Integrand size = 41, antiderivative size = 274 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (26 A+23 B+21 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac {a^3 (26 A+23 B+21 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d} \]
1/16*a^3*(26*A+23*B+21*C)*arctanh(sin(d*x+c))/d+1/35*a^3*(133*A+119*B+108* C)*tan(d*x+c)/d+1/16*a^3*(26*A+23*B+21*C)*sec(d*x+c)*tan(d*x+c)/d+1/280*a^ 3*(154*A+147*B+129*C)*sec(d*x+c)^3*tan(d*x+c)/d+1/7*C*sec(d*x+c)^3*(a+a*se c(d*x+c))^3*tan(d*x+c)/d+1/42*(7*B+3*C)*sec(d*x+c)^3*(a^2+a^2*sec(d*x+c))^ 2*tan(d*x+c)/a/d+1/15*(3*A+4*B+3*C)*sec(d*x+c)^3*(a^3+a^3*sec(d*x+c))*tan( d*x+c)/d+1/105*a^3*(133*A+119*B+108*C)*tan(d*x+c)^3/d
Time = 5.98 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.55 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (105 (26 A+23 B+21 C) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (105 (26 A+23 B+21 C)+16 (133 A+119 B+108 C) (2+\cos (2 (c+d x))) \sec (c+d x)+70 (18 A+23 B+21 C) \sec ^2(c+d x)+48 (7 A+21 B+27 C) \sec ^3(c+d x)+280 (B+3 C) \sec ^4(c+d x)+240 C \sec ^5(c+d x)\right ) \tan (c+d x)\right )}{1680 d} \]
(a^3*(105*(26*A + 23*B + 21*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(105*( 26*A + 23*B + 21*C) + 16*(133*A + 119*B + 108*C)*(2 + Cos[2*(c + d*x)])*Se c[c + d*x] + 70*(18*A + 23*B + 21*C)*Sec[c + d*x]^2 + 48*(7*A + 21*B + 27* C)*Sec[c + d*x]^3 + 280*(B + 3*C)*Sec[c + d*x]^4 + 240*C*Sec[c + d*x]^5)*T an[c + d*x]))/(1680*d)
Time = 1.72 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 4576, 3042, 4506, 3042, 4506, 27, 3042, 4485, 3042, 4274, 3042, 4254, 2009, 4255, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4576 |
| \(\displaystyle \frac {\int \sec ^3(c+d x) (\sec (c+d x) a+a)^3 (a (7 A+3 C)+a (7 B+3 C) \sec (c+d x))dx}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (7 A+3 C)+a (7 B+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{6} \int \sec ^3(c+d x) (\sec (c+d x) a+a)^2 \left (3 (14 A+7 B+9 C) a^2+14 (3 A+4 B+3 C) \sec (c+d x) a^2\right )dx+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (14 A+7 B+9 C) a^2+14 (3 A+4 B+3 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \int 3 \sec ^3(c+d x) (\sec (c+d x) a+a) \left ((112 A+91 B+87 C) a^3+(154 A+147 B+129 C) \sec (c+d x) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \sec ^3(c+d x) (\sec (c+d x) a+a) \left ((112 A+91 B+87 C) a^3+(154 A+147 B+129 C) \sec (c+d x) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((112 A+91 B+87 C) a^3+(154 A+147 B+129 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \sec ^3(c+d x) \left (35 (26 A+23 B+21 C) a^4+8 (133 A+119 B+108 C) \sec (c+d x) a^4\right )dx+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (35 (26 A+23 B+21 C) a^4+8 (133 A+119 B+108 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (8 a^4 (133 A+119 B+108 C) \int \sec ^4(c+d x)dx+35 a^4 (26 A+23 B+21 C) \int \sec ^3(c+d x)dx\right )+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+8 a^4 (133 A+119 B+108 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx\right )+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {8 a^4 (133 A+119 B+108 C) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {8 a^4 (133 A+119 B+108 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {8 a^4 (133 A+119 B+108 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {8 a^4 (133 A+119 B+108 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {8 a^4 (133 A+119 B+108 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
(C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(7*d) + (((7*B + 3* C)*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(6*d) + ((14*(3 *A + 4*B + 3*C)*Sec[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(5*d ) + (3*((a^4*(154*A + 147*B + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (35*a^4*(26*A + 23*B + 21*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]* Tan[c + d*x])/(2*d)) - (8*a^4*(133*A + 119*B + 108*C)*(-Tan[c + d*x] - Tan [c + d*x]^3/3))/d)/4))/5)/6)/(7*a)
