Integrand size = 15, antiderivative size = 104 \[ \int (a+a \cos (x))^4 (A+B \sec (x)) \, dx=\frac {1}{8} a^4 (35 A+48 B) x+a^4 B \text {arctanh}(\sin (x))+\frac {5}{8} a^4 (7 A+8 B) \sin (x)+\frac {1}{4} a A (a+a \cos (x))^3 \sin (x)+\frac {1}{12} (7 A+4 B) \left (a^2+a^2 \cos (x)\right )^2 \sin (x)+\frac {1}{24} (35 A+32 B) \left (a^4+a^4 \cos (x)\right ) \sin (x) \]
1/8*a^4*(35*A+48*B)*x+a^4*B*arctanh(sin(x))+5/8*a^4*(7*A+8*B)*sin(x)+1/4*a *A*(a+a*cos(x))^3*sin(x)+1/12*(7*A+4*B)*(a^2+a^2*cos(x))^2*sin(x)+1/24*(35 *A+32*B)*(a^4+a^4*cos(x))*sin(x)
Time = 0.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int (a+a \cos (x))^4 (A+B \sec (x)) \, dx=\frac {1}{96} a^4 \left (420 A x+576 B x-96 B \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+96 B \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+24 (28 A+27 B) \sin (x)+24 (7 A+4 B) \sin (2 x)+32 A \sin (3 x)+8 B \sin (3 x)+3 A \sin (4 x)\right ) \]
(a^4*(420*A*x + 576*B*x - 96*B*Log[Cos[x/2] - Sin[x/2]] + 96*B*Log[Cos[x/2 ] + Sin[x/2]] + 24*(28*A + 27*B)*Sin[x] + 24*(7*A + 4*B)*Sin[2*x] + 32*A*S in[3*x] + 8*B*Sin[3*x] + 3*A*Sin[4*x]))/96
Time = 1.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.11, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.133, Rules used = {3042, 3307, 3042, 3455, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3214, 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.
| \(\displaystyle \int (a \cos (x)+a)^4 (A+B \sec (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \csc \left (x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3307 |
| \(\displaystyle \int \sec (x) (a \cos (x)+a)^4 (A \cos (x)+B)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (x+\frac {\pi }{2}\right )+B\right )}{\sin \left (x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{4} \int (\cos (x) a+a)^3 (4 a B+a (7 A+4 B) \cos (x)) \sec (x)dx+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (\sin \left (x+\frac {\pi }{2}\right ) a+a\right )^3 \left (4 a B+a (7 A+4 B) \sin \left (x+\frac {\pi }{2}\right )\right )}{\sin \left (x+\frac {\pi }{2}\right )}dx+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (\cos (x) a+a)^2 \left (12 B a^2+(35 A+32 B) \cos (x) a^2\right ) \sec (x)dx+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (\sin \left (x+\frac {\pi }{2}\right ) a+a\right )^2 \left (12 B a^2+(35 A+32 B) \sin \left (x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (x+\frac {\pi }{2}\right )}dx+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int 3 (\cos (x) a+a) \left (8 B a^3+5 (7 A+8 B) \cos (x) a^3\right ) \sec (x)dx+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int (\cos (x) a+a) \left (8 B a^3+5 (7 A+8 B) \cos (x) a^3\right ) \sec (x)dx+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {\left (\sin \left (x+\frac {\pi }{2}\right ) a+a\right ) \left (8 B a^3+5 (7 A+8 B) \sin \left (x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (x+\frac {\pi }{2}\right )}dx+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \left (5 (7 A+8 B) \cos ^2(x) a^4+8 B a^4+\left (8 B a^4+5 (7 A+8 B) a^4\right ) \cos (x)\right ) \sec (x)dx+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {5 (7 A+8 B) \sin \left (x+\frac {\pi }{2}\right )^2 a^4+8 B a^4+\left (8 B a^4+5 (7 A+8 B) a^4\right ) \sin \left (x+\frac {\pi }{2}\right )}{\sin \left (x+\frac {\pi }{2}\right )}dx+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\int \left (8 B a^4+(35 A+48 B) \cos (x) a^4\right ) \sec (x)dx+5 a^4 (7 A+8 B) \sin (x)\right )+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\int \frac {8 B a^4+(35 A+48 B) \sin \left (x+\frac {\pi }{2}\right ) a^4}{\sin \left (x+\frac {\pi }{2}\right )}dx+5 a^4 (7 A+8 B) \sin (x)\right )+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^4 B \int \sec (x)dx+a^4 x (35 A+48 B)+5 a^4 (7 A+8 B) \sin (x)\right )+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^4 B \int \csc \left (x+\frac {\pi }{2}\right )dx+a^4 x (35 A+48 B)+5 a^4 (7 A+8 B) \sin (x)\right )+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (a^4 x (35 A+48 B)+5 a^4 (7 A+8 B) \sin (x)+8 a^4 B \text {arctanh}(\sin (x))\right )+\frac {1}{2} (35 A+32 B) \sin (x) \left (a^4 \cos (x)+a^4\right )\right )+\frac {1}{3} (7 A+4 B) \sin (x) \left (a^2 \cos (x)+a^2\right )^2\right )+\frac {1}{4} a A \sin (x) (a \cos (x)+a)^3\) |
(a*A*(a + a*Cos[x])^3*Sin[x])/4 + (((7*A + 4*B)*(a^2 + a^2*Cos[x])^2*Sin[x ])/3 + (((35*A + 32*B)*(a^4 + a^4*Cos[x])*Sin[x])/2 + (3*(a^4*(35*A + 48*B )*x + 8*a^4*B*ArcTanh[Sin[x]] + 5*a^4*(7*A + 8*B)*Sin[x]))/2)/3)/4
3.2.88.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])^n/Sin[e + f*x]^n), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ [n]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, 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[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*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] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.86 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.77
| method | result | size |
| parallelrisch | \(\frac {a^{4} \left (-32 B \ln \left (-\cot \left (x \right )+\csc \left (x \right )-1\right )+32 B \ln \left (\csc \left (x \right )-\cot \left (x \right )+1\right )+8 \left (7 A +4 B \right ) \sin \left (2 x \right )+\frac {8 \left (4 A +B \right ) \sin \left (3 x \right )}{3}+A \sin \left (4 x \right )+8 \left (28 A +27 B \right ) \sin \left (x \right )+140 x \left (A +\frac {48 B}{35}\right )\right )}{32}\) | \(80\) |
| default | \(a^{4} A \left (\frac {\left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{4}+\frac {3 x}{8}\right )+\frac {a^{4} B \left (2+\cos \left (x \right )^{2}\right ) \sin \left (x \right )}{3}+\frac {4 a^{4} A \left (2+\cos \left (x \right )^{2}\right ) \sin \left (x \right )}{3}+4 a^{4} B \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+6 a^{4} A \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+6 a^{4} B \sin \left (x \right )+4 a^{4} A \sin \left (x \right )+4 a^{4} B x +a^{4} A x +a^{4} B \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )\) | \(124\) |
