Integrand size = 31, antiderivative size = 176 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx=\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \text {arctanh}(\sin (e+f x))}{8 f}-\frac {a^2 \left (c^3-8 c^2 d-20 c d^2-8 d^3\right ) \tan (e+f x)}{6 d f}-\frac {a^2 \left (2 c (c-8 d)-21 d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}-\frac {a^2 (c-8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 d f}+\frac {a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{4 d f} \]
1/8*a^2*(12*c^2+16*c*d+7*d^2)*arctanh(sin(f*x+e))/f-1/6*a^2*(c^3-8*c^2*d-2 0*c*d^2-8*d^3)*tan(f*x+e)/d/f-1/24*a^2*(2*c*(c-8*d)-21*d^2)*sec(f*x+e)*tan (f*x+e)/f-1/12*a^2*(c-8*d)*(c+d*sec(f*x+e))^2*tan(f*x+e)/d/f+1/4*a^2*(c+d* sec(f*x+e))^3*tan(f*x+e)/d/f
Time = 1.68 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.57 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx=\frac {a^2 \left (3 \left (12 c^2+16 c d+7 d^2\right ) \text {arctanh}(\sin (e+f x))+\tan (e+f x) \left (3 \left (4 c^2+16 c d+7 d^2\right ) \sec (e+f x)+6 d^2 \sec ^3(e+f x)+16 (c+d) \left (3 (c+d)+d \tan ^2(e+f x)\right )\right )\right )}{24 f} \]
(a^2*(3*(12*c^2 + 16*c*d + 7*d^2)*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(3* (4*c^2 + 16*c*d + 7*d^2)*Sec[e + f*x] + 6*d^2*Sec[e + f*x]^3 + 16*(c + d)* (3*(c + d) + d*Tan[e + f*x]^2))))/(24*f)
Time = 0.40 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.53, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3042, 4475, 101, 25, 27, 90, 60, 60, 45, 218}
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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2 \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4475 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {(\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))^2}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (-\frac {\int -\frac {a^2 (\sec (e+f x) a+a)^{3/2} \left (4 c^2+2 d c+d^2+d (5 c+2 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{4 a^2}-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {a^2 (\sec (e+f x) a+a)^{3/2} \left (4 c^2+2 d c+d^2+d (5 c+2 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{4 a^2}-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{4} \int \frac {(\sec (e+f x) a+a)^{3/2} \left (4 c^2+2 d c+d^2+d (5 c+2 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{4} \left (\frac {1}{3} \left (12 c^2+16 c d+7 d^2\right ) \int \frac {(\sec (e+f x) a+a)^{3/2}}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {d (5 c+2 d) \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2}}{3 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{4} \left (\frac {1}{3} \left (12 c^2+16 c d+7 d^2\right ) \left (\frac {3}{2} a \int \frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d (5 c+2 d) \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2}}{3 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{4} \left (\frac {1}{3} \left (12 c^2+16 c d+7 d^2\right ) \left (\frac {3}{2} a \left (a \int \frac {1}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}d\sec (e+f x)-\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}{a}\right )-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d (5 c+2 d) \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2}}{3 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{4} \left (\frac {1}{3} \left (12 c^2+16 c d+7 d^2\right ) \left (\frac {3}{2} a \left (2 a \int \frac {1}{-\frac {(a-a \sec (e+f x)) a}{\sec (e+f x) a+a}-a}d\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {\sec (e+f x) a+a}}-\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}{a}\right )-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d (5 c+2 d) \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2}}{3 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{4} \left (\frac {1}{3} \left (12 c^2+16 c d+7 d^2\right ) \left (\frac {3}{2} a \left (-2 \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a \sec (e+f x)+a}}\right )-\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}{a}\right )-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d (5 c+2 d) \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2}}{3 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}{4 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^2*(-1/4*(d*Sqrt[a - a*Sec[e + f*x]]*(a + a*Sec[e + f*x])^(5/2)*(c + d *Sec[e + f*x]))/a^2 + (-1/3*(d*(5*c + 2*d)*Sqrt[a - a*Sec[e + f*x]]*(a + a *Sec[e + f*x])^(5/2))/a^2 + ((12*c^2 + 16*c*d + 7*d^2)*(-1/2*(Sqrt[a - a*S ec[e + f*x]]*(a + a*Sec[e + f*x])^(3/2))/a + (3*a*(-2*ArcTan[Sqrt[a - a*Se c[e + f*x]]/Sqrt[a + a*Sec[e + f*x]]] - (Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])/a))/2))/3)/4)*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]* Sqrt[a + a*Sec[e + f*x]]))
3.2.95.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[a ^2*g*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*((c + d*x)^n/Sqrt[a - b*x]), x ], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[ b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p, 1] || In tegerQ[m - 1/2])
Time = 4.36 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.15
| method | result | size |
| parts | \(-\frac {\left (2 a^{2} c d +2 a^{2} d^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (2 a^{2} c^{2}+2 a^{2} c d \right ) \tan \left (f x +e \right )}{f}+\frac {\left (a^{2} c^{2}+4 a^{2} c d +a^{2} d^{2}\right ) \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right ) a^{2} c^{2}}{f}+\frac {a^{2} d^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(202\) |
| parallelrisch | \(\frac {4 a^{2} \left (-\frac {3 \left (c^{2}+\frac {4}{3} c d +\frac {7}{12} d^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 f x +4 e \right )}{4}+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2}+\frac {3 \left (c^{2}+\frac {4}{3} c d +\frac {7}{12} d^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 f x +4 e \right )}{4}+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2}+\left (c +\frac {4 d}{3}\right ) \left (c +d \right ) \sin \left (2 f x +2 e \right )+\frac {\left (c +\frac {d}{2}\right ) \left (c +\frac {7 d}{2}\right ) \sin \left (3 f x +3 e \right )}{4}+\frac {\left (c +\frac {2 d}{3}\right ) \left (c +d \right ) \sin \left (4 f x +4 e \right )}{2}+\frac {\left (c +\frac {5 d}{2}\right ) \left (c +\frac {3 d}{2}\right ) \sin \left (f x +e \right )}{4}\right )}{f \left (3+\cos \left (4 f x +4 e \right )+4 \cos \left (2 f x +2 e \right )\right )}\) | \(208\) |
| norman | \(\frac {\frac {11 a^{2} \left (12 c^{2}+16 c d +7 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{12 f}-\frac {a^{2} \left (12 c^{2}+16 c d +7 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}+\frac {a^{2} \left (20 c^{2}+48 c d +25 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a^{2} \left (156 c^{2}+272 c d +83 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{12 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4}}-\frac {a^{2} \left (12 c^{2}+16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {a^{2} \left (12 c^{2}+16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(223\) |
| derivativedivides | \(\frac {a^{2} c^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 a^{2} c d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+a^{2} d^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+2 a^{2} c^{2} \tan \left (f x +e \right )+4 a^{2} c d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 a^{2} d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+a^{2} c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+2 a^{2} c d \tan \left (f x +e \right )+a^{2} d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}\) | \(270\) |
