\(\int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx\) [324]

Optimal result

Integrand size = 20, antiderivative size = 91 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=\frac {1}{2} \left (2 a^2+b^2+c^2\right ) x-\frac {3 a c \cos (d+e x)}{2 e}+\frac {3 a b \sin (d+e x)}{2 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{2 e} \] Output:

1/2*(2*a^2+b^2+c^2)*x-3/2*a*c*cos(e*x+d)/e+3/2*a*b*sin(e*x+d)/e-1/2*(c*cos 
(e*x+d)-b*sin(e*x+d))*(a+b*cos(e*x+d)+c*sin(e*x+d))/e
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.85 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=\frac {2 \left (2 a^2+b^2+c^2\right ) (d+e x)-8 a c \cos (d+e x)-2 b c \cos (2 (d+e x))+8 a b \sin (d+e x)+\left (b^2-c^2\right ) \sin (2 (d+e x))}{4 e} \] Input:

Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2,x]
 

Output:

(2*(2*a^2 + b^2 + c^2)*(d + e*x) - 8*a*c*Cos[d + e*x] - 2*b*c*Cos[2*(d + e 
*x)] + 8*a*b*Sin[d + e*x] + (b^2 - c^2)*Sin[2*(d + e*x)])/(4*e)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

\(\displaystyle \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \cos (d+e x)+c \sin (d+e x))^2dx\)

\(\Big \downarrow \) 3599

\(\displaystyle \frac {1}{2} \int \left (2 a^2+3 b \cos (d+e x) a+3 c \sin (d+e x) a+b^2+c^2\right )dx-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{2 e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (x \left (2 a^2+b^2+c^2\right )+\frac {3 a b \sin (d+e x)}{e}-\frac {3 a c \cos (d+e x)}{e}\right )-\frac {(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{2 e}\)

Input:

Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2,x]
 

Output:

-1/2*((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e* 
x]))/e + ((2*a^2 + b^2 + c^2)*x - (3*a*c*Cos[d + e*x])/e + (3*a*b*Sin[d + 
e*x])/e)/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3599

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]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97

method	result	size
parallelrisch	\(\frac {\left (b^{2}-c^{2}\right ) \sin \left (2 e x +2 d \right )-2 b c \cos \left (2 e x +2 d \right )-8 a c \cos \left (e x +d \right )+8 a b \sin \left (e x +d \right )+2 c^{2} e x +\left (8 a +2 b \right ) c +4 \left (a^{2}+\frac {b^{2}}{2}\right ) x e}{4 e}\)	\(88\)
risch	\(a^{2} x +\frac {b^{2} x}{2}+\frac {x \,c^{2}}{2}-\frac {2 a c \cos \left (e x +d \right )}{e}+\frac {2 a b \sin \left (e x +d \right )}{e}-\frac {b c \cos \left (2 e x +2 d \right )}{2 e}+\frac {\sin \left (2 e x +2 d \right ) b^{2}}{4 e}-\frac {\sin \left (2 e x +2 d \right ) c^{2}}{4 e}\)	\(95\)
derivativedivides	\(\frac {b^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-c b \cos \left (e x +d \right )^{2}+c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+2 a b \sin \left (e x +d \right )-2 a c \cos \left (e x +d \right )+a^{2} \left (e x +d \right )}{e}\)	\(99\)
default	\(\frac {b^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-c b \cos \left (e x +d \right )^{2}+c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+2 a b \sin \left (e x +d \right )-2 a c \cos \left (e x +d \right )+a^{2} \left (e x +d \right )}{e}\)	\(99\)
parts	\(a^{2} x +\frac {2 b \left (\frac {c \sin \left (e x +d \right )^{2}}{2}+a \sin \left (e x +d \right )\right )}{e}+\frac {b^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}-\frac {2 a c \cos \left (e x +d \right )}{e}\)	\(104\)
norman	\(\frac {\left (a^{2}+\frac {b^{2}}{2}+\frac {c^{2}}{2}\right ) x +\left (a^{2}+\frac {b^{2}}{2}+\frac {c^{2}}{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+\frac {4 a c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{e}+\frac {\left (4 b a -b^{2}+c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{e}+\frac {\left (4 b a +b^{2}-c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e}+\left (2 a^{2}+b^{2}+c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\frac {2 \left (2 a c +2 c b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{2}}\)	\(183\)
orering	\(x \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{2}-\frac {5 \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right ) \left (-b e \sin \left (e x +d \right )+c e \cos \left (e x +d \right )\right )}{2 e^{2}}+\frac {5 x \left (2 \left (-b e \sin \left (e x +d \right )+c e \cos \left (e x +d \right )\right )^{2}+2 \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right ) \left (-b \,e^{2} \cos \left (e x +d \right )-c \,e^{2} \sin \left (e x +d \right )\right )\right )}{4 e^{2}}-\frac {6 \left (-b e \sin \left (e x +d \right )+c e \cos \left (e x +d \right )\right ) \left (-b \,e^{2} \cos \left (e x +d \right )-c \,e^{2} \sin \left (e x +d \right )\right )+2 \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right ) \left (b \,e^{3} \sin \left (e x +d \right )-c \,e^{3} \cos \left (e x +d \right )\right )}{4 e^{4}}+\frac {x \left (6 \left (-b \,e^{2} \cos \left (e x +d \right )-c \,e^{2} \sin \left (e x +d \right )\right )^{2}+8 \left (-b e \sin \left (e x +d \right )+c e \cos \left (e x +d \right )\right ) \left (b \,e^{3} \sin \left (e x +d \right )-c \,e^{3} \cos \left (e x +d \right )\right )+2 \left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right ) \left (b \,e^{4} \cos \left (e x +d \right )+c \,e^{4} \sin \left (e x +d \right )\right )\right )}{4 e^{4}}\)	\(365\)