3.5.27.3.1 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_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[ e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Simp[1/(n + 1) Int[(d*Csc [e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x ], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && !LeQ[ n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x] )^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Cs c[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b* B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m , n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
Time = 1.03 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.02
| method | result | size |
| norman | \(\frac {-\frac {283 a^{3} \left (26 A +23 B +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}+\frac {5 a^{3} \left (26 A +23 B +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}-\frac {a^{3} \left (26 A +23 B +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}-\frac {a^{3} \left (102 A +105 B +107 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {16 a^{3} \left (203 A +189 B +163 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {a^{3} \left (286 A +237 B +183 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {a^{3} \left (10178 A +8979 B +9033 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {a^{3} \left (26 A +23 B +21 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {a^{3} \left (26 A +23 B +21 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(280\) |
| parts | \(-\frac {\left (3 a^{3} A +B \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,a^{3}+3 a^{3} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (a^{3} A +3 B \,a^{3}+3 a^{3} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a^{3} A +3 B \,a^{3}+a^{3} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {a^{3} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(310\) |
| parallelrisch | \(-\frac {13 \left (\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {23 B}{26}+\frac {21 C}{26}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {23 B}{26}+\frac {21 C}{26}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {345 C}{91}-\frac {226 A}{91}-\frac {841 B}{273}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {1744 B}{455}-\frac {1728 C}{455}-\frac {1648 A}{455}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {20 C}{13}-\frac {152 A}{91}-\frac {460 B}{273}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {272 B}{195}-\frac {304 A}{195}-\frac {576 C}{455}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {23 B}{91}-\frac {2 A}{7}-\frac {3 C}{13}\right ) \sin \left (6 d x +6 c \right )+\left (-\frac {576 C}{3185}-\frac {304 A}{1365}-\frac {272 B}{1365}\right ) \sin \left (7 d x +7 c \right )-\frac {16 \left (A +\frac {15 B}{13}+\frac {20 C}{13}\right ) \sin \left (d x +c \right )}{7}\right ) a^{3}}{8 d \left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right )}\) | \(311\) |
| derivativedivides | \(\frac {-a^{3} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{3} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+3 a^{3} A \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 )-3 B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \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 )-3 a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\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 )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \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}\) | \(504\) |
| default | \(\frac {-a^{3} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{3} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+3 a^{3} A \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 )-3 B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \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 )-3 a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\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 )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \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}\) | \(504\) |
