Integrand size = 33, antiderivative size = 223 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {(2 A+21 C) x}{2 a^4}-\frac {32 (5 A+54 C) \sin (c+d x)}{105 a^4 d}+\frac {(2 A+21 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(10 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {16 (5 A+54 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3} \]
1/2*(2*A+21*C)*x/a^4-32/105*(5*A+54*C)*sin(d*x+c)/a^4/d+1/2*(2*A+21*C)*cos (d*x+c)*sin(d*x+c)/a^4/d-1/105*(10*A+129*C)*cos(d*x+c)^3*sin(d*x+c)/a^4/d/ (1+cos(d*x+c))^2-16/105*(5*A+54*C)*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+cos(d* x+c))-1/7*(A+C)*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c))^4-2/5*C*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. \(513\) vs. \(2(223)=446\).
Time = 5.73 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.30 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (14700 (2 A+21 C) d x \cos \left (\frac {d x}{2}\right )+14700 (2 A+21 C) d x \cos \left (c+\frac {d x}{2}\right )+17640 A d x \cos \left (c+\frac {3 d x}{2}\right )+185220 C d x \cos \left (c+\frac {3 d x}{2}\right )+17640 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+185220 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+5880 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+5880 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+61740 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+840 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+840 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+8820 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-79520 A \sin \left (\frac {d x}{2}\right )-539490 C \sin \left (\frac {d x}{2}\right )+66080 A \sin \left (c+\frac {d x}{2}\right )+386190 C \sin \left (c+\frac {d x}{2}\right )-57120 A \sin \left (c+\frac {3 d x}{2}\right )-422478 C \sin \left (c+\frac {3 d x}{2}\right )+30240 A \sin \left (2 c+\frac {3 d x}{2}\right )+132930 C \sin \left (2 c+\frac {3 d x}{2}\right )-22400 A \sin \left (2 c+\frac {5 d x}{2}\right )-181461 C \sin \left (2 c+\frac {5 d x}{2}\right )+6720 A \sin \left (3 c+\frac {5 d x}{2}\right )+3675 C \sin \left (3 c+\frac {5 d x}{2}\right )-4160 A \sin \left (3 c+\frac {7 d x}{2}\right )-36003 C \sin \left (3 c+\frac {7 d x}{2}\right )-9555 C \sin \left (4 c+\frac {7 d x}{2}\right )-945 C \sin \left (4 c+\frac {9 d x}{2}\right )-945 C \sin \left (5 c+\frac {9 d x}{2}\right )+105 C \sin \left (5 c+\frac {11 d x}{2}\right )+105 C \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{6720 a^4 d (1+\cos (c+d x))^4} \]
(Cos[(c + d*x)/2]*Sec[c/2]*(14700*(2*A + 21*C)*d*x*Cos[(d*x)/2] + 14700*(2 *A + 21*C)*d*x*Cos[c + (d*x)/2] + 17640*A*d*x*Cos[c + (3*d*x)/2] + 185220* C*d*x*Cos[c + (3*d*x)/2] + 17640*A*d*x*Cos[2*c + (3*d*x)/2] + 185220*C*d*x *Cos[2*c + (3*d*x)/2] + 5880*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*C*d*x*Cos[ 2*c + (5*d*x)/2] + 5880*A*d*x*Cos[3*c + (5*d*x)/2] + 61740*C*d*x*Cos[3*c + (5*d*x)/2] + 840*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*C*d*x*Cos[3*c + (7*d*x )/2] + 840*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*C*d*x*Cos[4*c + (7*d*x)/2] - 79520*A*Sin[(d*x)/2] - 539490*C*Sin[(d*x)/2] + 66080*A*Sin[c + (d*x)/2] + 386190*C*Sin[c + (d*x)/2] - 57120*A*Sin[c + (3*d*x)/2] - 422478*C*Sin[c + (3*d*x)/2] + 30240*A*Sin[2*c + (3*d*x)/2] + 132930*C*Sin[2*c + (3*d*x)/2] - 22400*A*Sin[2*c + (5*d*x)/2] - 181461*C*Sin[2*c + (5*d*x)/2] + 6720*A*Si n[3*c + (5*d*x)/2] + 3675*C*Sin[3*c + (5*d*x)/2] - 4160*A*Sin[3*c + (7*d*x )/2] - 36003*C*Sin[3*c + (7*d*x)/2] - 9555*C*Sin[4*c + (7*d*x)/2] - 945*C* Sin[4*c + (9*d*x)/2] - 945*C*Sin[5*c + (9*d*x)/2] + 105*C*Sin[5*c + (11*d* x)/2] + 105*C*Sin[6*c + (11*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x])^4)