| parts | \(a^{4} A \left (\frac {\left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{4}+\frac {3 x}{8}\right )+\frac {a^{4} B \left (2+\cos \left (x \right )^{2}\right ) \sin \left (x \right )}{3}+\frac {4 a^{4} A \left (2+\cos \left (x \right )^{2}\right ) \sin \left (x \right )}{3}+4 a^{4} B \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+6 a^{4} A \left (\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+6 a^{4} B \sin \left (x \right )+4 a^{4} A \sin \left (x \right )+4 a^{4} B x +a^{4} A x +a^{4} B \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )\) | \(124\) |
| risch | \(\frac {35 a^{4} A x}{8}+6 a^{4} B x -\frac {7 i A \,{\mathrm e}^{i x} a^{4}}{2}-\frac {27 i B \,{\mathrm e}^{i x} a^{4}}{8}+\frac {7 i A \,{\mathrm e}^{-i x} a^{4}}{2}+\frac {27 i B \,{\mathrm e}^{-i x} a^{4}}{8}+a^{4} B \ln \left (i+{\mathrm e}^{i x}\right )-a^{4} B \ln \left ({\mathrm e}^{i x}-i\right )+\frac {a^{4} A \sin \left (4 x \right )}{32}+\frac {A \sin \left (3 x \right ) a^{4}}{3}+\frac {B \sin \left (3 x \right ) a^{4}}{12}+\frac {7 A \sin \left (2 x \right ) a^{4}}{4}+B \sin \left (2 x \right ) a^{4}\) | \(142\) |
| norman | \(\frac {\left (\frac {35}{4} a^{4} A +10 a^{4} B \right ) \tan \left (\frac {x}{2}\right )^{7}+\left (\frac {35}{8} a^{4} A +6 a^{4} B \right ) x +\left (\frac {93}{4} a^{4} A +18 a^{4} B \right ) \tan \left (\frac {x}{2}\right )+\left (\frac {385}{12} a^{4} A +\frac {106}{3} a^{4} B \right ) \tan \left (\frac {x}{2}\right )^{5}+\left (\frac {511}{12} a^{4} A +\frac {130}{3} a^{4} B \right ) \tan \left (\frac {x}{2}\right )^{3}+\left (\frac {35}{2} a^{4} A +24 a^{4} B \right ) x \tan \left (\frac {x}{2}\right )^{2}+\left (\frac {35}{2} a^{4} A +24 a^{4} B \right ) x \tan \left (\frac {x}{2}\right )^{6}+\left (\frac {35}{8} a^{4} A +6 a^{4} B \right ) x \tan \left (\frac {x}{2}\right )^{8}+\left (\frac {105}{4} a^{4} A +36 a^{4} B \right ) x \tan \left (\frac {x}{2}\right )^{4}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4}}+a^{4} B \ln \left (\tan \left (\frac {x}{2}\right )+1\right )-a^{4} B \ln \left (\tan \left (\frac {x}{2}\right )-1\right )\) | \(216\) |
1/32*a^4*(-32*B*ln(-cot(x)+csc(x)-1)+32*B*ln(csc(x)-cot(x)+1)+8*(7*A+4*B)* sin(2*x)+8/3*(4*A+B)*sin(3*x)+A*sin(4*x)+8*(28*A+27*B)*sin(x)+140*x*(A+48/ 35*B))
Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int (a+a \cos (x))^4 (A+B \sec (x)) \, dx=\frac {1}{8} \, {\left (35 \, A + 48 \, B\right )} a^{4} x + \frac {1}{2} \, B a^{4} \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, B a^{4} \log \left (-\sin \left (x\right ) + 1\right ) + \frac {1}{24} \, {\left (6 \, A a^{4} \cos \left (x\right )^{3} + 8 \, {\left (4 \, A + B\right )} a^{4} \cos \left (x\right )^{2} + 3 \, {\left (27 \, A + 16 \, B\right )} a^{4} \cos \left (x\right ) + 160 \, {\left (A + B\right )} a^{4}\right )} \sin \left (x\right ) \]
