Integrand size = 23, antiderivative size = 82 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=a^2 A x+\frac {a^2 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^2 (2 A+3 B) \tan (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d} \]
a^2*A*x+1/2*a^2*(4*A+3*B)*arctanh(sin(d*x+c))/d+1/2*a^2*(2*A+3*B)*tan(d*x+ c)/d+1/2*B*(a^2+a^2*sec(d*x+c))*tan(d*x+c)/d
Time = 0.50 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (2 A d x+(4 A+3 B) \text {arctanh}(\sin (c+d x))+(2 A+4 B+B \sec (c+d x)) \tan (c+d x))}{2 d} \]
(a^2*(2*A*d*x + (4*A + 3*B)*ArcTanh[Sin[c + d*x]] + (2*A + 4*B + B*Sec[c + d*x])*Tan[c + d*x]))/(2*d)
Time = 0.48 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {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.
| \(\displaystyle \int (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {1}{2} \int (\sec (c+d x) a+a) (2 a A+a (2 A+3 B) \sec (c+d x))dx+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (2 a A+a (2 A+3 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}\) |
| \(\Big \downarrow \) 4402 |
| \(\displaystyle \frac {1}{2} \left (a^2 (2 A+3 B) \int \sec ^2(c+d x)dx+a^2 (4 A+3 B) \int \sec (c+d x)dx+2 a^2 A x\right )+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (a^2 (4 A+3 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+a^2 (2 A+3 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+2 a^2 A x\right )+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a^2 (2 A+3 B) \int 1d(-\tan (c+d x))}{d}+a^2 (4 A+3 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+2 a^2 A x\right )+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{2} \left (a^2 (4 A+3 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {a^2 (2 A+3 B) \tan (c+d x)}{d}+2 a^2 A x\right )+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{2} \left (\frac {a^2 (4 A+3 B) \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 (2 A+3 B) \tan (c+d x)}{d}+2 a^2 A x\right )+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}\) |
(B*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d) + (2*a^2*A*x + (a^2*(4*A + 3*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B)*Tan[c + d*x])/d)/2
3.1.56.3.1 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_)]*(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]
Time = 2.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.21
| method | result | size |
| parts | \(a^{2} A x +\frac {\left (A \,a^{2}+2 B \,a^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (2 A \,a^{2}+B \,a^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(99\) |
| derivativedivides | \(\frac {A \,a^{2} \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 B \,a^{2} \tan \left (d x +c \right )+A \,a^{2} \left (d x +c \right )+B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(114\) |
| default | \(\frac {A \,a^{2} \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 B \,a^{2} \tan \left (d x +c \right )+A \,a^{2} \left (d x +c \right )+B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(114\) |
| parallelrisch | \(-\frac {2 \left (\left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {3 B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {3 B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {A x d \cos \left (2 d x +2 c \right )}{2}+\left (-\frac {A}{2}-B \right ) \sin \left (2 d x +2 c \right )-\frac {A x d}{2}-\frac {B \sin \left (d x +c \right )}{2}\right ) a^{2}}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(127\) |
| norman | \(\frac {a^{2} A x +a^{2} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {a^{2} \left (2 A +5 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-2 a^{2} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {a^{2} \left (2 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {a^{2} \left (4 A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{2} \left (4 A +3 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(163\) |
| risch | \(a^{2} A x -\frac {i a^{2} \left (B \,{\mathrm e}^{3 i \left (d x +c \right )}-2 A \,{\mathrm e}^{2 i \left (d x +c \right )}-4 B \,{\mathrm e}^{2 i \left (d x +c \right )}-B \,{\mathrm e}^{i \left (d x +c \right )}-2 A -4 B \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}\) | \(172\) |
a^2*A*x+(A*a^2+2*B*a^2)/d*tan(d*x+c)+(2*A*a^2+B*a^2)/d*ln(sec(d*x+c)+tan(d *x+c))+B*a^2/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.45 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {4 \, A a^{2} d x \cos \left (d x + c\right )^{2} + {\left (4 \, A + 3 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (4 \, A + 3 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right ) + B a^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
1/4*(4*A*a^2*d*x*cos(d*x + c)^2 + (4*A + 3*B)*a^2*cos(d*x + c)^2*log(sin(d *x + c) + 1) - (4*A + 3*B)*a^2*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*( 2*(A + 2*B)*a^2*cos(d*x + c) + B*a^2)*sin(d*x + c))/(d*cos(d*x + c)^2)
\[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=a^{2} \left (\int A\, dx + \int 2 A \sec {\left (c + d x \right )}\, dx + \int A \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sec {\left (c + d x \right )}\, dx + \int 2 B \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
a**2*(Integral(A, x) + Integral(2*A*sec(c + d*x), x) + Integral(A*sec(c + d*x)**2, x) + Integral(B*sec(c + d*x), x) + Integral(2*B*sec(c + d*x)**2, x) + Integral(B*sec(c + d*x)**3, x))
Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.56 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {4 \, {\left (d x + c\right )} A a^{2} - B a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, B a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, A a^{2} \tan \left (d x + c\right ) + 8 \, B a^{2} \tan \left (d x + c\right )}{4 \, d} \]
1/4*(4*(d*x + c)*A*a^2 - B*a^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log( sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 8*A*a^2*log(sec(d*x + c) + ta n(d*x + c)) + 4*B*a^2*log(sec(d*x + c) + tan(d*x + c)) + 4*A*a^2*tan(d*x + c) + 8*B*a^2*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (76) = 152\).
Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.88 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {2 \, {\left (d x + c\right )} A a^{2} + {\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
1/2*(2*(d*x + c)*A*a^2 + (4*A*a^2 + 3*B*a^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - (4*A*a^2 + 3*B*a^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(2*A*a^ 2*tan(1/2*d*x + 1/2*c)^3 + 3*B*a^2*tan(1/2*d*x + 1/2*c)^3 - 2*A*a^2*tan(1/ 2*d*x + 1/2*c) - 5*B*a^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1 )^2)/d
Time = 13.83 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.98 \[ \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {2\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,B\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {2\,B\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^2\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]
(2*A*a^2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (4*A*a^2*atanh(s in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (3*B*a^2*atanh(sin(c/2 + (d*x)/ 2)/cos(c/2 + (d*x)/2)))/d + (A*a^2*sin(c + d*x))/(d*cos(c + d*x)) + (2*B*a ^2*sin(c + d*x))/(d*cos(c + d*x)) + (B*a^2*sin(c + d*x))/(2*d*cos(c + d*x) ^2)