Time = 1.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3521, 3042, 3456, 25, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int \frac {\cos ^4(c+d x) (a (2 A-5 C)+a (2 A+9 C) \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (a (2 A-5 C)+a (2 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int -\frac {\cos ^3(c+d x) \left (56 a^2 C-a^2 (10 A+73 C) \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\cos ^3(c+d x) \left (56 a^2 C-a^2 (10 A+73 C) \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (56 a^2 C-a^2 (10 A+73 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\cos ^2(c+d x) \left (3 a^3 (10 A+129 C)-a^3 (50 A+477 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (3 a^3 (10 A+129 C)-a^3 (50 A+477 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \cos (c+d x) \left (32 a^4 (5 A+54 C)-105 a^4 (2 A+21 C) \cos (c+d x)\right )dx}{a^2}+\frac {16 a^3 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (32 a^4 (5 A+54 C)-105 a^4 (2 A+21 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {16 a^3 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {-\frac {\frac {\frac {16 a^3 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac {\frac {32 a^4 (5 A+54 C) \sin (c+d x)}{d}-\frac {105 a^4 (2 A+21 C) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {105}{2} a^4 x (2 A+21 C)}{a^2}}{3 a^2}+\frac {(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a C \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^4) + (( -14*a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (((10* A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(1 + Cos[c + d*x])^2) + ((16* a^3*(5*A + 54*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ( (-105*a^4*(2*A + 21*C)*x)/2 + (32*a^4*(5*A + 54*C)*Sin[c + d*x])/d - (105* a^4*(2*A + 21*C)*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])
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)]
Time = 2.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.53
| method | result | size |
| parallelrisch | \(\frac {-23360 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {31 A}{73}+\frac {1758 C}{365}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {13 A}{146}+\frac {23619 C}{23360}\right ) \cos \left (3 d x +3 c \right )+\frac {21 C \cos \left (4 d x +4 c \right )}{584}-\frac {21 C \cos \left (5 d x +5 c \right )}{4672}+\left (A +\frac {128643 C}{11680}\right ) \cos \left (d x +c \right )+\frac {47 A}{73}+\frac {20871 C}{2920}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+26880 \left (\frac {21 C}{2}+A \right ) x d}{26880 a^{4} d}\) | \(118\) |
| derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-72 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+8 \left (2 A +21 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(181\) |
| default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-72 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+8 \left (2 A +21 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(181\) |
| risch | \(\frac {x A}{a^{4}}+\frac {21 C x}{2 a^{4}}-\frac {i C \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i C \,{\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {2 i C \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i C \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (420 A \,{\mathrm e}^{6 i \left (d x +c \right )}+2100 C \,{\mathrm e}^{6 i \left (d x +c \right )}+1890 A \,{\mathrm e}^{5 i \left (d x +c \right )}+11025 C \,{\mathrm e}^{5 i \left (d x +c \right )}+4130 A \,{\mathrm e}^{4 i \left (d x +c \right )}+25515 C \,{\mathrm e}^{4 i \left (d x +c \right )}+4970 A \,{\mathrm e}^{3 i \left (d x +c \right )}+32340 C \,{\mathrm e}^{3 i \left (d x +c \right )}+3570 A \,{\mathrm e}^{2 i \left (d x +c \right )}+23688 C \,{\mathrm e}^{2 i \left (d x +c \right )}+1400 A \,{\mathrm e}^{i \left (d x +c \right )}+9471 C \,{\mathrm e}^{i \left (d x +c \right )}+260 A +1653 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(264\) |
1/26880*(-23360*tan(1/2*d*x+1/2*c)*((31/73*A+1758/365*C)*cos(2*d*x+2*c)+(1 3/146*A+23619/23360*C)*cos(3*d*x+3*c)+21/584*C*cos(4*d*x+4*c)-21/4672*C*co s(5*d*x+5*c)+(A+128643/11680*C)*cos(d*x+c)+47/73*A+20871/2920*C)*sec(1/2*d *x+1/2*c)^6+26880*(21/2*C+A)*x*d)/a^4/d
Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, {\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (2 \, A + 21 \, C\right )} d x + {\left (105 \, C \cos \left (d x + c\right )^{5} - 420 \, C \cos \left (d x + c\right )^{4} - 4 \, {\left (130 \, A + 1509 \, C\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (310 \, A + 3411 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (1070 \, A + 11619 \, C\right )} \cos \left (d x + c\right ) - 320 \, A - 3456 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
1/210*(105*(2*A + 21*C)*d*x*cos(d*x + c)^4 + 420*(2*A + 21*C)*d*x*cos(d*x + c)^3 + 630*(2*A + 21*C)*d*x*cos(d*x + c)^2 + 420*(2*A + 21*C)*d*x*cos(d* x + c) + 105*(2*A + 21*C)*d*x + (105*C*cos(d*x + c)^5 - 420*C*cos(d*x + c) ^4 - 4*(130*A + 1509*C)*cos(d*x + c)^3 - 4*(310*A + 3411*C)*cos(d*x + c)^2 - (1070*A + 11619*C)*cos(d*x + c) - 320*A - 3456*C)*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. 1086 vs. \(2 (209) = 418\).
Time = 7.60 (sec) , antiderivative size = 1086, normalized size of antiderivative = 4.87 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\text {Too large to display} \]
Piecewise((840*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) + 1680*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) + 840*A*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*A*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) - 75*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) + 190*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) - 910*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) - 2765*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) - 1575*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 + 84 0*a**4*d) + 8820*C*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*C*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*C*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*C*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* C*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan...
Time = 0.36 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.43 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=-\frac {3 \, C {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + 5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]
-1/840*(3*C*(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) + 5*A*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 336*arctan(sin (d*x + c)/(cos(d*x + c) + 1))/a^4))/d
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (d x + c\right )} {\left (2 \, A + 21 \, C\right )}}{a^{4}} - \frac {840 \, {\left (9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 189 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1365 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11655 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
1/840*(420*(d*x + c)*(2*A + 21*C)/a^4 - 840*(9*C*tan(1/2*d*x + 1/2*c)^3 + 7*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^4) + (15*A*a^2 4*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 105*A*a^24*t an(1/2*d*x + 1/2*c)^5 - 189*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 385*A*a^24*tan (1/2*d*x + 1/2*c)^3 + 1365*C*a^24*tan(1/2*d*x + 1/2*c)^3 - 1575*A*a^24*tan (1/2*d*x + 1/2*c) - 11655*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 1.08 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {x\,\left (2\,A+21\,C\right )}{2\,a^4}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{4\,a^4}-\frac {3\,\left (A-15\,C\right )}{8\,a^4}+\frac {3\,\left (2\,A+6\,C\right )}{4\,a^4}-\frac {4\,A-20\,C}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A+C\right )}{40\,a^4}+\frac {2\,A+6\,C}{40\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{4\,a^4}-\frac {A-15\,C}{24\,a^4}+\frac {2\,A+6\,C}{8\,a^4}\right )}{d}-\frac {9\,C\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+7\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \]
(x*(2*A + 21*C))/(2*a^4) - (tan(c/2 + (d*x)/2)*((5*(A + C))/(4*a^4) - (3*( A - 15*C))/(8*a^4) + (3*(2*A + 6*C))/(4*a^4) - (4*A - 20*C)/(8*a^4)))/d - (tan(c/2 + (d*x)/2)^5*((3*(A + C))/(40*a^4) + (2*A + 6*C)/(40*a^4)))/d + ( tan(c/2 + (d*x)/2)^3*((A + C)/(4*a^4) - (A - 15*C)/(24*a^4) + (2*A + 6*C)/ (8*a^4)))/d - (7*C*tan(c/2 + (d*x)/2) + 9*C*tan(c/2 + (d*x)/2)^3)/(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)^7*(A + C))/(56*a^4*d)