Integrand size = 33, antiderivative size = 146 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {(7 A+2 C) \text {arctanh}(\sin (c+d x))}{2 a^2 d}-\frac {4 (4 A+C) \tan (c+d x)}{3 a^2 d}+\frac {(7 A+2 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {2 (4 A+C) \sec (c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A+C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2} \] Output:
1/2*(7*A+2*C)*arctanh(sin(d*x+c))/a^2/d-4/3*(4*A+C)*tan(d*x+c)/a^2/d+1/2*( 7*A+2*C)*sec(d*x+c)*tan(d*x+c)/a^2/d-2/3*(4*A+C)*sec(d*x+c)*tan(d*x+c)/a^2 /d/(1+cos(d*x+c))-1/3*(A+C)*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^2
Leaf count is larger than twice the leaf count of optimal. \(484\) vs. \(2(146)=292\).
Time = 5.10 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.32 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {96 (7 A+2 C) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (-2 (7 A+10 C) \sin \left (\frac {d x}{2}\right )+(97 A+22 C) \sin \left (\frac {3 d x}{2}\right )-126 A \sin \left (c-\frac {d x}{2}\right )-36 C \sin \left (c-\frac {d x}{2}\right )+42 A \sin \left (c+\frac {d x}{2}\right )+36 C \sin \left (c+\frac {d x}{2}\right )-98 A \sin \left (2 c+\frac {d x}{2}\right )-20 C \sin \left (2 c+\frac {d x}{2}\right )-3 A \sin \left (c+\frac {3 d x}{2}\right )-18 C \sin \left (c+\frac {3 d x}{2}\right )+37 A \sin \left (2 c+\frac {3 d x}{2}\right )+22 C \sin \left (2 c+\frac {3 d x}{2}\right )-63 A \sin \left (3 c+\frac {3 d x}{2}\right )-18 C \sin \left (3 c+\frac {3 d x}{2}\right )+75 A \sin \left (c+\frac {5 d x}{2}\right )+18 C \sin \left (c+\frac {5 d x}{2}\right )+15 A \sin \left (2 c+\frac {5 d x}{2}\right )-6 C \sin \left (2 c+\frac {5 d x}{2}\right )+39 A \sin \left (3 c+\frac {5 d x}{2}\right )+18 C \sin \left (3 c+\frac {5 d x}{2}\right )-21 A \sin \left (4 c+\frac {5 d x}{2}\right )-6 C \sin \left (4 c+\frac {5 d x}{2}\right )+32 A \sin \left (2 c+\frac {7 d x}{2}\right )+8 C \sin \left (2 c+\frac {7 d x}{2}\right )+12 A \sin \left (3 c+\frac {7 d x}{2}\right )+20 A \sin \left (4 c+\frac {7 d x}{2}\right )+8 C \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{48 a^2 d (1+\cos (c+d x))^2} \] Input:
Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x ]
Output:
-1/48*(96*(7*A + 2*C)*Cos[(c + d*x)/2]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Se c[c/2]*Sec[c]*Sec[c + d*x]^2*(-2*(7*A + 10*C)*Sin[(d*x)/2] + (97*A + 22*C) *Sin[(3*d*x)/2] - 126*A*Sin[c - (d*x)/2] - 36*C*Sin[c - (d*x)/2] + 42*A*Si n[c + (d*x)/2] + 36*C*Sin[c + (d*x)/2] - 98*A*Sin[2*c + (d*x)/2] - 20*C*Si n[2*c + (d*x)/2] - 3*A*Sin[c + (3*d*x)/2] - 18*C*Sin[c + (3*d*x)/2] + 37*A *Sin[2*c + (3*d*x)/2] + 22*C*Sin[2*c + (3*d*x)/2] - 63*A*Sin[3*c + (3*d*x) /2] - 18*C*Sin[3*c + (3*d*x)/2] + 75*A*Sin[c + (5*d*x)/2] + 18*C*Sin[c + ( 5*d*x)/2] + 15*A*Sin[2*c + (5*d*x)/2] - 6*C*Sin[2*c + (5*d*x)/2] + 39*A*Si n[3*c + (5*d*x)/2] + 18*C*Sin[3*c + (5*d*x)/2] - 21*A*Sin[4*c + (5*d*x)/2] - 6*C*Sin[4*c + (5*d*x)/2] + 32*A*Sin[2*c + (7*d*x)/2] + 8*C*Sin[2*c + (7 *d*x)/2] + 12*A*Sin[3*c + (7*d*x)/2] + 20*A*Sin[4*c + (7*d*x)/2] + 8*C*Sin [4*c + (7*d*x)/2]))/(a^2*d*(1 + Cos[c + d*x])^2)
Time = 0.93 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 3521, 3042, 3457, 3042, 3227, 3042, 4254, 24, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int \frac {(a (5 A+2 C)-3 a A \cos (c+d x)) \sec ^3(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (5 A+2 C)-3 a A \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \left (3 a^2 (7 A+2 C)-4 a^2 (4 A+C) \cos (c+d x)\right ) \sec ^3(c+d x)dx}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 a^2 (7 A+2 C)-4 a^2 (4 A+C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {3 a^2 (7 A+2 C) \int \sec ^3(c+d x)dx-4 a^2 (4 A+C) \int \sec ^2(c+d x)dx}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 a^2 (7 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-4 a^2 (4 A+C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {4 a^2 (4 A+C) \int 1d(-\tan (c+d x))}{d}+3 a^2 (7 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {3 a^2 (7 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 a^2 (4 A+C) \tan (c+d x)}{d}}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {3 a^2 (7 A+2 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a^2 (4 A+C) \tan (c+d x)}{d}}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 a^2 (7 A+2 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a^2 (4 A+C) \tan (c+d x)}{d}}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {3 a^2 (7 A+2 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a^2 (4 A+C) \tan (c+d x)}{d}}{a^2}-\frac {2 (4 A+C) \tan (c+d x) \sec (c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
