Integrand size = 31, antiderivative size = 242 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx=\frac {3 a^2 (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) \text {arctanh}(\sin (e+f x))}{8 f}-\frac {a^2 \left (c^4-10 c^3 d-44 c^2 d^2-40 c d^3-12 d^4\right ) \tan (e+f x)}{10 d f}-\frac {a^2 \left (2 c^3-20 c^2 d-57 c d^2-30 d^3\right ) \sec (e+f x) \tan (e+f x)}{40 f}-\frac {a^2 \left (c^2-10 c d-12 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{20 d f}-\frac {a^2 (c-10 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{20 d f}+\frac {a^2 (c+d \sec (e+f x))^4 \tan (e+f x)}{5 d f} \]
3/8*a^2*(2*c+d)*(2*c^2+3*c*d+2*d^2)*arctanh(sin(f*x+e))/f-1/10*a^2*(c^4-10 *c^3*d-44*c^2*d^2-40*c*d^3-12*d^4)*tan(f*x+e)/d/f-1/40*a^2*(2*c^3-20*c^2*d -57*c*d^2-30*d^3)*sec(f*x+e)*tan(f*x+e)/f-1/20*a^2*(c^2-10*c*d-12*d^2)*(c+ d*sec(f*x+e))^2*tan(f*x+e)/d/f-1/20*a^2*(c-10*d)*(c+d*sec(f*x+e))^3*tan(f* x+e)/d/f+1/5*a^2*(c+d*sec(f*x+e))^4*tan(f*x+e)/d/f
Time = 3.45 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.58 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx=\frac {a^2 \left (15 \left (4 c^3+8 c^2 d+7 c d^2+2 d^3\right ) \text {arctanh}(\sin (e+f x))+\tan (e+f x) \left (5 \left (4 c^3+24 c^2 d+21 c d^2+6 d^3\right ) \sec (e+f x)+10 d^2 (3 c+2 d) \sec ^3(e+f x)+8 \left (10 (c+d)^3+5 d (c+d)^2 \tan ^2(e+f x)+d^3 \tan ^4(e+f x)\right )\right )\right )}{40 f} \]
(a^2*(15*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3)*ArcTanh[Sin[e + f*x]] + Tan[e + f*x]*(5*(4*c^3 + 24*c^2*d + 21*c*d^2 + 6*d^3)*Sec[e + f*x] + 10*d^2*(3* c + 2*d)*Sec[e + f*x]^3 + 8*(10*(c + d)^3 + 5*d*(c + d)^2*Tan[e + f*x]^2 + d^3*Tan[e + f*x]^4))))/(40*f)
Time = 0.46 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3042, 4475, 111, 25, 27, 164, 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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2 \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4475 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {(\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))^3}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (-\frac {\int -\frac {a^2 (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x)) \left (5 c^2+2 d c+2 d^2+d (7 c+2 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{5 a^2}-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {a^2 (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x)) \left (5 c^2+2 d c+2 d^2+d (7 c+2 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{5 a^2}-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{5} \int \frac {(\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x)) \left (5 c^2+2 d c+2 d^2+d (7 c+2 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{5} \left (\frac {5}{4} (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) \int \frac {(\sec (e+f x) a+a)^{3/2}}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} \left (2 \left (8 c^2+5 c d+2 d^2\right )+d (7 c+2 d) \sec (e+f x)\right )}{4 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{5} \left (\frac {5}{4} (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) \left (\frac {3}{2} a \int \frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} \left (2 \left (8 c^2+5 c d+2 d^2\right )+d (7 c+2 d) \sec (e+f x)\right )}{4 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{5} \left (\frac {5}{4} (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) \left (\frac {3}{2} a \left (a \int \frac {1}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}d\sec (e+f x)-\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}{a}\right )-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} \left (2 \left (8 c^2+5 c d+2 d^2\right )+d (7 c+2 d) \sec (e+f x)\right )}{4 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{5} \left (\frac {5}{4} (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) \left (\frac {3}{2} a \left (2 a \int \frac {1}{-\frac {(a-a \sec (e+f x)) a}{\sec (e+f x) a+a}-a}d\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {\sec (e+f x) a+a}}-\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}{a}\right )-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} \left (2 \left (8 c^2+5 c d+2 d^2\right )+d (7 c+2 d) \sec (e+f x)\right )}{4 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {1}{5} \left (\frac {5}{4} (2 c+d) \left (2 c^2+3 c d+2 d^2\right ) \left (\frac {3}{2} a \left (-2 \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a \sec (e+f x)+a}}\right )-\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}{a}\right )-\frac {\sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{3/2}}{2 a}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} \left (2 \left (8 c^2+5 c d+2 d^2\right )+d (7 c+2 d) \sec (e+f x)\right )}{4 a^2}\right )-\frac {d \sqrt {a-a \sec (e+f x)} (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))^2}{5 a^2}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^2*(-1/5*(d*Sqrt[a - a*Sec[e + f*x]]*(a + a*Sec[e + f*x])^(5/2)*(c + d *Sec[e + f*x])^2)/a^2 + (-1/4*(d*Sqrt[a - a*Sec[e + f*x]]*(a + a*Sec[e + f *x])^(5/2)*(2*(8*c^2 + 5*c*d + 2*d^2) + d*(7*c + 2*d)*Sec[e + f*x]))/a^2 + (5*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*(-1/2*(Sqrt[a - a*Sec[e + f*x]]*(a + a*Sec[e + f*x])^(3/2))/a + (3*a*(-2*ArcTan[Sqrt[a - a*Sec[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))/4)/5)*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))
3.2.94.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 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 = 5.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.11
| method | result | size |
| parts | \(\frac {\left (3 a^{2} c \,d^{2}+2 a^{2} d^{3}\right ) \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}+\frac {\left (2 a^{2} c^{3}+3 a^{2} c^{2} d \right ) \tan \left (f x +e \right )}{f}-\frac {\left (3 a^{2} c^{2} d +6 a^{2} c \,d^{2}+a^{2} d^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (a^{2} c^{3}+6 a^{2} c^{2} d +3 a^{2} c \,d^{2}\right ) \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right ) a^{2} c^{3}}{f}-\frac {a^{2} d^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}\) | \(269\) |
| norman | \(\frac {\frac {7 a^{2} \left (4 c^{3}+8 c^{2} d +7 c \,d^{2}+2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{2 f}-\frac {3 a^{2} \left (4 c^{3}+8 c^{2} d +7 c \,d^{2}+2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{4 f}-\frac {8 a^{2} \left (15 c^{3}+35 c^{2} d +25 c \,d^{2}+9 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5 f}-\frac {a^{2} \left (20 c^{3}+72 c^{2} d +75 c \,d^{2}+26 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{2} \left (36 c^{3}+104 c^{2} d +79 c \,d^{2}+18 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5}}-\frac {3 a^{2} \left (4 c^{3}+8 c^{2} d +7 c \,d^{2}+2 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {3 a^{2} \left (4 c^{3}+8 c^{2} d +7 c \,d^{2}+2 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(313\) |
| parallelrisch | \(\frac {4 \left (-\frac {15 \left (c +\frac {d}{2}\right ) \left (\frac {\cos \left (5 f x +5 e \right )}{10}+\frac {\cos \left (3 f x +3 e \right )}{2}+\cos \left (f x +e \right )\right ) \left (c^{2}+\frac {3}{2} c d +d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}+\frac {15 \left (c +\frac {d}{2}\right ) \left (\frac {\cos \left (5 f x +5 e \right )}{10}+\frac {\cos \left (3 f x +3 e \right )}{2}+\cos \left (f x +e \right )\right ) \left (c^{2}+\frac {3}{2} c d +d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}+\left (\frac {1}{2} c^{3}+\frac {7}{4} d^{3}+\frac {33}{8} c \,d^{2}+3 c^{2} d \right ) \sin \left (2 f x +2 e \right )+\left (\frac {3}{2} c^{3}+\frac {3}{2} d^{3}+5 c \,d^{2}+\frac {19}{4} c^{2} d \right ) \sin \left (3 f x +3 e \right )+\left (\frac {1}{4} c^{3}+\frac {3}{8} d^{3}+\frac {3}{2} c^{2} d +\frac {21}{16} c \,d^{2}\right ) \sin \left (4 f x +4 e \right )+\left (\frac {1}{2} c^{3}+\frac {3}{10} d^{3}+c \,d^{2}+\frac {5}{4} c^{2} d \right ) \sin \left (5 f x +5 e \right )+\sin \left (f x +e \right ) \left (c +2 d \right ) \left (c^{2}+\frac {3}{2} c d +d^{2}\right )\right ) a^{2}}{f \left (\cos \left (5 f x +5 e \right )+5 \cos \left (3 f x +3 e \right )+10 \cos \left (f x +e \right )\right )}\) | \(313\) |
