Integrand size = 31, antiderivative size = 229 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {(8 A-21 B) x}{2 a^4}+\frac {8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {4 (83 A-216 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))}+\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(A-2 B) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3} \]
-1/2*(8*A-21*B)*x/a^4+8/105*(83*A-216*B)*sin(d*x+c)/a^4/d-1/2*(8*A-21*B)*c os(d*x+c)*sin(d*x+c)/a^4/d+1/105*(52*A-129*B)*cos(d*x+c)^3*sin(d*x+c)/a^4/ d/(1+cos(d*x+c))^2+4/105*(83*A-216*B)*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+cos (d*x+c))+1/7*(A-B)*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c))^4+1/5*(A-2*B )*cos(d*x+c)^4*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3
Leaf count is larger than twice the leaf count of optimal. \(555\) vs. \(2(229)=458\).
Time = 5.03 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.42 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-14700 (8 A-21 B) d x \cos \left (\frac {d x}{2}\right )-14700 (8 A-21 B) d x \cos \left (c+\frac {d x}{2}\right )-70560 A d x \cos \left (c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (c+\frac {3 d x}{2}\right )-70560 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-23520 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-23520 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-3360 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (3 c+\frac {7 d x}{2}\right )-3360 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (4 c+\frac {7 d x}{2}\right )+243320 A \sin \left (\frac {d x}{2}\right )-539490 B \sin \left (\frac {d x}{2}\right )-184520 A \sin \left (c+\frac {d x}{2}\right )+386190 B \sin \left (c+\frac {d x}{2}\right )+184464 A \sin \left (c+\frac {3 d x}{2}\right )-422478 B \sin \left (c+\frac {3 d x}{2}\right )-72240 A \sin \left (2 c+\frac {3 d x}{2}\right )+132930 B \sin \left (2 c+\frac {3 d x}{2}\right )+77168 A \sin \left (2 c+\frac {5 d x}{2}\right )-181461 B \sin \left (2 c+\frac {5 d x}{2}\right )-8400 A \sin \left (3 c+\frac {5 d x}{2}\right )+3675 B \sin \left (3 c+\frac {5 d x}{2}\right )+15164 A \sin \left (3 c+\frac {7 d x}{2}\right )-36003 B \sin \left (3 c+\frac {7 d x}{2}\right )+2940 A \sin \left (4 c+\frac {7 d x}{2}\right )-9555 B \sin \left (4 c+\frac {7 d x}{2}\right )+420 A \sin \left (4 c+\frac {9 d x}{2}\right )-945 B \sin \left (4 c+\frac {9 d x}{2}\right )+420 A \sin \left (5 c+\frac {9 d x}{2}\right )-945 B \sin \left (5 c+\frac {9 d x}{2}\right )+105 B \sin \left (5 c+\frac {11 d x}{2}\right )+105 B \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{6720 a^4 d (1+\cos (c+d x))^4} \]
(Cos[(c + d*x)/2]*Sec[c/2]*(-14700*(8*A - 21*B)*d*x*Cos[(d*x)/2] - 14700*( 8*A - 21*B)*d*x*Cos[c + (d*x)/2] - 70560*A*d*x*Cos[c + (3*d*x)/2] + 185220 *B*d*x*Cos[c + (3*d*x)/2] - 70560*A*d*x*Cos[2*c + (3*d*x)/2] + 185220*B*d* x*Cos[2*c + (3*d*x)/2] - 23520*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*B*d*x*Co s[2*c + (5*d*x)/2] - 23520*A*d*x*Cos[3*c + (5*d*x)/2] + 61740*B*d*x*Cos[3* c + (5*d*x)/2] - 3360*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*B*d*x*Cos[3*c + (7 *d*x)/2] - 3360*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*B*d*x*Cos[4*c + (7*d*x)/ 2] + 243320*A*Sin[(d*x)/2] - 539490*B*Sin[(d*x)/2] - 184520*A*Sin[c + (d*x )/2] + 386190*B*Sin[c + (d*x)/2] + 184464*A*Sin[c + (3*d*x)/2] - 422478*B* Sin[c + (3*d*x)/2] - 72240*A*Sin[2*c + (3*d*x)/2] + 132930*B*Sin[2*c + (3* d*x)/2] + 77168*A*Sin[2*c + (5*d*x)/2] - 181461*B*Sin[2*c + (5*d*x)/2] - 8 400*A*Sin[3*c + (5*d*x)/2] + 3675*B*Sin[3*c + (5*d*x)/2] + 15164*A*Sin[3*c + (7*d*x)/2] - 36003*B*Sin[3*c + (7*d*x)/2] + 2940*A*Sin[4*c + (7*d*x)/2] - 9555*B*Sin[4*c + (7*d*x)/2] + 420*A*Sin[4*c + (9*d*x)/2] - 945*B*Sin[4* c + (9*d*x)/2] + 420*A*Sin[5*c + (9*d*x)/2] - 945*B*Sin[5*c + (9*d*x)/2] + 105*B*Sin[5*c + (11*d*x)/2] + 105*B*Sin[6*c + (11*d*x)/2]))/(6720*a^4*d*( 1 + Cos[c + d*x])^4)
Time = 1.27 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3042, 3456, 3042, 3456, 3042, 3456, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\int \frac {\cos ^4(c+d x) (5 a (A-B)-a (2 A-9 B) \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (5 a (A-B)-a (2 A-9 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^3(c+d x) \left (28 a^2 (A-2 B)-a^2 (24 A-73 B) \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {7 a (A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (28 a^2 (A-2 B)-a^2 (24 A-73 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {7 a (A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cos ^2(c+d x) \left (3 a^3 (52 A-129 B)-a^3 (176 A-477 B) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {7 a (A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (3 a^3 (52 A-129 B)-a^3 (176 A-477 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {7 