Integrand size = 31, antiderivative size = 145 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=b^3 B x+\frac {\left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a \left (2 a^2 A+8 A b^2+9 a b B\right ) \tan (c+d x)}{3 d}+\frac {a^2 (5 A b+3 a B) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {a A (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
b^3*B*x+1/2*(3*A*a^2*b+2*A*b^3+B*a^3+6*B*a*b^2)*arctanh(sin(d*x+c))/d+1/3* a*(2*A*a^2+8*A*b^2+9*B*a*b)*tan(d*x+c)/d+1/6*a^2*(5*A*b+3*B*a)*sec(d*x+c)* tan(d*x+c)/d+1/3*a*A*(a+b*cos(d*x+c))^2*sec(d*x+c)^2*tan(d*x+c)/d
Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.74 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {6 b^3 B d x+3 \left (3 a^2 A b+2 A b^3+a^3 B+6 a b^2 B\right ) \text {arctanh}(\sin (c+d x))+3 a \left (2 a^2 A+6 A b^2+6 a b B+a (3 A b+a B) \sec (c+d x)\right ) \tan (c+d x)+2 a^3 A \tan ^3(c+d x)}{6 d} \]
(6*b^3*B*d*x + 3*(3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*ArcTanh[Sin[c + d*x]] + 3*a*(2*a^2*A + 6*A*b^2 + 6*a*b*B + a*(3*A*b + a*B)*Sec[c + d*x])* Tan[c + d*x] + 2*a^3*A*Tan[c + d*x]^3)/(6*d)
Time = 1.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {3042, 3468, 3042, 3510, 25, 3042, 3500, 27, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {1}{3} \int (a+b \cos (c+d x)) \left (3 b^2 B \cos ^2(c+d x)+\left (2 A a^2+6 b B a+3 A b^2\right ) \cos (c+d x)+a (5 A b+3 a B)\right ) \sec ^3(c+d x)dx+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 b^2 B \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (2 A a^2+6 b B a+3 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (5 A b+3 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{3} \left (\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}-\frac {1}{2} \int -\left (\left (6 B \cos ^2(c+d x) b^3+2 a \left (2 A a^2+9 b B a+8 A b^2\right )+3 \left (B a^3+3 A b a^2+6 b^2 B a+2 A b^3\right ) \cos (c+d x)\right ) \sec ^2(c+d x)\right )dx\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \left (6 B \cos ^2(c+d x) b^3+2 a \left (2 A a^2+9 b B a+8 A b^2\right )+3 \left (B a^3+3 A b a^2+6 b^2 B a+2 A b^3\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {6 B \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3+2 a \left (2 A a^2+9 b B a+8 A b^2\right )+3 \left (B a^3+3 A b a^2+6 b^2 B a+2 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (B a^3+3 A b a^2+6 b^2 B a+2 A b^3+2 b^3 B \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 a \left (2 a^2 A+9 a b B+8 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (B a^3+3 A b a^2+6 b^2 B a+2 A b^3+2 b^3 B \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 a \left (2 a^2 A+9 a b B+8 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {B a^3+3 A b a^2+6 b^2 B a+2 A b^3+2 b^3 B \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a \left (2 a^2 A+9 a b B+8 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right ) \int \sec (c+d x)dx+2 b^3 B x\right )+\frac {2 a \left (2 a^2 A+9 a b B+8 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+2 b^3 B x\right )+\frac {2 a \left (2 a^2 A+9 a b B+8 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {1}{2} \left (\frac {2 a \left (2 a^2 A+9 a b B+8 A b^2\right ) \tan (c+d x)}{d}+3 \left (\frac {\left (a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right ) \text {arctanh}(\sin (c+d x))}{d}+2 b^3 B x\right )\right )\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
(a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((a^2*(5* A*b + 3*a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*(2*b^3*B*x + ((3*a^2*A* b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*ArcTanh[Sin[c + d*x]])/d) + (2*a*(2*a^2*A + 8*A*b^2 + 9*a*b*B)*Tan[c + d*x])/d)/2)/3
3.3.37.3.1 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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 4.88 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01
| method | result | size |
| parts | \(-\frac {A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 A a \,b^{2}+3 B \,a^{2} b \right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {B \,b^{3} \left (d x +c \right )}{d}\) | \(146\) |
| derivativedivides | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B \,a^{2} b \tan \left (d x +c \right )+3 A a \,b^{2} \tan \left (d x +c \right )+3 B a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{3} \left (d x +c \right )}{d}\) | \(180\) |
| default | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B \,a^{2} b \tan \left (d x +c \right )+3 A a \,b^{2} \tan \left (d x +c \right )+3 B a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{3} \left (d x +c \right )}{d}\) | \(180\) |
| parallelrisch | \(\frac {-27 \left (A \,a^{2} b +\frac {2}{3} A \,b^{3}+\frac {1}{3} B \,a^{3}+2 B a \,b^{2}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+27 \left (A \,a^{2} b +\frac {2}{3} A \,b^{3}+\frac {1}{3} B \,a^{3}+2 B a \,b^{2}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+6 B \,b^{3} d x \cos \left (3 d x +3 c \right )+\left (4 A \,a^{3}+18 A a \,b^{2}+18 B \,a^{2} b \right ) \sin \left (3 d x +3 c \right )+\left (18 A \,a^{2} b +6 B \,a^{3}\right ) \sin \left (2 d x +2 c \right )+18 B \,b^{3} d x \cos \left (d x +c \right )+12 \left (A \,a^{2}+\frac {3}{2} A \,b^{2}+\frac {3}{2} B a b \right ) a \sin \left (d x +c \right )}{6 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(255\) |
| risch | \(b^{3} B x -\frac {i a \left (9 A a b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 B \,a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-18 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 A a b \,{\mathrm e}^{i \left (d x +c \right )}-3 B \,a^{2} {\mathrm e}^{i \left (d x +c \right )}-4 A \,a^{2}-18 A \,b^{2}-18 B a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A b}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{3}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{2}}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A b}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{3}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{2}}{d}\) | \(356\) |