| derivativedivides | \(\frac {a^{2} c^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 a^{2} c^{2} d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-a^{2} d^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+2 a^{2} c^{3} \tan \left (f x +e \right )+6 a^{2} c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-6 a^{2} c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+2 a^{2} d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+a^{2} c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a^{2} c^{2} d \tan \left (f x +e \right )+3 a^{2} c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a^{2} d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(395\) |
| default | \(\frac {a^{2} c^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 a^{2} c^{2} d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{2} c \,d^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-a^{2} d^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+2 a^{2} c^{3} \tan \left (f x +e \right )+6 a^{2} c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-6 a^{2} c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+2 a^{2} d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+a^{2} c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a^{2} c^{2} d \tan \left (f x +e \right )+3 a^{2} c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a^{2} d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(395\) |
| risch | \(-\frac {i a^{2} \left (-80 c^{3}-160 c \,d^{2}-48 d^{3}-200 c^{2} d -105 c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-800 c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-330 c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-480 c \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-120 c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+330 c \,d^{2} {\mathrm e}^{7 i \left (f x +e \right )}-720 c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}-880 c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}-1280 c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+240 c^{2} d \,{\mathrm e}^{7 i \left (f x +e \right )}+120 c^{2} d \,{\mathrm e}^{9 i \left (f x +e \right )}+105 c \,d^{2} {\mathrm e}^{9 i \left (f x +e \right )}-120 c^{2} d \,{\mathrm e}^{8 i \left (f x +e \right )}-1120 c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-240 c^{2} d \,{\mathrm e}^{3 i \left (f x +e \right )}-240 d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-30 d^{3} {\mathrm e}^{i \left (f x +e \right )}+30 d^{3} {\mathrm e}^{9 i \left (f x +e \right )}-20 c^{3} {\mathrm e}^{i \left (f x +e \right )}-320 c^{3} {\mathrm e}^{6 i \left (f x +e \right )}+140 d^{3} {\mathrm e}^{7 i \left (f x +e \right )}-400 d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-320 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-140 d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-40 c^{3} {\mathrm e}^{3 i \left (f x +e \right )}-80 d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+20 c^{3} {\mathrm e}^{9 i \left (f x +e \right )}-80 c^{3} {\mathrm e}^{8 i \left (f x +e \right )}-480 c^{3} {\mathrm e}^{4 i \left (f x +e \right )}+40 c^{3} {\mathrm e}^{7 i \left (f x +e \right )}\right )}{20 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{5}}-\frac {3 a^{2} c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 f}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2} d}{f}-\frac {21 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c \,d^{2}}{8 f}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) d^{3}}{4 f}+\frac {3 a^{2} c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 f}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2} d}{f}+\frac {21 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c \,d^{2}}{8 f}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{3}}{4 f}\) | \(678\) |
(3*a^2*c*d^2+2*a^2*d^3)/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)))+(2*a^2*c^3+3*a^2*c^2*d)/f*tan(f*x+e)-(3*a^2 *c^2*d+6*a^2*c*d^2+a^2*d^3)/f*(-2/3-1/3*sec(f*x+e)^2)*tan(f*x+e)+(a^2*c^3+ 6*a^2*c^2*d+3*a^2*c*d^2)/f*(1/2*sec(f*x+e)*tan(f*x+e)+1/2*ln(sec(f*x+e)+ta n(f*x+e)))+1/f*ln(sec(f*x+e)+tan(f*x+e))*a^2*c^3-a^2*d^3/f*(-8/15-1/5*sec( f*x+e)^4-4/15*sec(f*x+e)^2)*tan(f*x+e)
