Integrand size = 41, antiderivative size = 207 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {(2 A-6 B+13 C) x}{2 a^3}-\frac {2 (11 A-36 B+76 C) \sin (c+d x)}{15 a^3 d}+\frac {(2 A-6 B+13 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(A-6 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(11 A-36 B+76 C) \cos ^2(c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )} \]
1/2*(2*A-6*B+13*C)*x/a^3-2/15*(11*A-36*B+76*C)*sin(d*x+c)/a^3/d+1/2*(2*A-6 *B+13*C)*cos(d*x+c)*sin(d*x+c)/a^3/d-1/5*(A-B+C)*cos(d*x+c)^4*sin(d*x+c)/d /(a+a*cos(d*x+c))^3-1/15*(A-6*B+11*C)*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos (d*x+c))^2-1/15*(11*A-36*B+76*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a^3+a^3*cos(d* x+c))
Leaf count is larger than twice the leaf count of optimal. \(565\) vs. \(2(207)=414\).
Time = 3.30 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.73 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (600 (2 A-6 B+13 C) d x \cos \left (\frac {d x}{2}\right )+600 (2 A-6 B+13 C) d x \cos \left (c+\frac {d x}{2}\right )+600 A d x \cos \left (c+\frac {3 d x}{2}\right )-1800 B d x \cos \left (c+\frac {3 d x}{2}\right )+3900 C d x \cos \left (c+\frac {3 d x}{2}\right )+600 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-1800 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+3900 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+120 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-360 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+120 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-360 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+780 C d x \cos \left (3 c+\frac {5 d x}{2}\right )-2960 A \sin \left (\frac {d x}{2}\right )+7020 B \sin \left (\frac {d x}{2}\right )-12760 C \sin \left (\frac {d x}{2}\right )+2160 A \sin \left (c+\frac {d x}{2}\right )-4500 B \sin \left (c+\frac {d x}{2}\right )+7560 C \sin \left (c+\frac {d x}{2}\right )-1840 A \sin \left (c+\frac {3 d x}{2}\right )+4860 B \sin \left (c+\frac {3 d x}{2}\right )-9230 C \sin \left (c+\frac {3 d x}{2}\right )+720 A \sin \left (2 c+\frac {3 d x}{2}\right )-900 B \sin \left (2 c+\frac {3 d x}{2}\right )+930 C \sin \left (2 c+\frac {3 d x}{2}\right )-512 A \sin \left (2 c+\frac {5 d x}{2}\right )+1452 B \sin \left (2 c+\frac {5 d x}{2}\right )-2782 C \sin \left (2 c+\frac {5 d x}{2}\right )+300 B \sin \left (3 c+\frac {5 d x}{2}\right )-750 C \sin \left (3 c+\frac {5 d x}{2}\right )+60 B \sin \left (3 c+\frac {7 d x}{2}\right )-105 C \sin \left (3 c+\frac {7 d x}{2}\right )+60 B \sin \left (4 c+\frac {7 d x}{2}\right )-105 C \sin \left (4 c+\frac {7 d x}{2}\right )+15 C \sin \left (4 c+\frac {9 d x}{2}\right )+15 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{480 a^3 d (1+\cos (c+d x))^3} \]
(Cos[(c + d*x)/2]*Sec[c/2]*(600*(2*A - 6*B + 13*C)*d*x*Cos[(d*x)/2] + 600* (2*A - 6*B + 13*C)*d*x*Cos[c + (d*x)/2] + 600*A*d*x*Cos[c + (3*d*x)/2] - 1 800*B*d*x*Cos[c + (3*d*x)/2] + 3900*C*d*x*Cos[c + (3*d*x)/2] + 600*A*d*x*C os[2*c + (3*d*x)/2] - 1800*B*d*x*Cos[2*c + (3*d*x)/2] + 3900*C*d*x*Cos[2*c + (3*d*x)/2] + 120*A*d*x*Cos[2*c + (5*d*x)/2] - 360*B*d*x*Cos[2*c + (5*d* x)/2] + 780*C*d*x*Cos[2*c + (5*d*x)/2] + 120*A*d*x*Cos[3*c + (5*d*x)/2] - 360*B*d*x*Cos[3*c + (5*d*x)/2] + 780*C*d*x*Cos[3*c + (5*d*x)/2] - 2960*A*S in[(d*x)/2] + 7020*B*Sin[(d*x)/2] - 12760*C*Sin[(d*x)/2] + 2160*A*Sin[c + (d*x)/2] - 4500*B*Sin[c + (d*x)/2] + 7560*C*Sin[c + (d*x)/2] - 1840*A*Sin[ c + (3*d*x)/2] + 4860*B*Sin[c + (3*d*x)/2] - 9230*C*Sin[c + (3*d*x)/2] + 7 20*A*Sin[2*c + (3*d*x)/2] - 900*B*Sin[2*c + (3*d*x)/2] + 930*C*Sin[2*c + ( 3*d*x)/2] - 512*A*Sin[2*c + (5*d*x)/2] + 1452*B*Sin[2*c + (5*d*x)/2] - 278 2*C*Sin[2*c + (5*d*x)/2] + 300*B*Sin[3*c + (5*d*x)/2] - 750*C*Sin[3*c + (5 *d*x)/2] + 60*B*Sin[3*c + (7*d*x)/2] - 105*C*Sin[3*c + (7*d*x)/2] + 60*B*S in[4*c + (7*d*x)/2] - 105*C*Sin[4*c + (7*d*x)/2] + 15*C*Sin[4*c + (9*d*x)/ 2] + 15*C*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)