| risch | \(-\frac {i a^{3} \left (-3456 C -4256 A -3808 B -52640 A \,{\mathrm e}^{8 i \left (d x +c \right )}-23730 A \,{\mathrm e}^{5 i \left (d x +c \right )}-15960 A \,{\mathrm e}^{3 i \left (d x +c \right )}-2730 A \,{\mathrm e}^{i \left (d x +c \right )}-76608 B \,{\mathrm e}^{4 i \left (d x +c \right )}-96320 A \,{\mathrm e}^{6 i \left (d x +c \right )}-94080 C \,{\mathrm e}^{6 i \left (d x +c \right )}-79296 A \,{\mathrm e}^{4 i \left (d x +c \right )}-72576 C \,{\mathrm e}^{4 i \left (d x +c \right )}-29792 A \,{\mathrm e}^{2 i \left (d x +c \right )}-24192 C \,{\mathrm e}^{2 i \left (d x +c \right )}-36225 C \,{\mathrm e}^{5 i \left (d x +c \right )}-14700 C \,{\mathrm e}^{3 i \left (d x +c \right )}-26656 B \,{\mathrm e}^{2 i \left (d x +c \right )}-2205 C \,{\mathrm e}^{i \left (d x +c \right )}+2205 C \,{\mathrm e}^{13 i \left (d x +c \right )}+2730 A \,{\mathrm e}^{13 i \left (d x +c \right )}+15960 A \,{\mathrm e}^{11 i \left (d x +c \right )}+14700 C \,{\mathrm e}^{11 i \left (d x +c \right )}-10080 A \,{\mathrm e}^{10 i \left (d x +c \right )}+23730 A \,{\mathrm e}^{9 i \left (d x +c \right )}+36225 C \,{\mathrm e}^{9 i \left (d x +c \right )}-26880 C \,{\mathrm e}^{8 i \left (d x +c \right )}-2415 B \,{\mathrm e}^{i \left (d x +c \right )}-29435 B \,{\mathrm e}^{5 i \left (d x +c \right )}-41440 B \,{\mathrm e}^{8 i \left (d x +c \right )}+16100 B \,{\mathrm e}^{11 i \left (d x +c \right )}+29435 B \,{\mathrm e}^{9 i \left (d x +c \right )}-3360 B \,{\mathrm e}^{10 i \left (d x +c \right )}-91840 B \,{\mathrm e}^{6 i \left (d x +c \right )}+2415 B \,{\mathrm e}^{13 i \left (d x +c \right )}-16100 B \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{840 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{16 d}-\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{16 d}+\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\) | \(550\) |
(-283/120*a^3*(26*A+23*B+21*C)/d*tan(1/2*d*x+1/2*c)^9+5/6*a^3*(26*A+23*B+2 1*C)/d*tan(1/2*d*x+1/2*c)^11-1/8*a^3*(26*A+23*B+21*C)/d*tan(1/2*d*x+1/2*c) ^13-1/8*a^3*(102*A+105*B+107*C)/d*tan(1/2*d*x+1/2*c)+16/35*a^3*(203*A+189* B+163*C)/d*tan(1/2*d*x+1/2*c)^7+1/6*a^3*(286*A+237*B+183*C)/d*tan(1/2*d*x+ 1/2*c)^3-1/120*a^3*(10178*A+8979*B+9033*C)/d*tan(1/2*d*x+1/2*c)^5)/(tan(1/ 2*d*x+1/2*c)^2-1)^7-1/16*a^3*(26*A+23*B+21*C)/d*ln(tan(1/2*d*x+1/2*c)-1)+1 /16*a^3*(26*A+23*B+21*C)/d*ln(tan(1/2*d*x+1/2*c)+1)
Time = 0.27 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.82 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (32 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} + 105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 16 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \, {\left (18 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 48 \, {\left (7 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 280 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 240 \, C a^{3}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]
integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/3360*(105*(26*A + 23*B + 21*C)*a^3*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(26*A + 23*B + 21*C)*a^3*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 2*( 32*(133*A + 119*B + 108*C)*a^3*cos(d*x + c)^6 + 105*(26*A + 23*B + 21*C)*a ^3*cos(d*x + c)^5 + 16*(133*A + 119*B + 108*C)*a^3*cos(d*x + c)^4 + 70*(18 *A + 23*B + 21*C)*a^3*cos(d*x + c)^3 + 48*(7*A + 21*B + 27*C)*a^3*cos(d*x + c)^2 + 280*(B + 3*C)*a^3*cos(d*x + c) + 240*C*a^3)*sin(d*x + c))/(d*cos( d*x + c)^7)
\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(A*sec(c + d*x)**3, x) + Integral(3*A*sec(c + d*x)**4, x) + Integral(3*A*sec(c + d*x)**5, x) + Integral(A*sec(c + d*x)**6, x) + Integr al(B*sec(c + d*x)**4, x) + Integral(3*B*sec(c + d*x)**5, x) + Integral(3*B *sec(c + d*x)**6, x) + Integral(B*sec(c + d*x)**7, x) + Integral(C*sec(c + d*x)**5, x) + Integral(3*C*sec(c + d*x)**6, x) + Integral(3*C*sec(c + d*x )**7, x) + Integral(C*sec(c + d*x)**8, x))
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (258) = 516\).