Time = 0.29 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.21 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx=\frac {15 \, {\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (8 \, a^{2} d^{3} + 8 \, {\left (10 \, a^{2} c^{3} + 25 \, a^{2} c^{2} d + 20 \, a^{2} c d^{2} + 6 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} + 5 \, {\left (4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 21 \, a^{2} c d^{2} + 6 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 8 \, {\left (5 \, a^{2} c^{2} d + 10 \, a^{2} c d^{2} + 3 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 10 \, {\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{80 \, f \cos \left (f x + e\right )^{5}} \]
1/80*(15*(4*a^2*c^3 + 8*a^2*c^2*d + 7*a^2*c*d^2 + 2*a^2*d^3)*cos(f*x + e)^ 5*log(sin(f*x + e) + 1) - 15*(4*a^2*c^3 + 8*a^2*c^2*d + 7*a^2*c*d^2 + 2*a^ 2*d^3)*cos(f*x + e)^5*log(-sin(f*x + e) + 1) + 2*(8*a^2*d^3 + 8*(10*a^2*c^ 3 + 25*a^2*c^2*d + 20*a^2*c*d^2 + 6*a^2*d^3)*cos(f*x + e)^4 + 5*(4*a^2*c^3 + 24*a^2*c^2*d + 21*a^2*c*d^2 + 6*a^2*d^3)*cos(f*x + e)^3 + 8*(5*a^2*c^2* d + 10*a^2*c*d^2 + 3*a^2*d^3)*cos(f*x + e)^2 + 10*(3*a^2*c*d^2 + 2*a^2*d^3 )*cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^5)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx=a^{2} \left (\int c^{3} \sec {\left (e + f x \right )}\, dx + \int 2 c^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int c^{3} \sec ^{3}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int 2 d^{3} \sec ^{5}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{6}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 6 c d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{2}{\left (e + f x \right )}\, dx + \int 6 c^{2} d \sec ^{3}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
a**2*(Integral(c**3*sec(e + f*x), x) + Integral(2*c**3*sec(e + f*x)**2, x) + Integral(c**3*sec(e + f*x)**3, x) + Integral(d**3*sec(e + f*x)**4, x) + Integral(2*d**3*sec(e + f*x)**5, x) + Integral(d**3*sec(e + f*x)**6, x) + Integral(3*c*d**2*sec(e + f*x)**3, x) + Integral(6*c*d**2*sec(e + f*x)**4 , x) + Integral(3*c*d**2*sec(e + f*x)**5, x) + Integral(3*c**2*d*sec(e + f *x)**2, x) + Integral(6*c**2*d*sec(e + f*x)**3, x) + Integral(3*c**2*d*sec (e + f*x)**4, x))
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (230) = 460\).
Time = 0.22 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.94 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx=\frac {240 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{2} d + 480 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c d^{2} + 16 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{2} d^{3} + 80 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} d^{3} - 45 \, a^{2} c 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 )} - 30 \, a^{2} d^{3} {\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 )} - 60 \, a^{2} c^{3} {\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 )} - 360 \, a^{2} c^{2} 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 )} - 180 \, a^{2} c 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 )} + 240 \, a^{2} c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 480 \, a^{2} c^{3} \tan \left (f x + e\right ) + 720 \, a^{2} c^{2} d \tan \left (f x + e\right )}{240 \, f} \]
1/240*(240*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^2*c^2*d + 480*(tan(f*x + e) ^3 + 3*tan(f*x + e))*a^2*c*d^2 + 16*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^2*d^3 + 80*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^2*d^3 - 45*a^2*c*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)) - 30*a^2*d^3*(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)) - 60 *a^2*c^3*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + lo g(sin(f*x + e) - 1)) - 360*a^2*c^2*d*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) - 180*a^2*c*d^2*(2*sin(f* x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1 )) + 240*a^2*c^3*log(sec(f*x + e) + tan(f*x + e)) + 480*a^2*c^3*tan(f*x + e) + 720*a^2*c^2*d*tan(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (230) = 460\).