Time = 1.05 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {3042, 3520, 3042, 3456, 25, 3042, 3456, 3042, 3213}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (a (A+4 B-4 C)+a (2 A-2 B+7 C) \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a (A+4 B-4 C)+a (2 A-2 B+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int -\frac {\cos ^2(c+d x) \left (3 a^2 (A-6 B+11 C)-a^2 (8 A-18 B+43 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {a (A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\cos ^2(c+d x) \left (3 a^2 (A-6 B+11 C)-a^2 (8 A-18 B+43 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {a (A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (3 a^2 (A-6 B+11 C)-a^2 (8 A-18 B+43 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {a (A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {-\frac {\frac {\int \cos (c+d x) \left (2 a^3 (11 A-36 B+76 C)-15 a^3 (2 A-6 B+13 C) \cos (c+d x)\right )dx}{a^2}+\frac {a^2 (11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 a^3 (11 A-36 B+76 C)-15 a^3 (2 A-6 B+13 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {a^2 (11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {-\frac {\frac {a^2 (11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac {\frac {2 a^3 (11 A-36 B+76 C) \sin (c+d x)}{d}-\frac {15 a^3 (2 A-6 B+13 C) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {15}{2} a^3 x (2 A-6 B+13 C)}{a^2}}{3 a^2}-\frac {a (A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
-1/5*((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^3) + (-1/3*(a*(A - 6*B + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^2) - ((a^2*(11*A - 36*B + 76*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((-15*a^3*(2*A - 6*B + 13*C)*x)/2 + (2*a^3*(11*A - 36 *B + 76*C)*Sin[c + d*x])/d - (15*a^3*(2*A - 6*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d))/a^2)/(3*a^2))/(5*a^2)
3.4.56.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
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[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, 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] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(a*A - b*B + 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)) Int[(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)) + B*(b*c*m + a *d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(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, B, 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)]
Time = 2.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.57
| method | result | size |
| parallelrisch | \(\frac {-816 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {16 A}{51}-\frac {39 B}{34}+\frac {116 C}{51}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {5 B}{68}+\frac {15 C}{136}\right ) \cos \left (3 d x +3 c \right )-\frac {5 C \cos \left (4 d x +4 c \right )}{272}+\left (A -\frac {243 B}{68}+\frac {1001 C}{136}\right ) \cos \left (d x +c \right )+\frac {38 A}{51}-\frac {87 B}{34}+\frac {4303 C}{816}\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+960 x \left (\frac {13 C}{2}-3 B +A \right ) d}{960 a^{3} d}\) | \(118\) |
| derivativedivides | \(\frac {-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}-7 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 \left (B -\frac {7 C}{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (-\frac {5 C}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+4 \left (2 A -6 B +13 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(202\) |
