Integrand size = 21, antiderivative size = 130 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 b \left (4 a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \left (a^2+4 b^2\right ) \tan (c+d x)}{2 d}+\frac {b \left (2 a^2+3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a (a+b \sec (c+d x))^2 \tan (c+d x)}{4 d}+\frac {(a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]
3/8*b*(4*a^2+b^2)*arctanh(sin(d*x+c))/d+1/2*a*(a^2+4*b^2)*tan(d*x+c)/d+1/8 *b*(2*a^2+3*b^2)*sec(d*x+c)*tan(d*x+c)/d+1/4*a*(a+b*sec(d*x+c))^2*tan(d*x+ c)/d+1/4*(a+b*sec(d*x+c))^3*tan(d*x+c)/d
Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.69 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 b \left (4 a^2+b^2\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (3 b \left (4 a^2+b^2\right ) \sec (c+d x)+2 b^3 \sec ^3(c+d x)+8 a \left (a^2+3 b^2+b^2 \tan ^2(c+d x)\right )\right )}{8 d} \]
(3*b*(4*a^2 + b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*b*(4*a^2 + b^2) *Sec[c + d*x] + 2*b^3*Sec[c + d*x]^3 + 8*a*(a^2 + 3*b^2 + b^2*Tan[c + d*x] ^2)))/(8*d)
Time = 0.87 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4322, 3042, 4490, 3042, 4485, 3042, 4274, 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.
| \(\displaystyle \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4322 |
| \(\displaystyle \frac {3}{4} \int \sec (c+d x) (b+a \sec (c+d x)) (a+b \sec (c+d x))^2dx+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2dx+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \int \sec (c+d x) (a+b \sec (c+d x)) \left (5 a b+\left (2 a^2+3 b^2\right ) \sec (c+d x)\right )dx+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (5 a b+\left (2 a^2+3 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \sec (c+d x) \left (3 b \left (4 a^2+b^2\right )+4 a \left (a^2+4 b^2\right ) \sec (c+d x)\right )dx+\frac {b \left (2 a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (3 b \left (4 a^2+b^2\right )+4 a \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b \left (2 a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (4 a \left (a^2+4 b^2\right ) \int \sec ^2(c+d x)dx+3 b \left (4 a^2+b^2\right ) \int \sec (c+d x)dx\right )+\frac {b \left (2 a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 b \left (4 a^2+b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+4 a \left (a^2+4 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {b \left (2 a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 b \left (4 a^2+b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {4 a \left (a^2+4 b^2\right ) \int 1d(-\tan (c+d x))}{d}\right )+\frac {b \left (2 a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 b \left (4 a^2+b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 a \left (a^2+4 b^2\right ) \tan (c+d x)}{d}\right )+\frac {b \left (2 a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {3 b \left (4 a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {4 a \left (a^2+4 b^2\right ) \tan (c+d x)}{d}\right )+\frac {b \left (2 a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\) |
((a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (3*((a*(a + b*Sec[c + d*x])^ 2*Tan[c + d*x])/(3*d) + ((b*(2*a^2 + 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2* d) + ((3*b*(4*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/d + (4*a*(a^2 + 4*b^2)*Tan [c + d*x])/d)/2)/3))/4
3.5.67.3.1 Defintions of rubi rules used
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_)]*(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_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[m/(m + 1) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(b + a*Csc [e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
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_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*(( a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[1/(m + 1) Int[Csc[e + f*x]* (a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1 ))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a* B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
Time = 1.23 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(\frac {a^{3} \tan \left (d x +c \right )+3 a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(125\) |
| default | \(\frac {a^{3} \tan \left (d x +c \right )+3 a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(125\) |
| parts | \(\frac {a^{3} \tan \left (d x +c \right )}{d}+\frac {b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {3 a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {3 a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(133\) |