1/8*(35*A + 48*B)*a^4*x + 1/2*B*a^4*log(sin(x) + 1) - 1/2*B*a^4*log(-sin(x ) + 1) + 1/24*(6*A*a^4*cos(x)^3 + 8*(4*A + B)*a^4*cos(x)^2 + 3*(27*A + 16* B)*a^4*cos(x) + 160*(A + B)*a^4)*sin(x)
Time = 6.66 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.12 \[ \int (a+a \cos (x))^4 (A+B \sec (x)) \, dx=\frac {35 A a^{4} x}{8} - \frac {4 A a^{4} \sin ^{3}{\left (x \right )}}{3} + 8 A a^{4} \sin {\left (x \right )} + \frac {7 A a^{4} \sin {\left (2 x \right )}}{4} + \frac {A a^{4} \sin {\left (4 x \right )}}{32} + 6 B a^{4} x + B a^{4} \log {\left (\tan {\left (x \right )} + \sec {\left (x \right )} \right )} - \frac {B a^{4} \sin ^{3}{\left (x \right )}}{3} + 2 B a^{4} \sin {\left (x \right )} \cos {\left (x \right )} + 7 B a^{4} \sin {\left (x \right )} \]
35*A*a**4*x/8 - 4*A*a**4*sin(x)**3/3 + 8*A*a**4*sin(x) + 7*A*a**4*sin(2*x) /4 + A*a**4*sin(4*x)/32 + 6*B*a**4*x + B*a**4*log(tan(x) + sec(x)) - B*a** 4*sin(x)**3/3 + 2*B*a**4*sin(x)*cos(x) + 7*B*a**4*sin(x)
Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.13 \[ \int (a+a \cos (x))^4 (A+B \sec (x)) \, dx=-\frac {4}{3} \, {\left (\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )\right )} A a^{4} - \frac {1}{3} \, {\left (\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )\right )} B a^{4} + \frac {1}{32} \, A a^{4} {\left (12 \, x + \sin \left (4 \, x\right ) + 8 \, \sin \left (2 \, x\right )\right )} + \frac {3}{2} \, A a^{4} {\left (2 \, x + \sin \left (2 \, x\right )\right )} + B a^{4} {\left (2 \, x + \sin \left (2 \, x\right )\right )} + A a^{4} x + 4 \, B a^{4} x + B a^{4} \log \left (\sec \left (x\right ) + \tan \left (x\right )\right ) + 4 \, A a^{4} \sin \left (x\right ) + 6 \, B a^{4} \sin \left (x\right ) \]
-4/3*(sin(x)^3 - 3*sin(x))*A*a^4 - 1/3*(sin(x)^3 - 3*sin(x))*B*a^4 + 1/32* A*a^4*(12*x + sin(4*x) + 8*sin(2*x)) + 3/2*A*a^4*(2*x + sin(2*x)) + B*a^4* (2*x + sin(2*x)) + A*a^4*x + 4*B*a^4*x + B*a^4*log(sec(x) + tan(x)) + 4*A* a^4*sin(x) + 6*B*a^4*sin(x)
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.43 \[ \int (a+a \cos (x))^4 (A+B \sec (x)) \, dx=B a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) - B a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) + \frac {1}{8} \, {\left (35 \, A a^{4} + 48 \, B a^{4}\right )} x + \frac {105 \, A a^{4} \tan \left (\frac {1}{2} \, x\right )^{7} + 120 \, B a^{4} \tan \left (\frac {1}{2} \, x\right )^{7} + 385 \, A a^{4} \tan \left (\frac {1}{2} \, x\right )^{5} + 424 \, B a^{4} \tan \left (\frac {1}{2} \, x\right )^{5} + 511 \, A a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} + 520 \, B a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} + 279 \, A a^{4} \tan \left (\frac {1}{2} \, x\right ) + 216 \, B a^{4} \tan \left (\frac {1}{2} \, x\right )}{12 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{4}} \]