| default | \(\frac {-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}-7 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 \left (B -\frac {7 C}{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (-\frac {5 C}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+4 \left (2 A -6 B +13 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(202\) |
| risch | \(\frac {A x}{a^{3}}-\frac {3 B x}{a^{3}}+\frac {13 C x}{2 a^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C}{8 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{3} d}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{3} d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{3} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C}{8 a^{3} d}-\frac {2 i \left (45 A \,{\mathrm e}^{4 i \left (d x +c \right )}-90 B \,{\mathrm e}^{4 i \left (d x +c \right )}+150 C \,{\mathrm e}^{4 i \left (d x +c \right )}+135 A \,{\mathrm e}^{3 i \left (d x +c \right )}-300 B \,{\mathrm e}^{3 i \left (d x +c \right )}+525 C \,{\mathrm e}^{3 i \left (d x +c \right )}+185 A \,{\mathrm e}^{2 i \left (d x +c \right )}-420 B \,{\mathrm e}^{2 i \left (d x +c \right )}+745 C \,{\mathrm e}^{2 i \left (d x +c \right )}+115 A \,{\mathrm e}^{i \left (d x +c \right )}-270 B \,{\mathrm e}^{i \left (d x +c \right )}+485 C \,{\mathrm e}^{i \left (d x +c \right )}+32 A -72 B +127 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(312\) |
| norman | \(\frac {\frac {\left (2 A -6 B +13 C \right ) x}{2 a}+\frac {\left (A -3 B +5 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {\left (A -B +C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {5 \left (2 A -6 B +13 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {5 \left (2 A -6 B +13 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 \left (2 A -6 B +13 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 \left (2 A -6 B +13 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {\left (2 A -6 B +13 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {\left (7 A -27 B +59 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {\left (7 A -25 B +51 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {\left (71 A -225 B +475 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {\left (101 A -345 B +721 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {\left (173 A -549 B +1165 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {\left (953 A -3123 B +6613 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2}}\) | \(396\) |
1/960*(-816*tan(1/2*d*x+1/2*c)*((16/51*A-39/34*B+116/51*C)*cos(2*d*x+2*c)+ (-5/68*B+15/136*C)*cos(3*d*x+3*c)-5/272*C*cos(4*d*x+4*c)+(A-243/68*B+1001/ 136*C)*cos(d*x+c)+38/51*A-87/34*B+4303/816*C)*sec(1/2*d*x+1/2*c)^4+960*x*( 13/2*C-3*B+A)*d)/a^3/d
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {15 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (2 \, A - 6 \, B + 13 \, C\right )} d x + {\left (15 \, C \cos \left (d x + c\right )^{4} + 15 \, {\left (2 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} - {\left (64 \, A - 234 \, B + 479 \, C\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (34 \, A - 114 \, B + 239 \, C\right )} \cos \left (d x + c\right ) - 44 \, A + 144 \, B - 304 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3, x, algorithm="fricas")
1/30*(15*(2*A - 6*B + 13*C)*d*x*cos(d*x + c)^3 + 45*(2*A - 6*B + 13*C)*d*x *cos(d*x + c)^2 + 45*(2*A - 6*B + 13*C)*d*x*cos(d*x + c) + 15*(2*A - 6*B + 13*C)*d*x + (15*C*cos(d*x + c)^4 + 15*(2*B - 3*C)*cos(d*x + c)^3 - (64*A - 234*B + 479*C)*cos(d*x + c)^2 - 3*(34*A - 114*B + 239*C)*cos(d*x + c) - 44*A + 144*B - 304*C)*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d* x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (199) = 398\).
