Integrand size = 25, antiderivative size = 147 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^3 A x+\frac {a^3 (28 A+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 (4 A+3 C) \tan (c+d x)}{8 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{8 d} \]
a^3*A*x+1/8*a^3*(28*A+15*C)*arctanh(sin(d*x+c))/d+5/8*a^3*(4*A+3*C)*tan(d* x+c)/d+1/4*C*(a+a*sec(d*x+c))^3*tan(d*x+c)/d+1/4*C*(a^2+a^2*sec(d*x+c))^2* tan(d*x+c)/a/d+1/8*(4*A+5*C)*(a^3+a^3*sec(d*x+c))*tan(d*x+c)/d
Time = 2.93 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.63 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (8 A d x+(28 A+15 C) \text {arctanh}(\sin (c+d x))+\left (24 A+26 C+(4 A+15 C) \sec (c+d x)+6 C \sec ^2(c+d x)+2 C \sec ^3(c+d x)\right ) \tan (c+d x)+2 C \tan ^3(c+d x)\right )}{8 d} \]
(a^3*(8*A*d*x + (28*A + 15*C)*ArcTanh[Sin[c + d*x]] + (24*A + 26*C + (4*A + 15*C)*Sec[c + d*x] + 6*C*Sec[c + d*x]^2 + 2*C*Sec[c + d*x]^3)*Tan[c + d* x] + 2*C*Tan[c + d*x]^3))/(8*d)
Time = 0.87 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4543, 3042, 4405, 27, 3042, 4405, 3042, 4402, 3042, 4254, 24, 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.
| \(\displaystyle \int (a \sec (c+d x)+a)^3 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4543 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^3 (4 a A+3 a C \sec (c+d x))dx}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (4 a A+3 a C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {1}{3} \int 3 (\sec (c+d x) a+a)^2 \left (4 A a^2+(4 A+5 C) \sec (c+d x) a^2\right )dx+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^2 \left (4 A a^2+(4 A+5 C) \sec (c+d x) a^2\right )dx+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (4 A a^2+(4 A+5 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {1}{2} \int (\sec (c+d x) a+a) \left (8 A a^3+5 (4 A+3 C) \sec (c+d x) a^3\right )dx+\frac {(4 A+5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (8 A a^3+5 (4 A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {(4 A+5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 4402 |
| \(\displaystyle \frac {\frac {1}{2} \left (5 a^4 (4 A+3 C) \int \sec ^2(c+d x)dx+a^4 (28 A+15 C) \int \sec (c+d x)dx+8 a^4 A x\right )+\frac {(4 A+5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (a^4 (28 A+15 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+5 a^4 (4 A+3 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+8 a^4 A x\right )+\frac {(4 A+5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {5 a^4 (4 A+3 C) \int 1d(-\tan (c+d x))}{d}+a^4 (28 A+15 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+8 a^4 A x\right )+\frac {(4 A+5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{2} \left (a^4 (28 A+15 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^4 (4 A+3 C) \tan (c+d x)}{d}+8 a^4 A x\right )+\frac {(4 A+5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {a^4 (28 A+15 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (4 A+3 C) \tan (c+d x)}{d}+8 a^4 A x\right )+\frac {(4 A+5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}+\frac {C \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{d}}{4 a}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}\) |
(C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + ((C*(a^2 + a^2*Sec[c + d*x ])^2*Tan[c + d*x])/d + ((4*A + 5*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x]) /(2*d) + (8*a^4*A*x + (a^4*(28*A + 15*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^4 *(4*A + 3*C)*Tan[c + d*x])/d)/2)/(4*a)
3.2.4.3.1 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[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[a*c*x, x] + (Simp[b*d Int[Csc[e + f*x]^2, x], x ] + Simp[(b*c + a*d) Int[Csc[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f }, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[(-b)*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[1/m Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*c*m + (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2 *m]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_))^(m_.), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^ m/(f*(m + 1))), x] + Simp[1/(b*(m + 1)) Int[(a + b*Csc[e + f*x])^m*Simp[A *b*(m + 1) + a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 0.44 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.22
| method | result | size |
| parts | \(a^{3} A x +\frac {\left (a^{3} A +3 a^{3} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 a^{3} A +a^{3} C \right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {3 A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{3}}{d}-\frac {3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(179\) |
| derivativedivides | \(\frac {a^{3} A \left (d x +c \right )+a^{3} C \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(205\) |
| default | \(\frac {a^{3} A \left (d x +c \right )+a^{3} C \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(205\) |
| parallelrisch | \(\frac {6 a^{3} \left (-\frac {7 \left (A +\frac {15 C}{28}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{3}+\frac {7 \left (A +\frac {15 C}{28}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{3}+\frac {2 d x A \cos \left (2 d x +2 c \right )}{3}+\frac {d x A \cos \left (4 d x +4 c \right )}{6}+\left (A +\frac {5 C}{3}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (\frac {A}{3}+\frac {5 C}{4}\right ) \sin \left (3 d x +3 c \right )}{2}+\frac {\left (A +C \right ) \sin \left (4 d x +4 c \right )}{2}+\frac {\left (A +\frac {23 C}{4}\right ) \sin \left (d x +c \right )}{6}+\frac {d x A}{2}\right )}{d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(209\) |