B*a^4*log(abs(tan(1/2*x) + 1)) - B*a^4*log(abs(tan(1/2*x) - 1)) + 1/8*(35* A*a^4 + 48*B*a^4)*x + 1/12*(105*A*a^4*tan(1/2*x)^7 + 120*B*a^4*tan(1/2*x)^ 7 + 385*A*a^4*tan(1/2*x)^5 + 424*B*a^4*tan(1/2*x)^5 + 511*A*a^4*tan(1/2*x) ^3 + 520*B*a^4*tan(1/2*x)^3 + 279*A*a^4*tan(1/2*x) + 216*B*a^4*tan(1/2*x)) /(tan(1/2*x)^2 + 1)^4
Time = 26.38 (sec) , antiderivative size = 460, normalized size of antiderivative = 4.42 \[ \int (a+a \cos (x))^4 (A+B \sec (x)) \, dx=\frac {\left (\frac {35\,A\,a^4}{4}+10\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7+\left (\frac {385\,A\,a^4}{12}+\frac {106\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\left (\frac {511\,A\,a^4}{12}+\frac {130\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+18\,B\,a^4\right )\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\mathrm {tan}\left (\frac {x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}+\frac {a^4\,\mathrm {atan}\left (\frac {42875\,A^3\,a^{12}\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,\left (\frac {42875\,A^3\,a^{12}}{8}+22050\,A^2\,B\,a^{12}+30520\,A\,B^2\,a^{12}+14208\,B^3\,a^{12}\right )}+\frac {14208\,B^3\,a^{12}\,\mathrm {tan}\left (\frac {x}{2}\right )}{\frac {42875\,A^3\,a^{12}}{8}+22050\,A^2\,B\,a^{12}+30520\,A\,B^2\,a^{12}+14208\,B^3\,a^{12}}+\frac {30520\,A\,B^2\,a^{12}\,\mathrm {tan}\left (\frac {x}{2}\right )}{\frac {42875\,A^3\,a^{12}}{8}+22050\,A^2\,B\,a^{12}+30520\,A\,B^2\,a^{12}+14208\,B^3\,a^{12}}+\frac {22050\,A^2\,B\,a^{12}\,\mathrm {tan}\left (\frac {x}{2}\right )}{\frac {42875\,A^3\,a^{12}}{8}+22050\,A^2\,B\,a^{12}+30520\,A\,B^2\,a^{12}+14208\,B^3\,a^{12}}\right )\,\left (35\,A+48\,B\right )}{4}+2\,B\,a^4\,\mathrm {atanh}\left (\frac {2368\,B^3\,a^{12}\,\mathrm {tan}\left (\frac {x}{2}\right )}{1225\,A^2\,B\,a^{12}+3360\,A\,B^2\,a^{12}+2368\,B^3\,a^{12}}+\frac {3360\,A\,B^2\,a^{12}\,\mathrm {tan}\left (\frac {x}{2}\right )}{1225\,A^2\,B\,a^{12}+3360\,A\,B^2\,a^{12}+2368\,B^3\,a^{12}}+\frac {1225\,A^2\,B\,a^{12}\,\mathrm {tan}\left (\frac {x}{2}\right )}{1225\,A^2\,B\,a^{12}+3360\,A\,B^2\,a^{12}+2368\,B^3\,a^{12}}\right ) \]
(tan(x/2)^7*((35*A*a^4)/4 + 10*B*a^4) + tan(x/2)^5*((385*A*a^4)/12 + (106* B*a^4)/3) + tan(x/2)^3*((511*A*a^4)/12 + (130*B*a^4)/3) + tan(x/2)*((93*A* a^4)/4 + 18*B*a^4))/(4*tan(x/2)^2 + 6*tan(x/2)^4 + 4*tan(x/2)^6 + tan(x/2) ^8 + 1) + (a^4*atan((42875*A^3*a^12*tan(x/2))/(8*((42875*A^3*a^12)/8 + 142 08*B^3*a^12 + 30520*A*B^2*a^12 + 22050*A^2*B*a^12)) + (14208*B^3*a^12*tan( x/2))/((42875*A^3*a^12)/8 + 14208*B^3*a^12 + 30520*A*B^2*a^12 + 22050*A^2* B*a^12) + (30520*A*B^2*a^12*tan(x/2))/((42875*A^3*a^12)/8 + 14208*B^3*a^12 + 30520*A*B^2*a^12 + 22050*A^2*B*a^12) + (22050*A^2*B*a^12*tan(x/2))/((42 875*A^3*a^12)/8 + 14208*B^3*a^12 + 30520*A*B^2*a^12 + 22050*A^2*B*a^12))*( 35*A + 48*B))/4 + 2*B*a^4*atanh((2368*B^3*a^12*tan(x/2))/(2368*B^3*a^12 + 3360*A*B^2*a^12 + 1225*A^2*B*a^12) + (3360*A*B^2*a^12*tan(x/2))/(2368*B^3* a^12 + 3360*A*B^2*a^12 + 1225*A^2*B*a^12) + (1225*A^2*B*a^12*tan(x/2))/(23 68*B^3*a^12 + 3360*A*B^2*a^12 + 1225*A^2*B*a^12))