| default | \(\frac {a^{2} c^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 a^{2} c d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+a^{2} d^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+2 a^{2} c^{2} \tan \left (f x +e \right )+4 a^{2} c d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 a^{2} d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+a^{2} c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+2 a^{2} c d \tan \left (f x +e \right )+a^{2} d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}\) | \(270\) |
| risch | \(-\frac {i a^{2} \left (12 c^{2} {\mathrm e}^{7 i \left (f x +e \right )}+48 c d \,{\mathrm e}^{7 i \left (f x +e \right )}+21 d^{2} {\mathrm e}^{7 i \left (f x +e \right )}-48 c^{2} {\mathrm e}^{6 i \left (f x +e \right )}-48 c d \,{\mathrm e}^{6 i \left (f x +e \right )}+12 c^{2} {\mathrm e}^{5 i \left (f x +e \right )}+48 c d \,{\mathrm e}^{5 i \left (f x +e \right )}+45 d^{2} {\mathrm e}^{5 i \left (f x +e \right )}-144 c^{2} {\mathrm e}^{4 i \left (f x +e \right )}-240 c d \,{\mathrm e}^{4 i \left (f x +e \right )}-96 d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-12 c^{2} {\mathrm e}^{3 i \left (f x +e \right )}-48 c d \,{\mathrm e}^{3 i \left (f x +e \right )}-45 d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-144 c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-272 c d \,{\mathrm e}^{2 i \left (f x +e \right )}-128 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-12 c^{2} {\mathrm e}^{i \left (f x +e \right )}-48 d \,{\mathrm e}^{i \left (f x +e \right )} c -21 d^{2} {\mathrm e}^{i \left (f x +e \right )}-48 c^{2}-80 c d -32 d^{2}\right )}{12 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{4}}+\frac {3 a^{2} c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 f}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c d}{f}+\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{2}}{8 f}-\frac {3 a^{2} c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 f}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c d}{f}-\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) d^{2}}{8 f}\) | \(454\) |
-(2*a^2*c*d+2*a^2*d^2)/f*(-2/3-1/3*sec(f*x+e)^2)*tan(f*x+e)+(2*a^2*c^2+2*a ^2*c*d)/f*tan(f*x+e)+(a^2*c^2+4*a^2*c*d+a^2*d^2)/f*(1/2*sec(f*x+e)*tan(f*x +e)+1/2*ln(sec(f*x+e)+tan(f*x+e)))+1/f*ln(sec(f*x+e)+tan(f*x+e))*a^2*c^2+a ^2*d^2/f*(-(-1/4*sec(f*x+e)^3-3/8*sec(f*x+e))*tan(f*x+e)+3/8*ln(sec(f*x+e) +tan(f*x+e)))
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.19 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx=\frac {3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (6 \, a^{2} d^{2} + 16 \, {\left (3 \, a^{2} c^{2} + 5 \, a^{2} c d + 2 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 16 \, {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \]
1/48*(3*(12*a^2*c^2 + 16*a^2*c*d + 7*a^2*d^2)*cos(f*x + e)^4*log(sin(f*x + e) + 1) - 3*(12*a^2*c^2 + 16*a^2*c*d + 7*a^2*d^2)*cos(f*x + e)^4*log(-sin (f*x + e) + 1) + 2*(6*a^2*d^2 + 16*(3*a^2*c^2 + 5*a^2*c*d + 2*a^2*d^2)*cos (f*x + e)^3 + 3*(4*a^2*c^2 + 16*a^2*c*d + 7*a^2*d^2)*cos(f*x + e)^2 + 16*( a^2*c*d + a^2*d^2)*cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx=a^{2} \left (\int c^{2} \sec {\left (e + f x \right )}\, dx + \int 2 c^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int c^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 2 d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{2}{\left (e + f x \right )}\, dx + \int 4 c d \sec ^{3}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
a**2*(Integral(c**2*sec(e + f*x), x) + Integral(2*c**2*sec(e + f*x)**2, x) + Integral(c**2*sec(e + f*x)**3, x) + Integral(d**2*sec(e + f*x)**3, x) + Integral(2*d**2*sec(e + f*x)**4, x) + Integral(d**2*sec(e + f*x)**5, x) + Integral(2*c*d*sec(e + f*x)**2, x) + Integral(4*c*d*sec(e + f*x)**3, x) + Integral(2*c*d*sec(e + f*x)**4, x))
