Integrand size = 19, antiderivative size = 57 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=a^2 x+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{d}-\frac {a^2 \sin ^3(c+d x)}{3 d} \] Output:
a^2*x+2*a^2*sin(d*x+c)/d+a^2*cos(d*x+c)*sin(d*x+c)/d-1/3*a^2*sin(d*x+c)^3/ d
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.72 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {a^2 (12 d x+21 \sin (c+d x)+6 \sin (2 (c+d x))+\sin (3 (c+d x)))}{12 d} \] Input:
Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^2,x]
Output:
(a^2*(12*d*x + 21*Sin[c + d*x] + 6*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/( 12*d)
Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.32, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3230, 3042, 3123}
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.
\(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2dx\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle \frac {2}{3} \int (\cos (c+d x) a+a)^2dx+\frac {\sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2dx+\frac {\sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
\(\Big \downarrow \) 3123 |
\(\displaystyle \frac {2}{3} \left (\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3 a^2 x}{2}\right )+\frac {\sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
Input:
Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^2,x]
Output:
((a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (2*((3*a^2*x)/2 + (2*a^2*Sin [c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)))/3
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(2*a^2 + b^ 2)*(x/2), x] + (-Simp[2*a*b*(Cos[c + d*x]/d), x] - Simp[b^2*Cos[c + d*x]*(S in[c + d*x]/(2*d)), x]) /; FreeQ[{a, b, c, d}, x]
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[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 3.51 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(\frac {a^{2} \left (12 d x +21 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )+6 \sin \left (2 d x +2 c \right )\right )}{12 d}\) | \(42\) |
risch | \(a^{2} x +\frac {7 a^{2} \sin \left (d x +c \right )}{4 d}+\frac {a^{2} \sin \left (3 d x +3 c \right )}{12 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{2 d}\) | \(55\) |
derivativedivides | \(\frac {\frac {a^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+2 a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} \sin \left (d x +c \right )}{d}\) | \(64\) |
default | \(\frac {\frac {a^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+2 a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} \sin \left (d x +c \right )}{d}\) | \(64\) |
parts | \(\frac {a^{2} \sin \left (d x +c \right )}{d}+\frac {a^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}+\frac {2 a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(69\) |
norman | \(\frac {a^{2} x +a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {6 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {16 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+3 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(128\) |
orering | \(x \cos \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right )^{2}-\frac {49 \left (-d \sin \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right )^{2}-2 \cos \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right ) d \sin \left (d x +c \right ) a \right )}{36 d^{2}}+\frac {49 x \left (-d^{2} \cos \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right )^{2}+4 d^{2} \sin \left (d x +c \right )^{2} \left (a +a \cos \left (d x +c \right )\right ) a +2 \cos \left (d x +c \right ) d^{2} \sin \left (d x +c \right )^{2} a^{2}-2 \cos \left (d x +c \right )^{2} \left (a +a \cos \left (d x +c \right )\right ) d^{2} a \right )}{36 d^{2}}-\frac {7 \left (d^{3} \sin \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right )^{2}+14 d^{3} \cos \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right ) \sin \left (d x +c \right ) a -6 \sin \left (d x +c \right )^{3} a^{2} d^{3}+6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} a^{2} d^{3}\right )}{18 d^{4}}+\frac {7 x \left (d^{4} \cos \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right )^{2}-16 d^{4} \sin \left (d x +c \right )^{2} \left (a +a \cos \left (d x +c \right )\right ) a -44 d^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}+14 d^{4} \cos \left (d x +c \right )^{2} \left (a +a \cos \left (d x +c \right )\right ) a +6 d^{4} \cos \left (d x +c \right )^{3} a^{2}\right )}{18 d^{4}}-\frac {-d^{5} \sin \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right )^{2}-62 d^{5} \cos \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right ) \sin \left (d x +c \right ) a +60 \sin \left (d x +c \right )^{3} a^{2} d^{5}-120 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} a^{2} d^{5}}{36 d^{6}}+\frac {x \left (-d^{6} \cos \left (d x +c \right ) \left (a +a \cos \left (d x +c \right )\right )^{2}+64 d^{6} \sin \left (d x +c \right )^{2} \left (a +a \cos \left (d x +c \right )\right ) a +482 d^{6} \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-62 d^{6} \cos \left (d x +c \right )^{2} \left (a +a \cos \left (d x +c \right )\right ) a -120 d^{6} \cos \left (d x +c \right )^{3} a^{2}\right )}{36 d^{6}}\) | \(595\) |
Input:
int(cos(d*x+c)*(a+a*cos(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/12*a^2*(12*d*x+21*sin(d*x+c)+sin(3*d*x+3*c)+6*sin(2*d*x+2*c))/d
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {3 \, a^{2} d x + {\left (a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} \cos \left (d x + c\right ) + 5 \, a^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^2,x, algorithm="fricas")
Output:
1/3*(3*a^2*d*x + (a^2*cos(d*x + c)^2 + 3*a^2*cos(d*x + c) + 5*a^2)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (51) = 102\).
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.88 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=\begin {cases} a^{2} x \sin ^{2}{\left (c + d x \right )} + a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {a^{2} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right )^{2} \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))**2,x)
Output:
Piecewise((a**2*x*sin(c + d*x)**2 + a**2*x*cos(c + d*x)**2 + 2*a**2*sin(c + d*x)**3/(3*d) + a**2*sin(c + d*x)*cos(c + d*x)**2/d + a**2*sin(c + d*x)* cos(c + d*x)/d + a**2*sin(c + d*x)/d, Ne(d, 0)), (x*(a*cos(c) + a)**2*cos( c), True))
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=-\frac {2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 6 \, a^{2} \sin \left (d x + c\right )}{6 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^2,x, algorithm="maxima")
Output:
-1/6*(2*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^2 - 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^2 - 6*a^2*sin(d*x + c))/d
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=a^{2} x + \frac {a^{2} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {7 \, a^{2} \sin \left (d x + c\right )}{4 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^2,x, algorithm="giac")
Output:
a^2*x + 1/12*a^2*sin(3*d*x + 3*c)/d + 1/2*a^2*sin(2*d*x + 2*c)/d + 7/4*a^2 *sin(d*x + c)/d
Time = 44.39 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=a^2\,x+\frac {5\,a^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \] Input:
int(cos(c + d*x)*(a + a*cos(c + d*x))^2,x)
Output:
a^2*x + (5*a^2*sin(c + d*x))/(3*d) + (a^2*cos(c + d*x)^2*sin(c + d*x))/(3* d) + (a^2*cos(c + d*x)*sin(c + d*x))/d
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {a^{2} \left (3 \cos \left (d x +c \right ) \sin \left (d x +c \right )-\sin \left (d x +c \right )^{3}+6 \sin \left (d x +c \right )+3 d x \right )}{3 d} \] Input:
int(cos(d*x+c)*(a+a*cos(d*x+c))^2,x)
Output:
(a**2*(3*cos(c + d*x)*sin(c + d*x) - sin(c + d*x)**3 + 6*sin(c + d*x) + 3* d*x))/(3*d)