Input:

int((a+b*cos(e*x+d)+c*sin(e*x+d))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4*((b^2-c^2)*sin(2*e*x+2*d)-2*b*c*cos(2*e*x+2*d)-8*a*c*cos(e*x+d)+8*a*b* 
sin(e*x+d)+2*c^2*e*x+(8*a+2*b)*c+4*(a^2+1/2*b^2)*x*e)/e
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=-\frac {2 \, b c \cos \left (e x + d\right )^{2} - {\left (2 \, a^{2} + b^{2} + c^{2}\right )} e x + 4 \, a c \cos \left (e x + d\right ) - {\left (4 \, a b + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{2 \, e} \] Input:

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))^2,x, algorithm="fricas")
 

Output:

-1/2*(2*b*c*cos(e*x + d)^2 - (2*a^2 + b^2 + c^2)*e*x + 4*a*c*cos(e*x + d) 
- (4*a*b + (b^2 - c^2)*cos(e*x + d))*sin(e*x + d))/e
 

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.78 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=\begin {cases} a^{2} x + \frac {2 a b \sin {\left (d + e x \right )}}{e} - \frac {2 a c \cos {\left (d + e x \right )}}{e} + \frac {b^{2} x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {b^{2} x \cos ^{2}{\left (d + e x \right )}}{2} + \frac {b^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} - \frac {b c \cos ^{2}{\left (d + e x \right )}}{e} + \frac {c^{2} x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {c^{2} x \cos ^{2}{\left (d + e x \right )}}{2} - \frac {c^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} & \text {for}\: e \neq 0 \\x \left (a + b \cos {\left (d \right )} + c \sin {\left (d \right )}\right )^{2} & \text {otherwise} \end {cases} \] Input:

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))**2,x)
 

Output:

Piecewise((a**2*x + 2*a*b*sin(d + e*x)/e - 2*a*c*cos(d + e*x)/e + b**2*x*s 
in(d + e*x)**2/2 + b**2*x*cos(d + e*x)**2/2 + b**2*sin(d + e*x)*cos(d + e* 
x)/(2*e) - b*c*cos(d + e*x)**2/e + 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)/(2*e), Ne(e, 0)), (x*(a + b* 
cos(d) + c*sin(d))**2, True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.10 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=a^{2} x - \frac {b c \cos \left (e x + d\right )^{2}}{e} + \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{4 \, e} + \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{4 \, e} - 2 \, a {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {b \sin \left (e x + d\right )}{e}\right )} \] Input:

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))^2,x, algorithm="maxima")
 

Output:

a^2*x - b*c*cos(e*x + d)^2/e + 1/4*(2*e*x + 2*d + sin(2*e*x + 2*d))*b^2/e 
+ 1/4*(2*e*x + 2*d - sin(2*e*x + 2*d))*c^2/e - 2*a*(c*cos(e*x + d)/e - b*s 
in(e*x + d)/e)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.89 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=\frac {1}{2} \, {\left (2 \, a^{2} + b^{2} + c^{2}\right )} x - \frac {b c \cos \left (2 \, e x + 2 \, d\right )}{2 \, e} - \frac {2 \, a c \cos \left (e x + d\right )}{e} + \frac {2 \, a b \sin \left (e x + d\right )}{e} + \frac {{\left (b^{2} - c^{2}\right )} \sin \left (2 \, e x + 2 \, d\right )}{4 \, e} \] Input:

integrate((a+b*cos(e*x+d)+c*sin(e*x+d))^2,x, algorithm="giac")
 

Output:

1/2*(2*a^2 + b^2 + c^2)*x - 1/2*b*c*cos(2*e*x + 2*d)/e - 2*a*c*cos(e*x + d 
)/e + 2*a*b*sin(e*x + d)/e + 1/4*(b^2 - c^2)*sin(2*e*x + 2*d)/e
 

Mupad [B] (verification not implemented)

Time = 16.73 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.37 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=\frac {x\,\left (2\,a^2+b^2+c^2\right )}{2}-\frac {\left (b^2-4\,a\,b-c^2\right )\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3+\left (4\,a\,c-4\,b\,c\right )\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+\left (-b^2-4\,a\,b+c^2\right )\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+4\,a\,c}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \] Input:

int((a + b*cos(d + e*x) + c*sin(d + e*x))^2,x)
 

Output:

(x*(2*a^2 + b^2 + c^2))/2 - (4*a*c + tan(d/2 + (e*x)/2)^2*(4*a*c - 4*b*c) 
- tan(d/2 + (e*x)/2)*(4*a*b + b^2 - c^2) - tan(d/2 + (e*x)/2)^3*(4*a*b - b 
^2 + c^2))/(e*(2*tan(d/2 + (e*x)/2)^2 + tan(d/2 + (e*x)/2)^4 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.47 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx=\frac {\cos \left (e x +d \right )^{2} b^{2} e x -2 \cos \left (e x +d \right )^{2} b c +\cos \left (e x +d \right )^{2} c^{2} e x +\cos \left (e x +d \right ) \sin \left (e x +d \right ) b^{2}-\cos \left (e x +d \right ) \sin \left (e x +d \right ) c^{2}-4 \cos \left (e x +d \right ) a c +\sin \left (e x +d \right )^{2} b^{2} e x +\sin \left (e x +d \right )^{2} c^{2} e x +4 \sin \left (e x +d \right ) a b +2 a^{2} e x}{2 e} \] Input:

int((a+b*cos(e*x+d)+c*sin(e*x+d))^2,x)
 

Output:

(cos(d + e*x)**2*b**2*e*x - 2*cos(d + e*x)**2*b*c + cos(d + e*x)**2*c**2*e 
*x + cos(d + e*x)*sin(d + e*x)*b**2 - cos(d + e*x)*sin(d + e*x)*c**2 - 4*c 
os(d + e*x)*a*c + sin(d + e*x)**2*b**2*e*x + sin(d + e*x)**2*c**2*e*x + 4* 
sin(d + e*x)*a*b + 2*a**2*e*x)/(2*e)