Time = 0.25 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.37 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {224 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{3} + 3360 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 672 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{3} + 1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 96 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C a^{3} + 672 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{3} - 35 \, B a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 105 \, C a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 630 \, A 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 )} - 630 \, B 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 )} - 210 \, 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 )} - 840 \, 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 )}}{3360 \, d} \]
integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
1/3360*(224*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^3 + 3360*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3 + 672*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^3 + 1120*(tan(d*x + c)^3 + 3*tan( d*x + c))*B*a^3 + 96*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*C*a^3 + 672*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^3 - 35*B*a^3*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c )^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c) ^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 105*C*a^3 *(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c )^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 630*A*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)) - 630*B*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)) - 210*C*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(si n(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)) - 840*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)))/d
Time = 0.41 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.62 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, 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 (2730 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2415 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2205 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 18200 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 16100 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 14700 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 51506 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45563 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 41601 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 77952 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72576 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 62592 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 71246 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 62853 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63231 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40040 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 33180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 25620 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10710 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11025 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11235 \, 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 )}^{7}}}{1680 \, d} \]
integrate(sec(d*x+c)^3*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/1680*(105*(26*A*a^3 + 23*B*a^3 + 21*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(26*A*a^3 + 23*B*a^3 + 21*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(2730*A*a^3*tan(1/2*d*x + 1/2*c)^13 + 2415*B*a^3*tan(1/2*d*x + 1 /2*c)^13 + 2205*C*a^3*tan(1/2*d*x + 1/2*c)^13 - 18200*A*a^3*tan(1/2*d*x + 1/2*c)^11 - 16100*B*a^3*tan(1/2*d*x + 1/2*c)^11 - 14700*C*a^3*tan(1/2*d*x + 1/2*c)^11 + 51506*A*a^3*tan(1/2*d*x + 1/2*c)^9 + 45563*B*a^3*tan(1/2*d*x + 1/2*c)^9 + 41601*C*a^3*tan(1/2*d*x + 1/2*c)^9 - 77952*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 72576*B*a^3*tan(1/2*d*x + 1/2*c)^7 - 62592*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 71246*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 62853*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 63231*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 40040*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 33180*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 25620*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 10710*A*a^3*tan(1/2*d*x + 1/2*c) + 11025*B*a^3*tan(1/2*d*x + 1/2*c) + 11235*C*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^7 )/d
Time = 20.61 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.39 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,A+23\,B+21\,C\right )}{4\,\left (\frac {13\,A\,a^3}{2}+\frac {23\,B\,a^3}{4}+\frac {21\,C\,a^3}{4}\right )}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,d}-\frac {\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}+\frac {21\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {65\,A\,a^3}{3}-\frac {115\,B\,a^3}{6}-\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3679\,A\,a^3}{60}+\frac {6509\,B\,a^3}{120}+\frac {1981\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {464\,A\,a^3}{5}-\frac {432\,B\,a^3}{5}-\frac {2608\,C\,a^3}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {5089\,A\,a^3}{60}+\frac {2993\,B\,a^3}{40}+\frac {3011\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {143\,A\,a^3}{3}-\frac {79\,B\,a^3}{2}-\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {105\,B\,a^3}{8}+\frac {107\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
(a^3*atanh((a^3*tan(c/2 + (d*x)/2)*(26*A + 23*B + 21*C))/(4*((13*A*a^3)/2 + (23*B*a^3)/4 + (21*C*a^3)/4)))*(26*A + 23*B + 21*C))/(8*d) - (tan(c/2 + (d*x)/2)^13*((13*A*a^3)/4 + (23*B*a^3)/8 + (21*C*a^3)/8) - tan(c/2 + (d*x) /2)^11*((65*A*a^3)/3 + (115*B*a^3)/6 + (35*C*a^3)/2) - tan(c/2 + (d*x)/2)^ 3*((143*A*a^3)/3 + (79*B*a^3)/2 + (61*C*a^3)/2) - tan(c/2 + (d*x)/2)^7*((4 64*A*a^3)/5 + (432*B*a^3)/5 + (2608*C*a^3)/35) + tan(c/2 + (d*x)/2)^5*((50 89*A*a^3)/60 + (2993*B*a^3)/40 + (3011*C*a^3)/40) + tan(c/2 + (d*x)/2)^9*( (3679*A*a^3)/60 + (6509*B*a^3)/120 + (1981*C*a^3)/40) + tan(c/2 + (d*x)/2) *((51*A*a^3)/4 + (105*B*a^3)/8 + (107*C*a^3)/8))/(d*(7*tan(c/2 + (d*x)/2)^ 2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x) /2)^8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d* x)/2)^14 - 1))