Input:
Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]
Output:
-1/3*((A + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])^2) + ((-2 *(4*A + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(1 + Cos[c + d*x])) + ((-4*a^2*(4 *A + C)*Tan[c + d*x])/d + 3*a^2*(7*A + 2*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/a^2)/(3*a^2)
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[a*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x] + Simp[1/(b*(b*c - a*d)*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2) + C*(b* c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.50 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.07
| method | result | size |
| parallelrisch | \(\frac {-7 \left (\cos \left (2 d x +2 c \right )+1\right ) \left (A +\frac {2 C}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+7 \left (\cos \left (2 d x +2 c \right )+1\right ) \left (A +\frac {2 C}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-10 \left (\frac {\left (\frac {43 A}{10}+C \right ) \cos \left (2 d x +2 c \right )}{6}+\frac {\left (4 A +C \right ) \cos \left (3 d x +3 c \right )}{15}+\left (A +\frac {C}{5}\right ) \cos \left (d x +c \right )+\frac {37 A}{60}+\frac {C}{6}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2} \left (\cos \left (2 d x +2 c \right )+1\right )}\) | \(156\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (-7 A -2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{2 d \,a^{2}}\) | \(165\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (-7 A -2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{2 d \,a^{2}}\) | \(165\) |
| norman | \(\frac {\frac {\left (14 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}-\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 a d}-\frac {\left (7 A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}-\frac {\left (7 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{2 a d}-\frac {\left (13 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {\left (16 A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2} a}-\frac {\left (7 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{2} d}+\frac {\left (7 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{2} d}\) | \(236\) |
| risch | \(-\frac {i \left (21 A \,{\mathrm e}^{6 i \left (d x +c \right )}+6 C \,{\mathrm e}^{6 i \left (d x +c \right )}+63 A \,{\mathrm e}^{5 i \left (d x +c \right )}+18 C \,{\mathrm e}^{5 i \left (d x +c \right )}+98 A \,{\mathrm e}^{4 i \left (d x +c \right )}+20 C \,{\mathrm e}^{4 i \left (d x +c \right )}+126 A \,{\mathrm e}^{3 i \left (d x +c \right )}+36 C \,{\mathrm e}^{3 i \left (d x +c \right )}+97 A \,{\mathrm e}^{2 i \left (d x +c \right )}+22 C \,{\mathrm e}^{2 i \left (d x +c \right )}+75 A \,{\mathrm e}^{i \left (d x +c \right )}+18 C \,{\mathrm e}^{i \left (d x +c \right )}+32 A +8 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {7 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{2} d}+\frac {7 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{2} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{2} d}\) | \(275\) |
Input:
int((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^2,x,method=_RETURNVER BOSE)
Output:
1/2*(-7*(cos(2*d*x+2*c)+1)*(A+2/7*C)*ln(tan(1/2*d*x+1/2*c)-1)+7*(cos(2*d*x +2*c)+1)*(A+2/7*C)*ln(tan(1/2*d*x+1/2*c)+1)-10*(1/6*(43/10*A+C)*cos(2*d*x+ 2*c)+1/15*(4*A+C)*cos(3*d*x+3*c)+(A+1/5*C)*cos(d*x+c)+37/60*A+1/6*C)*sec(1 /2*d*x+1/2*c)^2*tan(1/2*d*x+1/2*c))/d/a^2/(cos(2*d*x+2*c)+1)
Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.52 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {3 \, {\left ({\left (7 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (7 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (7 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (7 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (8 \, {\left (4 \, A + C\right )} \cos \left (d x + c\right )^{3} + {\left (43 \, A + 10 \, C\right )} \cos \left (d x + c\right )^{2} + 6 \, A \cos \left (d x + c\right ) - 3 \, A\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2}\right )}} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^2,x, algorithm= "fricas")
Output:
1/12*(3*((7*A + 2*C)*cos(d*x + c)^4 + 2*(7*A + 2*C)*cos(d*x + c)^3 + (7*A + 2*C)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 3*((7*A + 2*C)*cos(d*x + c) ^4 + 2*(7*A + 2*C)*cos(d*x + c)^3 + (7*A + 2*C)*cos(d*x + c)^2)*log(-sin(d *x + c) + 1) - 2*(8*(4*A + C)*cos(d*x + c)^3 + (43*A + 10*C)*cos(d*x + c)^ 2 + 6*A*cos(d*x + c) - 3*A)*sin(d*x + c))/(a^2*d*cos(d*x + c)^4 + 2*a^2*d* cos(d*x + c)^3 + a^2*d*cos(d*x + c)^2)