Time = 4.97 (sec) , antiderivative size = 1445, normalized size of antiderivative = 6.98 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\text {Too large to display} \]
Piecewise((60*A*d*x*tan(c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x/2)**4 + 1 20*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 120*A*d*x*tan(c/2 + d*x/2)**2 /(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3 *d) + 60*A*d*x/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2 )**2 + 60*a**3*d) - 3*A*tan(c/2 + d*x/2)**9/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 14*A*tan(c/2 + d*x/2)**7/ (60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3* d) - 68*A*tan(c/2 + d*x/2)**5/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d* tan(c/2 + d*x/2)**2 + 60*a**3*d) - 190*A*tan(c/2 + d*x/2)**3/(60*a**3*d*ta n(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 105*A*ta n(c/2 + d*x/2)/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2 )**2 + 60*a**3*d) - 180*B*d*x*tan(c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x /2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 360*B*d*x*tan(c/2 + d*x/2)**2/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 180*B*d*x/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan( c/2 + d*x/2)**2 + 60*a**3*d) + 3*B*tan(c/2 + d*x/2)**9/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 24*B*tan(c/2 + d*x/2)**7/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 198*B*tan(c/2 + d*x/2)**5/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 600*B*tan(c/2 + d*x/2)**...
Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (195) = 390\).
Time = 0.29 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.99 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {C {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {780 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - 3 \, B {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \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 {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + A {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3, x, algorithm="maxima")
-1/60*(C*(60*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 7*sin(d*x + c)^3/(cos(d* x + c) + 1)^3)/(a^3 + 2*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^3*sin( d*x + c)^4/(cos(d*x + c) + 1)^4) + (465*sin(d*x + c)/(cos(d*x + c) + 1) - 40*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 780*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3) - 3*B*(40*sin (d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (85*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 120*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3) + A*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 20*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 120*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3))/d
Time = 0.32 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\frac {30 \, {\left (d x + c\right )} {\left (2 \, A - 6 \, B + 13 \, C\right )}}{a^{3}} + \frac {60 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 20 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 255 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 465 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3, x, algorithm="giac")
1/60*(30*(d*x + c)*(2*A - 6*B + 13*C)/a^3 + 60*(2*B*tan(1/2*d*x + 1/2*c)^3 - 7*C*tan(1/2*d*x + 1/2*c)^3 + 2*B*tan(1/2*d*x + 1/2*c) - 5*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^3) - (3*A*a^12*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^12*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^12*tan(1/2*d*x + 1/2*c) ^5 - 20*A*a^12*tan(1/2*d*x + 1/2*c)^3 + 30*B*a^12*tan(1/2*d*x + 1/2*c)^3 - 40*C*a^12*tan(1/2*d*x + 1/2*c)^3 + 105*A*a^12*tan(1/2*d*x + 1/2*c) - 255* B*a^12*tan(1/2*d*x + 1/2*c) + 465*C*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d
Time = 1.42 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-3\,B+5\,C}{12\,a^3}+\frac {A-B+C}{4\,a^3}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A-3\,B+5\,C\right )}{4\,a^3}-\frac {2\,A+2\,B-10\,C}{4\,a^3}+\frac {3\,\left (A-B+C\right )}{2\,a^3}\right )}{d}+\frac {\left (2\,B-7\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B-5\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {x\,\left (2\,A-6\,B+13\,C\right )}{2\,a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B+C\right )}{20\,a^3\,d} \]
(tan(c/2 + (d*x)/2)^3*((A - 3*B + 5*C)/(12*a^3) + (A - B + C)/(4*a^3)))/d - (tan(c/2 + (d*x)/2)*((3*(A - 3*B + 5*C))/(4*a^3) - (2*A + 2*B - 10*C)/(4 *a^3) + (3*(A - B + C))/(2*a^3)))/d + (tan(c/2 + (d*x)/2)^3*(2*B - 7*C) + tan(c/2 + (d*x)/2)*(2*B - 5*C))/(d*(2*a^3*tan(c/2 + (d*x)/2)^2 + a^3*tan(c /2 + (d*x)/2)^4 + a^3)) + (x*(2*A - 6*B + 13*C))/(2*a^3) - (tan(c/2 + (d*x )/2)^5*(A - B + C))/(20*a^3*d)