering | \(x \left (2 a +2 a \cos \left (e x +d \right )+2 c \sin \left (e x +d \right )\right )^{2}-\frac {5 \left (2 a +2 a \cos \left (e x +d \right )+2 c \sin \left (e x +d \right )\right ) \left (-2 a e \sin \left (e x +d \right )+2 c e \cos \left (e x +d \right )\right )}{2 e^{2}}+\frac {5 x \left (2 \left (-2 a e \sin \left (e x +d \right )+2 c e \cos \left (e x +d \right )\right )^{2}+2 \left (2 a +2 a \cos \left (e x +d \right )+2 c \sin \left (e x +d \right )\right ) \left (-2 a \,e^{2} \cos \left (e x +d \right )-2 c \,e^{2} \sin \left (e x +d \right )\right )\right )}{4 e^{2}}-\frac {6 \left (-2 a e \sin \left (e x +d \right )+2 c e \cos \left (e x +d \right )\right ) \left (-2 a \,e^{2} \cos \left (e x +d \right )-2 c \,e^{2} \sin \left (e x +d \right )\right )+2 \left (2 a +2 a \cos \left (e x +d \right )+2 c \sin \left (e x +d \right )\right ) \left (2 a \,e^{3} \sin \left (e x +d \right )-2 c \,e^{3} \cos \left (e x +d \right )\right )}{4 e^{4}}+\frac {x \left (6 \left (-2 a \,e^{2} \cos \left (e x +d \right )-2 c \,e^{2} \sin \left (e x +d \right )\right )^{2}+8 \left (-2 a e \sin \left (e x +d \right )+2 c e \cos \left (e x +d \right )\right ) \left (2 a \,e^{3} \sin \left (e x +d \right )-2 c \,e^{3} \cos \left (e x +d \right )\right )+2 \left (2 a +2 a \cos \left (e x +d \right )+2 c \sin \left (e x +d \right )\right ) \left (2 a \,e^{4} \cos \left (e x +d \right )+2 c \,e^{4} \sin \left (e x +d \right )\right )\right )}{4 e^{4}}\) | \(393\) |
Input:
int((2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^2,x,method=_RETURNVERBOSE)
Output:
((a^2-c^2)*sin(2*e*x+2*d)+6*a^2*e*x+2*c^2*e*x+8*a^2*sin(e*x+d)-8*a*c*cos(e *x+d)-2*a*c*cos(2*e*x+2*d)-6*a*c)/e
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=-\frac {2 \, {\left (2 \, a c \cos \left (e x + d\right )^{2} - {\left (3 \, a^{2} + c^{2}\right )} e x + 4 \, a c \cos \left (e x + d\right ) - {\left (4 \, a^{2} + {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )}}{e} \] Input:
integrate((2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^2,x, algorithm="fricas")
Output:
-2*(2*a*c*cos(e*x + d)^2 - (3*a^2 + c^2)*e*x + 4*a*c*cos(e*x + d) - (4*a^2 + (a^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. 170 vs. \(2 (78) = 156\).
Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.10 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=\begin {cases} 2 a^{2} x \sin ^{2}{\left (d + e x \right )} + 2 a^{2} x \cos ^{2}{\left (d + e x \right )} + 4 a^{2} x + \frac {2 a^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} + \frac {8 a^{2} \sin {\left (d + e x \right )}}{e} - \frac {4 a c \cos ^{2}{\left (d + e x \right )}}{e} - \frac {8 a c \cos {\left (d + e x \right )}}{e} + 2 c^{2} x \sin ^{2}{\left (d + e x \right )} + 2 c^{2} x \cos ^{2}{\left (d + e x \right )} - \frac {2 c^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \left (2 a \cos {\left (d \right )} + 2 a + 2 c \sin {\left (d \right )}\right )^{2} & \text {otherwise} \end {cases} \] Input:
integrate((2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))**2,x)
Output:
Piecewise((2*a**2*x*sin(d + e*x)**2 + 2*a**2*x*cos(d + e*x)**2 + 4*a**2*x + 2*a**2*sin(d + e*x)*cos(d + e*x)/e + 8*a**2*sin(d + e*x)/e - 4*a*c*cos(d + e*x)**2/e - 8*a*c*cos(d + e*x)/e + 2*c**2*x*sin(d + e*x)**2 + 2*c**2*x* cos(d + e*x)**2 - 2*c**2*sin(d + e*x)*cos(d + e*x)/e, Ne(e, 0)), (x*(2*a*c os(d) + 2*a + 2*c*sin(d))**2, True))
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.22 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=4 \, a^{2} x - \frac {4 \, a c \cos \left (e x + d\right )^{2}}{e} + \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} a^{2}}{e} + \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e} - 8 \, a {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {a \sin \left (e x + d\right )}{e}\right )} \] Input:
integrate((2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^2,x, algorithm="maxima")
Output:
4*a^2*x - 4*a*c*cos(e*x + d)^2/e + (2*e*x + 2*d + sin(2*e*x + 2*d))*a^2/e + (2*e*x + 2*d - sin(2*e*x + 2*d))*c^2/e - 8*a*(c*cos(e*x + d)/e - a*sin(e *x + d)/e)
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=2 \, {\left (3 \, a^{2} + c^{2}\right )} x - \frac {2 \, a c \cos \left (2 \, e x + 2 \, d\right )}{e} - \frac {8 \, a c \cos \left (e x + d\right )}{e} + \frac {8 \, a^{2} \sin \left (e x + d\right )}{e} + \frac {{\left (a^{2} - c^{2}\right )} \sin \left (2 \, e x + 2 \, d\right )}{e} \] Input:
integrate((2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^2,x, algorithm="giac")
Output:
2*(3*a^2 + c^2)*x - 2*a*c*cos(2*e*x + 2*d)/e - 8*a*c*cos(e*x + d)/e + 8*a^ 2*sin(e*x + d)/e + (a^2 - c^2)*sin(2*e*x + 2*d)/e
Time = 15.97 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.19 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=\frac {x\,\left (12\,a^2+4\,c^2\right )}{2}+\frac {\left (12\,a^2+4\,c^2\right )\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3+\left (20\,a^2-4\,c^2\right )\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )-16\,a\,c}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \] Input:
int((2*a + 2*a*cos(d + e*x) + 2*c*sin(d + e*x))^2,x)
Output:
(x*(12*a^2 + 4*c^2))/2 + (tan(d/2 + (e*x)/2)^3*(12*a^2 + 4*c^2) - 16*a*c + tan(d/2 + (e*x)/2)*(20*a^2 - 4*c^2))/(e*(2*tan(d/2 + (e*x)/2)^2 + tan(d/2 + (e*x)/2)^4 + 1))
Time = 0.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx=\frac {2 \cos \left (e x +d \right )^{2} a^{2} e x -4 \cos \left (e x +d \right )^{2} a c +2 \cos \left (e x +d \right )^{2} c^{2} e x +2 \cos \left (e x +d \right ) \sin \left (e x +d \right ) a^{2}-2 \cos \left (e x +d \right ) \sin \left (e x +d \right ) c^{2}-8 \cos \left (e x +d \right ) a c +2 \sin \left (e x +d \right )^{2} a^{2} e x +2 \sin \left (e x +d \right )^{2} c^{2} e x +8 \sin \left (e x +d \right ) a^{2}+4 a^{2} e x}{e} \] Input:
int((2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^2,x)
Output:
(2*(cos(d + e*x)**2*a**2*e*x - 2*cos(d + e*x)**2*a*c + cos(d + e*x)**2*c** 2*e*x + cos(d + e*x)*sin(d + e*x)*a**2 - cos(d + e*x)*sin(d + e*x)*c**2 - 4*cos(d + e*x)*a*c + sin(d + e*x)**2*a**2*e*x + sin(d + e*x)**2*c**2*e*x + 4*sin(d + e*x)*a**2 + 2*a**2*e*x))/e