Time = 0.21 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.84 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx=\frac {32 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c d + 32 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} d^{2} - 3 \, a^{2} d^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{2} c^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 48 \, a^{2} c d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{2} d^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 48 \, a^{2} c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 96 \, a^{2} c^{2} \tan \left (f x + e\right ) + 96 \, a^{2} c d \tan \left (f x + e\right )}{48 \, f} \]
1/48*(32*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^2*c*d + 32*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^2*d^2 - 3*a^2*d^2*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e)) /(sin(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log (sin(f*x + e) - 1)) - 12*a^2*c^2*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - lo g(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) - 48*a^2*c*d*(2*sin(f*x + e)/ (sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) - 12 *a^2*d^2*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + lo g(sin(f*x + e) - 1)) + 48*a^2*c^2*log(sec(f*x + e) + tan(f*x + e)) + 96*a^ 2*c^2*tan(f*x + e) + 96*a^2*c*d*tan(f*x + e))/f
Time = 0.35 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.82 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx=\frac {3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (36 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 48 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 21 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 132 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 176 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 77 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 156 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 272 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 83 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 60 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 144 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 75 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \]
1/24*(3*(12*a^2*c^2 + 16*a^2*c*d + 7*a^2*d^2)*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 3*(12*a^2*c^2 + 16*a^2*c*d + 7*a^2*d^2)*log(abs(tan(1/2*f*x + 1/2 *e) - 1)) - 2*(36*a^2*c^2*tan(1/2*f*x + 1/2*e)^7 + 48*a^2*c*d*tan(1/2*f*x + 1/2*e)^7 + 21*a^2*d^2*tan(1/2*f*x + 1/2*e)^7 - 132*a^2*c^2*tan(1/2*f*x + 1/2*e)^5 - 176*a^2*c*d*tan(1/2*f*x + 1/2*e)^5 - 77*a^2*d^2*tan(1/2*f*x + 1/2*e)^5 + 156*a^2*c^2*tan(1/2*f*x + 1/2*e)^3 + 272*a^2*c*d*tan(1/2*f*x + 1/2*e)^3 + 83*a^2*d^2*tan(1/2*f*x + 1/2*e)^3 - 60*a^2*c^2*tan(1/2*f*x + 1/ 2*e) - 144*a^2*c*d*tan(1/2*f*x + 1/2*e) - 75*a^2*d^2*tan(1/2*f*x + 1/2*e)) /(tan(1/2*f*x + 1/2*e)^2 - 1)^4)/f
Time = 17.20 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.35 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx=\frac {\left (-3\,a^2\,c^2-4\,a^2\,c\,d-\frac {7\,a^2\,d^2}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (11\,a^2\,c^2+\frac {44\,a^2\,c\,d}{3}+\frac {77\,a^2\,d^2}{12}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-13\,a^2\,c^2-\frac {68\,a^2\,c\,d}{3}-\frac {83\,a^2\,d^2}{12}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (5\,a^2\,c^2+12\,a^2\,c\,d+\frac {25\,a^2\,d^2}{4}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{2\,\left (6\,c^2+8\,c\,d+\frac {7\,d^2}{2}\right )}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{4\,f} \]
(tan(e/2 + (f*x)/2)*(5*a^2*c^2 + (25*a^2*d^2)/4 + 12*a^2*c*d) - tan(e/2 + (f*x)/2)^7*(3*a^2*c^2 + (7*a^2*d^2)/4 + 4*a^2*c*d) + tan(e/2 + (f*x)/2)^5* (11*a^2*c^2 + (77*a^2*d^2)/12 + (44*a^2*c*d)/3) - tan(e/2 + (f*x)/2)^3*(13 *a^2*c^2 + (83*a^2*d^2)/12 + (68*a^2*c*d)/3))/(f*(6*tan(e/2 + (f*x)/2)^4 - 4*tan(e/2 + (f*x)/2)^2 - 4*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^8 + 1)) + (a^2*atanh((tan(e/2 + (f*x)/2)*(16*c*d + 12*c^2 + 7*d^2))/(2*(8*c*d + 6*c^2 + (7*d^2)/2)))*(16*c*d + 12*c^2 + 7*d^2))/(4*f)