| norman | \(\frac {b^{3} B x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b^{3} B x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b^{3} B x -\frac {8 a \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \left (2 A \,a^{2}-3 A a b +6 A \,b^{2}-B \,a^{2}+6 B a b \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \left (2 A \,a^{2}+3 A a b +6 A \,b^{2}+B \,a^{2}+6 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 a \left (5 A \,a^{2}-9 A a b +9 A \,b^{2}-3 B \,a^{2}+9 B a b \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 a \left (5 A \,a^{2}+9 A a b +9 A \,b^{2}+3 B \,a^{2}+9 B a b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a \left (26 A \,a^{2}-45 A a b -18 A \,b^{2}-15 B \,a^{2}-18 B a b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a \left (26 A \,a^{2}+45 A a b -18 A \,b^{2}+15 B \,a^{2}-18 B a b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-b^{3} B x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b^{3} B x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b^{3} B x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{3} B x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{3} B x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {\left (3 A \,a^{2} b +2 A \,b^{3}+B \,a^{3}+6 B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (3 A \,a^{2} b +2 A \,b^{3}+B \,a^{3}+6 B a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(558\) |
-A*a^3/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(A*b^3+3*B*a*b^2)/d*ln(sec(d*x +c)+tan(d*x+c))+(3*A*a*b^2+3*B*a^2*b)/d*tan(d*x+c)+(3*A*a^2*b+B*a^3)/d*(1/ 2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+B*b^3/d*(d*x+c)
Time = 0.31 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.30 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {12 \, B b^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{3} + 2 \, {\left (2 \, A a^{3} + 9 \, B a^{2} b + 9 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
1/12*(12*B*b^3*d*x*cos(d*x + c)^3 + 3*(B*a^3 + 3*A*a^2*b + 6*B*a*b^2 + 2*A *b^3)*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*(B*a^3 + 3*A*a^2*b + 6*B*a* b^2 + 2*A*b^3)*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*(2*A*a^3 + 2*(2*A *a^3 + 9*B*a^2*b + 9*A*a*b^2)*cos(d*x + c)^2 + 3*(B*a^3 + 3*A*a^2*b)*cos(d *x + c))*sin(d*x + c))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.49 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 12 \, {\left (d x + c\right )} B b^{3} - 3 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 9 \, A a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, A b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{2} b \tan \left (d x + c\right ) + 36 \, A a b^{2} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(4*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3 + 12*(d*x + c)*B*b^3 - 3*B *a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(si n(d*x + c) - 1)) - 9*A*a^2*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(si n(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 18*B*a*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 6*A*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*B*a^2*b*tan(d*x + c) + 36*A*a*b^2*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (137) = 274\).
Time = 0.32 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.32 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {6 \, {\left (d x + c\right )} B b^{3} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
1/6*(6*(d*x + c)*B*b^3 + 3*(B*a^3 + 3*A*a^2*b + 6*B*a*b^2 + 2*A*b^3)*log(a bs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(B*a^3 + 3*A*a^2*b + 6*B*a*b^2 + 2*A*b^3 )*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(6*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 9*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 18*B* a^2*b*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 4*A*a^3 *tan(1/2*d*x + 1/2*c)^3 - 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^2*t an(1/2*d*x + 1/2*c)^3 + 6*A*a^3*tan(1/2*d*x + 1/2*c) + 3*B*a^3*tan(1/2*d*x + 1/2*c) + 9*A*a^2*b*tan(1/2*d*x + 1/2*c) + 18*B*a^2*b*tan(1/2*d*x + 1/2* c) + 18*A*a*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^3)/d
Time = 2.17 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.63 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {\frac {A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{6}+\frac {B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{2}+\frac {3\,A\,a\,b^2\,\sin \left (c+d\,x\right )}{4}+\frac {3\,B\,a^2\,b\,\sin \left (c+d\,x\right )}{4}-\frac {A\,b^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{2}-\frac {B\,a^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{4}+\frac {3\,B\,b^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {3\,A\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,A\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {3\,B\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4}-\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{2}-\frac {B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{4}+\frac {B\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}-\frac {A\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{4}-\frac {B\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{2}-\frac {A\,a^2\,b\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,9{}\mathrm {i}}{4}-\frac {B\,a\,b^2\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,9{}\mathrm {i}}{2}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
((A*a^3*sin(3*c + 3*d*x))/6 + (B*a^3*sin(2*c + 2*d*x))/4 + (A*a^3*sin(c + d*x))/2 + (3*A*a*b^2*sin(c + d*x))/4 + (3*B*a^2*b*sin(c + d*x))/4 - (A*b^3 *cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*3i)/2 - (B* a^3*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*3i)/4 + (3*B*b^3*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + (3* A*a^2*b*sin(2*c + 2*d*x))/4 + (3*A*a*b^2*sin(3*c + 3*d*x))/4 + (3*B*a^2*b* sin(3*c + 3*d*x))/4 - (A*b^3*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/ 2))*cos(3*c + 3*d*x)*1i)/2 - (B*a^3*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*1i)/4 + (B*b^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/2 - (A*a^2*b*atan((sin(c/2 + (d*x)/2)*1i)/c os(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/4 - (B*a*b^2*atan((sin(c/2 + (d*x) /2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/2 - (A*a^2*b*cos(c + d*x) *atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*9i)/4 - (B*a*b^2*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*9i)/2)/(d*((3*cos(c + d*x))/4 + cos(3*c + 3*d*x)/4))