a (A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \cos (c+d x) \left (8 a^4 (83 A-216 B)-105 a^4 (8 A-21 B) \cos (c+d x)\right )dx}{a^2}+\frac {4 a^3 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {7 a (A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 a^4 (83 A-216 B)-105 a^4 (8 A-21 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {4 a^3 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {7 a (A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a^3 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac {\frac {8 a^4 (83 A-216 B) \sin (c+d x)}{d}-\frac {105 a^4 (8 A-21 B) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {105}{2} a^4 x (8 A-21 B)}{a^2}}{3 a^2}+\frac {(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {7 a (A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
((A - B)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((7*a *(A - 2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((( 52*A - 129*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(1 + Cos[c + d*x])^2) + (( 4*a^3*(83*A - 216*B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((-105*a^4*(8*A - 21*B)*x)/2 + (8*a^4*(83*A - 216*B)*Sin[c + d*x])/d - (105*a^4*(8*A - 21*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d))/a^2)/(3*a^2))/(5*a ^2))/(7*a^2)
3.1.65.3.1 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])
Time = 1.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.54
| method | result | size |
| parallelrisch | \(\frac {420 \left (\left (\frac {10964 A}{105}-\frac {9376 B}{35}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {2368 A}{105}-\frac {7873 B}{140}\right ) \cos \left (3 d x +3 c \right )+\left (A -2 B \right ) \cos \left (4 d x +4 c \right )+\frac {B \cos \left (5 d x +5 c \right )}{4}+\left (\frac {24992 A}{105}-\frac {42881 B}{70}\right ) \cos \left (d x +c \right )+\frac {16171 A}{105}-\frac {13914 B}{35}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-107520 \left (A -\frac {21 B}{8}\right ) x d}{26880 a^{4} d}\) | \(123\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-A +\frac {9 B}{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-8 \left (8 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(191\) |
| default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-A +\frac {9 B}{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-8 \left (8 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(191\) |
| risch | \(-\frac {4 x A}{a^{4}}+\frac {21 B x}{2 a^{4}}-\frac {i B \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A}{2 a^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B}{a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B}{a^{4} d}+\frac {i B \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i \left (1050 A \,{\mathrm e}^{6 i \left (d x +c \right )}-2100 B \,{\mathrm e}^{6 i \left (d x +c \right )}+5250 A \,{\mathrm e}^{5 i \left (d x +c \right )}-11025 B \,{\mathrm e}^{5 i \left (d x +c \right )}+11900 A \,{\mathrm e}^{4 i \left (d x +c \right )}-25515 B \,{\mathrm e}^{4 i \left (d x +c \right )}+14840 A \,{\mathrm e}^{3 i \left (d x +c \right )}-32340 B \,{\mathrm e}^{3 i \left (d x +c \right )}+10794 A \,{\mathrm e}^{2 i \left (d x +c \right )}-23688 B \,{\mathrm e}^{2 i \left (d x +c \right )}+4298 A \,{\mathrm e}^{i \left (d x +c \right )}-9471 B \,{\mathrm e}^{i \left (d x +c \right )}+764 A -1653 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(303\) |
1/26880*(420*((10964/105*A-9376/35*B)*cos(2*d*x+2*c)+(2368/105*A-7873/140* B)*cos(3*d*x+3*c)+(A-2*B)*cos(4*d*x+4*c)+1/4*B*cos(5*d*x+5*c)+(24992/105*A -42881/70*B)*cos(d*x+c)+16171/105*A-13914/35*B)*tan(1/2*d*x+1/2*c)*sec(1/2 *d*x+1/2*c)^6-107520*(A-21/8*B)*x*d)/a^4/d
Time = 0.32 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.04 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {105 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (8 \, A - 21 \, B\right )} d x - {\left (105 \, B \cos \left (d x + c\right )^{5} + 210 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (592 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (1318 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (4472 \, A - 11619 \, B\right )} \cos \left (d x + c\right ) + 1328 \, A - 3456 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
-1/210*(105*(8*A - 21*B)*d*x*cos(d*x + c)^4 + 420*(8*A - 21*B)*d*x*cos(d*x + c)^3 + 630*(8*A - 21*B)*d*x*cos(d*x + c)^2 + 420*(8*A - 21*B)*d*x*cos(d *x + c) + 105*(8*A - 21*B)*d*x - (105*B*cos(d*x + c)^5 + 210*(A - 2*B)*cos (d*x + c)^4 + 4*(592*A - 1509*B)*cos(d*x + c)^3 + 4*(1318*A - 3411*B)*cos( d*x + c)^2 + (4472*A - 11619*B)*cos(d*x + c) + 1328*A - 3456*B)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^ 2 + 4*a^4*d*cos(d*x + c) + a^4*d)
Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (216) = 432\).
