Integrand size = 25, antiderivative size = 48 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {C x}{a}+\frac {C \sin (c+d x)}{a d}+\frac {(A+C) \sin (c+d x)}{a d (1+\cos (c+d x))} \] Output:
-C*x/a+C*sin(d*x+c)/a/d+(A+C)*sin(d*x+c)/a/d/(1+cos(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(48)=96\).
Time = 0.76 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.25 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-2 C d x \cos \left (\frac {d x}{2}\right )-2 C d x \cos \left (c+\frac {d x}{2}\right )+4 A \sin \left (\frac {d x}{2}\right )+5 C \sin \left (\frac {d x}{2}\right )+C \sin \left (c+\frac {d x}{2}\right )+C \sin \left (c+\frac {3 d x}{2}\right )+C \sin \left (2 c+\frac {3 d x}{2}\right )\right )}{4 a d} \] Input:
Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]
Output:
(Sec[c/2]*Sec[(c + d*x)/2]*(-2*C*d*x*Cos[(d*x)/2] - 2*C*d*x*Cos[c + (d*x)/ 2] + 4*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2] + C*Sin[c + (d*x)/2] + C*Sin[c + (3*d*x)/2] + C*Sin[2*c + (3*d*x)/2]))/(4*a*d)
Time = 0.38 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3503, 3042, 3214, 3042, 3127}
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.
\(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{a \cos (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 3503 |
\(\displaystyle \frac {\int \frac {a A-a C \cos (c+d x)}{\cos (c+d x) a+a}dx}{a}+\frac {C \sin (c+d x)}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a A-a C \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{a}+\frac {C \sin (c+d x)}{a d}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {a (A+C) \int \frac {1}{\cos (c+d x) a+a}dx-C x}{a}+\frac {C \sin (c+d x)}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a (A+C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-C x}{a}+\frac {C \sin (c+d x)}{a d}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {\frac {a (A+C) \sin (c+d x)}{d (a \cos (c+d x)+a)}-C x}{a}+\frac {C \sin (c+d x)}{a d}\) |
Input:
Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]
Output:
(C*Sin[c + d*x])/(a*d) + (-(C*x) + (a*(A + C)*Sin[c + d*x])/(d*(a + a*Cos[ c + d*x])))/a
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^ (m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^ m*Simp[A*b*(m + 2) + b*C*(m + 1) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a , b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77
method | result | size |
parallelrisch | \(\frac {-d x C +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right ) C +A +2 C \right )}{a d}\) | \(37\) |
derivativedivides | \(\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 C \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{d a}\) | \(73\) |
default | \(\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 C \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{d a}\) | \(73\) |
risch | \(-\frac {C x}{a}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a d}+\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {2 i C}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}\) | \(93\) |
norman | \(\frac {\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}+\frac {\left (A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {C x}{a}-\frac {2 C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}+\frac {2 \left (A +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}\) | \(127\) |
Input:
int((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/a/d*(-d*x*C+tan(1/2*d*x+1/2*c)*(cos(d*x+c)*C+A+2*C))
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {C d x \cos \left (d x + c\right ) + C d x - {\left (C \cos \left (d x + c\right ) + A + 2 \, C\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="fricas")
Output:
-(C*d*x*cos(d*x + c) + C*d*x - (C*cos(d*x + c) + A + 2*C)*sin(d*x + c))/(a *d*cos(d*x + c) + a*d)
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (37) = 74\).
Time = 0.55 (sec) , antiderivative size = 202, normalized size of antiderivative = 4.21 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=\begin {cases} \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {C d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {C d x}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {3 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \left (A + C \cos ^{2}{\left (c \right )}\right )}{a \cos {\left (c \right )} + a} & \text {otherwise} \end {cases} \] Input:
integrate((A+C*cos(d*x+c)**2)/(a+a*cos(d*x+c)),x)
Output:
Piecewise((A*tan(c/2 + d*x/2)**3/(a*d*tan(c/2 + d*x/2)**2 + a*d) + A*tan(c /2 + d*x/2)/(a*d*tan(c/2 + d*x/2)**2 + a*d) - C*d*x*tan(c/2 + d*x/2)**2/(a *d*tan(c/2 + d*x/2)**2 + a*d) - C*d*x/(a*d*tan(c/2 + d*x/2)**2 + a*d) + C* tan(c/2 + d*x/2)**3/(a*d*tan(c/2 + d*x/2)**2 + a*d) + 3*C*tan(c/2 + d*x/2) /(a*d*tan(c/2 + d*x/2)**2 + a*d), Ne(d, 0)), (x*(A + C*cos(c)**2)/(a*cos(c ) + a), True))
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (48) = 96\).
Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.44 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {C {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - \frac {A \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="maxima")
Output:
-(C*(2*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - 2*sin(d*x + c)/((a + a* sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) - sin(d*x + c)/(a *(cos(d*x + c) + 1))) - A*sin(d*x + c)/(a*(cos(d*x + c) + 1)))/d
Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.54 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\frac {{\left (d x + c\right )} C}{a} - \frac {A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a}}{d} \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="giac")
Output:
-((d*x + c)*C/a - (A*tan(1/2*d*x + 1/2*c) + C*tan(1/2*d*x + 1/2*c))/a - 2* C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a))/d
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {C\,x}{a}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+C\right )}{a\,d} \] Input:
int((A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x)),x)
Output:
(2*C*tan(c/2 + (d*x)/2))/(d*(a + a*tan(c/2 + (d*x)/2)^2)) - (C*x)/a + (tan (c/2 + (d*x)/2)*(A + C))/(a*d)
Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \frac {A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\sin \left (d x +c \right ) c +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) c -c d x}{a d} \] Input:
int((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x)
Output:
(sin(c + d*x)*c + tan((c + d*x)/2)*a + tan((c + d*x)/2)*c - c*d*x)/(a*d)