\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\int \frac {A \sec ^{3}{\left (c + d x \right )}}{\cos ^{2}{\left (c + d x \right )} + 2 \cos {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{\cos ^{2}{\left (c + d x \right )} + 2 \cos {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+a*cos(d*x+c))**2,x)
Output:
(Integral(A*sec(c + d*x)**3/(cos(c + d*x)**2 + 2*cos(c + d*x) + 1), x) + I ntegral(C*cos(c + d*x)**2*sec(c + d*x)**3/(cos(c + d*x)**2 + 2*cos(c + d*x ) + 1), x))/a**2
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (136) = 272\).
Time = 0.05 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.97 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {A {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} + C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )}}{6 \, d} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^2,x, algorithm= "maxima")
Output:
-1/6*(A*(6*(3*sin(d*x + c)/(cos(d*x + c) + 1) - 5*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^2 - 2*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^2*sin(d* x + c)^4/(cos(d*x + c) + 1)^4) + (21*sin(d*x + c)/(cos(d*x + c) + 1) + sin (d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 21*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^2 + 21*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^2) + C*((9*s in(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 6*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^2 + 6*log(sin(d*x + c)/(cos (d*x + c) + 1) - 1)/a^2))/d
Time = 0.36 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.17 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (7 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (7 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (5 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{2}} - \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^2,x, algorithm= "giac")
Output:
1/6*(3*(7*A + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^2 - 3*(7*A + 2*C)* log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^2 + 6*(5*A*tan(1/2*d*x + 1/2*c)^3 - 3 *A*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*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 + 21*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 = 0.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {7\,A}{2}+C\right )}{a^2\,d}-\frac {3\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-5\,A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A+C\right )}{2\,a^2}+\frac {2\,A}{a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A+C\right )}{6\,a^2\,d} \] Input:
int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(a + a*cos(c + d*x))^2),x)
Output:
(2*atanh(tan(c/2 + (d*x)/2))*((7*A)/2 + C))/(a^2*d) - (3*A*tan(c/2 + (d*x) /2) - 5*A*tan(c/2 + (d*x)/2)^3)/(d*(a^2*tan(c/2 + (d*x)/2)^4 - 2*a^2*tan(c /2 + (d*x)/2)^2 + a^2)) - (tan(c/2 + (d*x)/2)*((3*(A + C))/(2*a^2) + (2*A) /a^2))/d - (tan(c/2 + (d*x)/2)^3*(A + C))/(6*a^2*d)
Time = 0.16 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.83 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {-21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} c +42 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} c -21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a -6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) c +21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} c -42 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} c +21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a +6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) c -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} c -19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} c +71 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} c -39 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) c}{6 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^2,x)
Output:
( - 21*log(tan((c + d*x)/2) - 1)*tan((c + d*x)/2)**4*a - 6*log(tan((c + d* x)/2) - 1)*tan((c + d*x)/2)**4*c + 42*log(tan((c + d*x)/2) - 1)*tan((c + d *x)/2)**2*a + 12*log(tan((c + d*x)/2) - 1)*tan((c + d*x)/2)**2*c - 21*log( tan((c + d*x)/2) - 1)*a - 6*log(tan((c + d*x)/2) - 1)*c + 21*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**4*a + 6*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**4*c - 42*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**2*a - 12*log (tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**2*c + 21*log(tan((c + d*x)/2) + 1 )*a + 6*log(tan((c + d*x)/2) + 1)*c - tan((c + d*x)/2)**7*a - tan((c + d*x )/2)**7*c - 19*tan((c + d*x)/2)**5*a - 7*tan((c + d*x)/2)**5*c + 71*tan((c + d*x)/2)**3*a + 17*tan((c + d*x)/2)**3*c - 39*tan((c + d*x)/2)*a - 9*tan ((c + d*x)/2)*c)/(6*a**2*d*(tan((c + d*x)/2)**4 - 2*tan((c + d*x)/2)**2 + 1))