Time = 7.97 (sec) , antiderivative size = 1085, normalized size of antiderivative = 4.74 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\text {Too large to display} \]
Piecewise((-3360*A*d*x*tan(c/2 + d*x/2)**4/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 6720*A*d*x*tan(c/2 + d* x/2)**2/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 3360*A*d*x/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*t an(c/2 + d*x/2)**2 + 840*a**4*d) - 15*A*tan(c/2 + d*x/2)**11/(840*a**4*d*t an(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 117*A *tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 526*A*tan(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 3682*A*tan( c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d* x/2)**2 + 840*a**4*d) + 11165*A*tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 6825*A*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)** 2 + 840*a**4*d) + 8820*B*d*x*tan(c/2 + d*x/2)**4/(840*a**4*d*tan(c/2 + d*x /2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 17640*B*d*x*tan(c /2 + d*x/2)**2/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x /2)**2 + 840*a**4*d) + 8820*B*d*x/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a **4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 15*B*tan(c/2 + d*x/2)**11/(840*a **4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 159*B*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4...
Time = 0.30 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.59 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {3 \, B {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]
-1/840*(3*B*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^3/(co s(d*x + c) + 1)^3)/(a^4 + 2*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4* sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) - 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 5880*arctan(sin (d*x + c)/(cos(d*x + c) + 1))/a^4) - A*(1680*sin(d*x + c)/((a^4 + a^4*sin( d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (5145*sin(d*x + c)/ (cos(d*x + c) + 1) - 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 6720*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4))/d
Time = 0.35 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {420 \, {\left (d x + c\right )} {\left (8 \, A - 21 \, B\right )}}{a^{4}} - \frac {840 \, {\left (2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 189 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11655 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
-1/840*(420*(d*x + c)*(8*A - 21*B)/a^4 - 840*(2*A*tan(1/2*d*x + 1/2*c)^3 - 9*B*tan(1/2*d*x + 1/2*c)^3 + 2*A*tan(1/2*d*x + 1/2*c) - 7*B*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^4) + (15*A*a^24*tan(1/2*d*x + 1 /2*c)^7 - 15*B*a^24*tan(1/2*d*x + 1/2*c)^7 - 147*A*a^24*tan(1/2*d*x + 1/2* c)^5 + 189*B*a^24*tan(1/2*d*x + 1/2*c)^5 + 805*A*a^24*tan(1/2*d*x + 1/2*c) ^3 - 1365*B*a^24*tan(1/2*d*x + 1/2*c)^3 - 5145*A*a^24*tan(1/2*d*x + 1/2*c) + 11655*B*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 0.34 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{4\,a^4}-\frac {5\,B}{2\,a^4}+\frac {3\,\left (4\,A-6\,B\right )}{4\,a^4}+\frac {3\,\left (5\,A-15\,B\right )}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^4}+\frac {4\,A-6\,B}{8\,a^4}+\frac {5\,A-15\,B}{24\,a^4}\right )}{d}-\frac {x\,\left (8\,A-21\,B\right )}{2\,a^4}+\frac {\left (2\,A-9\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A-7\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A-B\right )}{40\,a^4}+\frac {4\,A-6\,B}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d} \]
(tan(c/2 + (d*x)/2)*((5*(A - B))/(4*a^4) - (5*B)/(2*a^4) + (3*(4*A - 6*B)) /(4*a^4) + (3*(5*A - 15*B))/(8*a^4)))/d - (tan(c/2 + (d*x)/2)^3*((A - B)/( 4*a^4) + (4*A - 6*B)/(8*a^4) + (5*A - 15*B)/(24*a^4)))/d - (x*(8*A - 21*B) )/(2*a^4) + (tan(c/2 + (d*x)/2)^3*(2*A - 9*B) + tan(c/2 + (d*x)/2)*(2*A - 7*B))/(d*(2*a^4*tan(c/2 + (d*x)/2)^2 + a^4*tan(c/2 + (d*x)/2)^4 + a^4)) + (tan(c/2 + (d*x)/2)^5*((3*(A - B))/(40*a^4) + (4*A - 6*B)/(40*a^4)))/d - ( tan(c/2 + (d*x)/2)^7*(A - B))/(56*a^4*d)