| norman | \(\frac {a^{3} A x +a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {5 a^{3} \left (4 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {7 a^{3} \left (4 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (68 A +55 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {a^{3} \left (76 A +73 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {a^{3} \left (28 A +15 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{3} \left (28 A +15 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(252\) |
| risch | \(a^{3} A x -\frac {i a^{3} \left (4 A \,{\mathrm e}^{7 i \left (d x +c \right )}+15 C \,{\mathrm e}^{7 i \left (d x +c \right )}-24 A \,{\mathrm e}^{6 i \left (d x +c \right )}-8 C \,{\mathrm e}^{6 i \left (d x +c \right )}+4 A \,{\mathrm e}^{5 i \left (d x +c \right )}+23 C \,{\mathrm e}^{5 i \left (d x +c \right )}-72 A \,{\mathrm e}^{4 i \left (d x +c \right )}-72 C \,{\mathrm e}^{4 i \left (d x +c \right )}-4 A \,{\mathrm e}^{3 i \left (d x +c \right )}-23 C \,{\mathrm e}^{3 i \left (d x +c \right )}-72 A \,{\mathrm e}^{2 i \left (d x +c \right )}-88 C \,{\mathrm e}^{2 i \left (d x +c \right )}-4 A \,{\mathrm e}^{i \left (d x +c \right )}-15 C \,{\mathrm e}^{i \left (d x +c \right )}-24 A -24 C \right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(293\) |
a^3*A*x+(A*a^3+3*C*a^3)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan (d*x+c)))+(3*A*a^3+C*a^3)/d*tan(d*x+c)+a^3*C/d*(-(-1/4*sec(d*x+c)^3-3/8*se c(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))+3/d*A*ln(sec(d*x+c)+ta n(d*x+c))*a^3-3*a^3*C/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)
Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {16 \, A a^{3} d x \cos \left (d x + c\right )^{4} + {\left (28 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (28 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, {\left (A + C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (4 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 8 \, C a^{3} \cos \left (d x + c\right ) + 2 \, C a^{3}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
1/16*(16*A*a^3*d*x*cos(d*x + c)^4 + (28*A + 15*C)*a^3*cos(d*x + c)^4*log(s in(d*x + c) + 1) - (28*A + 15*C)*a^3*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(24*(A + C)*a^3*cos(d*x + c)^3 + (4*A + 15*C)*a^3*cos(d*x + c)^2 + 8* C*a^3*cos(d*x + c) + 2*C*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4)
\[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A\, dx + \int 3 A \sec {\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int C \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(A, x) + Integral(3*A*sec(c + d*x), x) + Integral(3*A*sec(c + d*x)**2, x) + Integral(A*sec(c + d*x)**3, x) + Integral(C*sec(c + d*x)** 2, x) + Integral(3*C*sec(c + d*x)**3, x) + Integral(3*C*sec(c + d*x)**4, x ) + Integral(C*sec(c + d*x)**5, x))
Time = 0.20 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.70 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {16 \, {\left (d x + c\right )} A a^{3} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, A a^{3} \tan \left (d x + c\right ) + 16 \, C a^{3} \tan \left (d x + c\right )}{16 \, d} \]
1/16*(16*(d*x + c)*A*a^3 + 16*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 - C* a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c )^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 4*A*a^3*(2 *sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 12*C*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 48*A*a^3*log(sec(d*x + c) + tan(d*x + c )) + 48*A*a^3*tan(d*x + c) + 16*C*a^3*tan(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.51 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 \, {\left (d x + c\right )} A a^{3} + {\left (28 \, A a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (28 \, A a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (20 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 68 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 55 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 73 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 28 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 49 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{8 \, d} \]
1/8*(8*(d*x + c)*A*a^3 + (28*A*a^3 + 15*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c ) + 1)) - (28*A*a^3 + 15*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(20 *A*a^3*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 68*A*a^3 *tan(1/2*d*x + 1/2*c)^5 - 55*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 76*A*a^3*tan(1 /2*d*x + 1/2*c)^3 + 73*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 28*A*a^3*tan(1/2*d*x + 1/2*c) - 49*C*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4) /d
Time = 15.89 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.57 \[ \int (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {15\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {3\,C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {15\,C\,a^3\,\sin \left (c+d\,x\right )}{8\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^3}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{4\,d\,{\cos \left (c+d\,x\right )}^4} \]
(2*A*a^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (7*A*a^3*atanh(s in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (15*C*a^3*atanh(sin(c/2 + (d*x) /2)/cos(c/2 + (d*x)/2)))/(4*d) + (3*A*a^3*sin(c + d*x))/(d*cos(c + d*x)) + (A*a^3*sin(c + d*x))/(2*d*cos(c + d*x)^2) + (3*C*a^3*sin(c + d*x))/(d*cos (c + d*x)) + (15*C*a^3*sin(c + d*x))/(8*d*cos(c + d*x)^2) + (C*a^3*sin(c + d*x))/(d*cos(c + d*x)^3) + (C*a^3*sin(c + d*x))/(4*d*cos(c + d*x)^4)