| parallelrisch | \(\frac {-24 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {b^{2}}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+24 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {b^{2}}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+8 \left (a^{3}+4 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+3 \left (4 a^{2} b +b^{3}\right ) \sin \left (3 d x +3 c \right )+4 \left (a^{3}+2 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (12 a^{2} b +11 b^{3}\right ) \sin \left (d x +c \right )}{4 d \left (3+\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )\right )}\) | \(205\) |
| norman | \(\frac {-\frac {\left (8 a^{3}-12 a^{2} b +24 a \,b^{2}-5 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (8 a^{3}+12 a^{2} b +24 a \,b^{2}+5 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (24 a^{3}-12 a^{2} b +40 a \,b^{2}+3 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {\left (24 a^{3}+12 a^{2} b +40 a \,b^{2}-3 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {3 b \left (4 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {3 b \left (4 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(227\) |
| risch | \(-\frac {i \left (12 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+3 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-8 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+11 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-24 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-48 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-11 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-24 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-64 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-3 b^{3} {\mathrm e}^{i \left (d x +c \right )}-8 a^{3}-16 a \,b^{2}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{2 d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{2 d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}\) | \(307\) |
1/d*(a^3*tan(d*x+c)+3*a^2*b*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+t an(d*x+c)))-3*a*b^2*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+b^3*(-(-1/4*sec(d*x +c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c))))
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.08 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, a b^{2} \cos \left (d x + c\right ) + 8 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 2 \, b^{3} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
1/16*(3*(4*a^2*b + b^3)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(4*a^2*b + b^3)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(8*a*b^2*cos(d*x + c) + 8 *(a^3 + 2*a*b^2)*cos(d*x + c)^3 + 2*b^3 + 3*(4*a^2*b + b^3)*cos(d*x + c)^2 )*sin(d*x + c))/(d*cos(d*x + c)^4)
\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{2} - b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 16 \, a^{3} \tan \left (d x + c\right )}{16 \, d} \]
1/16*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*a*b^2 - b^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)) - 12*a^2*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 16*a^3*tan (d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (120) = 240\).
Time = 0.34 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.54 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{8 \, d} \]
1/8*(3*(4*a^2*b + b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(4*a^2*b + b ^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(8*a^3*tan(1/2*d*x + 1/2*c)^7 - 12*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 24*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 5*b^3 *tan(1/2*d*x + 1/2*c)^7 - 24*a^3*tan(1/2*d*x + 1/2*c)^5 + 12*a^2*b*tan(1/2 *d*x + 1/2*c)^5 - 40*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 3*b^3*tan(1/2*d*x + 1/ 2*c)^5 + 24*a^3*tan(1/2*d*x + 1/2*c)^3 + 12*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 40*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 3*b^3*tan(1/2*d*x + 1/2*c)^3 - 8*a^3*ta n(1/2*d*x + 1/2*c) - 12*a^2*b*tan(1/2*d*x + 1/2*c) - 24*a*b^2*tan(1/2*d*x + 1/2*c) - 5*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d
Time = 17.08 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.74 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3\,b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,a^2+b^2\right )}{4\,d}-\frac {\left (2\,a^3-3\,a^2\,b+6\,a\,b^2-\frac {5\,b^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-6\,a^3+3\,a^2\,b-10\,a\,b^2-\frac {3\,b^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (6\,a^3+3\,a^2\,b+10\,a\,b^2-\frac {3\,b^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-2\,a^3-3\,a^2\,b-6\,a\,b^2-\frac {5\,b^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(3*b*atanh(tan(c/2 + (d*x)/2))*(4*a^2 + b^2))/(4*d) - (tan(c/2 + (d*x)/2)^ 7*(6*a*b^2 - 3*a^2*b + 2*a^3 - (5*b^3)/4) + tan(c/2 + (d*x)/2)^3*(10*a*b^2 + 3*a^2*b + 6*a^3 - (3*b^3)/4) - tan(c/2 + (d*x)/2)^5*(10*a*b^2 - 3*a^2*b + 6*a^3 + (3*b^3)/4) - tan(c/2 + (d*x)/2)*(6*a*b^2 + 3*a^2*b + 2*a^3 + (5 *b^3)/4))/(d*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1))