Integrand size = 25, antiderivative size = 66 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {C x}{a^2}+\frac {(A-5 C) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}+\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \] Output:
C*x/a^2+1/3*(A-5*C)*sin(d*x+c)/a^2/d/(1+cos(d*x+c))+1/3*(A+C)*sin(d*x+c)/d /(a+a*cos(d*x+c))^2
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(66)=132\).
Time = 0.89 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.14 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (9 C d x \cos \left (\frac {d x}{2}\right )+9 C d x \cos \left (c+\frac {d x}{2}\right )+3 C d x \cos \left (c+\frac {3 d x}{2}\right )+3 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+6 A \sin \left (\frac {d x}{2}\right )-18 C \sin \left (\frac {d x}{2}\right )+12 C \sin \left (c+\frac {d x}{2}\right )+2 A \sin \left (c+\frac {3 d x}{2}\right )-10 C \sin \left (c+\frac {3 d x}{2}\right )\right )}{24 a^2 d} \] Input:
Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]
Output:
(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*C*d*x*Cos[(d*x)/2] + 9*C*d*x*Cos[c + (d*x) /2] + 3*C*d*x*Cos[c + (3*d*x)/2] + 3*C*d*x*Cos[2*c + (3*d*x)/2] + 6*A*Sin[ (d*x)/2] - 18*C*Sin[(d*x)/2] + 12*C*Sin[c + (d*x)/2] + 2*A*Sin[c + (3*d*x) /2] - 10*C*Sin[c + (3*d*x)/2]))/(24*a^2*d)
Time = 0.42 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3499, 25, 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.
\(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{(a \cos (c+d x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
\(\Big \downarrow \) 3499 |
\(\displaystyle \frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}-\frac {\int -\frac {a (A-2 C)+3 a C \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a (A-2 C)+3 a C \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (A-2 C)+3 a C \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {a (A-5 C) \int \frac {1}{\cos (c+d x) a+a}dx+3 C x}{3 a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a (A-5 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+3 C x}{3 a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {\frac {a (A-5 C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+3 C x}{3 a^2}+\frac {(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
Input:
Int[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]
Output:
((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (3*C*x + (a*(A - 5*C )*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])))/(3*a^2)
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[b*(A + C)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[1/(a^2*(2*m + 1)) Int[(a + b*Sin[e + f* x])^(m + 1)*Simp[a*A*(m + 1) - a*C*m + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.71
method | result | size |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (A +C \right )+\left (3 A -9 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+6 d x C}{6 a^{2} d}\) | \(47\) |
risch | \(\frac {C x}{a^{2}}-\frac {2 i \left (6 C \,{\mathrm e}^{2 i \left (d x +c \right )}-3 A \,{\mathrm e}^{i \left (d x +c \right )}+9 C \,{\mathrm e}^{i \left (d x +c \right )}-A +5 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(73\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(74\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(74\) |
norman | \(\frac {\frac {C x}{a}+\frac {C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}+\frac {2 C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (A -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 a d}+\frac {\left (5 A -7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 a d}+\frac {\left (7 A -17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} a}\) | \(158\) |
Input:
int((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/6*(tan(1/2*d*x+1/2*c)^3*(A+C)+(3*A-9*C)*tan(1/2*d*x+1/2*c)+6*d*x*C)/a^2/ d
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {3 \, C d x \cos \left (d x + c\right )^{2} + 6 \, C d x \cos \left (d x + c\right ) + 3 \, C d x + {\left ({\left (A - 5 \, C\right )} \cos \left (d x + c\right ) + 2 \, A - 4 \, C\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^2,x, algorithm="fricas")
Output:
1/3*(3*C*d*x*cos(d*x + c)^2 + 6*C*d*x*cos(d*x + c) + 3*C*d*x + ((A - 5*C)* cos(d*x + c) + 2*A - 4*C)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*co s(d*x + c) + a^2*d)
Time = 0.64 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.58 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\begin {cases} \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d} + \frac {C x}{a^{2}} + \frac {C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d} - \frac {3 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + C \cos ^{2}{\left (c \right )}\right )}{\left (a \cos {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \] Input:
integrate((A+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**2,x)
Output:
Piecewise((A*tan(c/2 + d*x/2)**3/(6*a**2*d) + A*tan(c/2 + d*x/2)/(2*a**2*d ) + C*x/a**2 + C*tan(c/2 + d*x/2)**3/(6*a**2*d) - 3*C*tan(c/2 + d*x/2)/(2* a**2*d), Ne(d, 0)), (x*(A + C*cos(c)**2)/(a*cos(c) + a)**2, True))
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.80 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} - \frac {A {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^2,x, algorithm="maxima")
Output:
-1/6*(C*((9*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 12*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2) - A*(3*sin( d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2)/d
Time = 0.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.27 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\frac {6 \, {\left (d x + c\right )} C}{a^{2}} + \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \] Input:
integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^2,x, algorithm="giac")
Output:
1/6*(6*(d*x + c)*C/a^2 + (A*a^4*tan(1/2*d*x + 1/2*c)^3 + C*a^4*tan(1/2*d*x + 1/2*c)^3 + 3*A*a^4*tan(1/2*d*x + 1/2*c) - 9*C*a^4*tan(1/2*d*x + 1/2*c)) /a^6)/d
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {3\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+C\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,C\,d\,x}{6\,a^2\,d} \] Input:
int((A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2,x)
Output:
(3*A*tan(c/2 + (d*x)/2) - 9*C*tan(c/2 + (d*x)/2) + A*tan(c/2 + (d*x)/2)^3 + C*tan(c/2 + (d*x)/2)^3 + 6*C*d*x)/(6*a^2*d)
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a +\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} c +3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) c +6 c d x}{6 a^{2} d} \] Input:
int((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^2,x)
Output:
(tan((c + d*x)/2)**3*a + tan((c + d*x)/2)**3*c + 3*tan((c + d*x)/2)*a - 9* tan((c + d*x)/2)*c + 6*c*d*x)/(6*a**2*d)