Time = 0.39 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.09 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx=\frac {15 \, {\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, {\left (4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (60 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 120 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 105 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 30 \, a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 280 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 560 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 490 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 140 \, a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 480 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1120 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 800 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 288 \, a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 360 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1040 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 790 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 180 \, a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 100 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 360 \, a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 375 \, a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 130 \, a^{2} d^{3} \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 )}^{5}}}{40 \, f} \]
1/40*(15*(4*a^2*c^3 + 8*a^2*c^2*d + 7*a^2*c*d^2 + 2*a^2*d^3)*log(abs(tan(1 /2*f*x + 1/2*e) + 1)) - 15*(4*a^2*c^3 + 8*a^2*c^2*d + 7*a^2*c*d^2 + 2*a^2* d^3)*log(abs(tan(1/2*f*x + 1/2*e) - 1)) - 2*(60*a^2*c^3*tan(1/2*f*x + 1/2* e)^9 + 120*a^2*c^2*d*tan(1/2*f*x + 1/2*e)^9 + 105*a^2*c*d^2*tan(1/2*f*x + 1/2*e)^9 + 30*a^2*d^3*tan(1/2*f*x + 1/2*e)^9 - 280*a^2*c^3*tan(1/2*f*x + 1 /2*e)^7 - 560*a^2*c^2*d*tan(1/2*f*x + 1/2*e)^7 - 490*a^2*c*d^2*tan(1/2*f*x + 1/2*e)^7 - 140*a^2*d^3*tan(1/2*f*x + 1/2*e)^7 + 480*a^2*c^3*tan(1/2*f*x + 1/2*e)^5 + 1120*a^2*c^2*d*tan(1/2*f*x + 1/2*e)^5 + 800*a^2*c*d^2*tan(1/ 2*f*x + 1/2*e)^5 + 288*a^2*d^3*tan(1/2*f*x + 1/2*e)^5 - 360*a^2*c^3*tan(1/ 2*f*x + 1/2*e)^3 - 1040*a^2*c^2*d*tan(1/2*f*x + 1/2*e)^3 - 790*a^2*c*d^2*t an(1/2*f*x + 1/2*e)^3 - 180*a^2*d^3*tan(1/2*f*x + 1/2*e)^3 + 100*a^2*c^3*t an(1/2*f*x + 1/2*e) + 360*a^2*c^2*d*tan(1/2*f*x + 1/2*e) + 375*a^2*c*d^2*t an(1/2*f*x + 1/2*e) + 130*a^2*d^3*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2 *e)^2 - 1)^5)/f
Time = 17.06 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.63 \[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \, dx=\frac {3\,a^2\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{2\,\left (6\,c^3+12\,c^2\,d+\frac {21\,c\,d^2}{2}+3\,d^3\right )}\right )\,\left (2\,c+d\right )\,\left (2\,c^2+3\,c\,d+2\,d^2\right )}{4\,f}-\frac {\left (3\,a^2\,c^3+6\,a^2\,c^2\,d+\frac {21\,a^2\,c\,d^2}{4}+\frac {3\,a^2\,d^3}{2}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+\left (-14\,a^2\,c^3-28\,a^2\,c^2\,d-\frac {49\,a^2\,c\,d^2}{2}-7\,a^2\,d^3\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (24\,a^2\,c^3+56\,a^2\,c^2\,d+40\,a^2\,c\,d^2+\frac {72\,a^2\,d^3}{5}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-18\,a^2\,c^3-52\,a^2\,c^2\,d-\frac {79\,a^2\,c\,d^2}{2}-9\,a^2\,d^3\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (5\,a^2\,c^3+18\,a^2\,c^2\,d+\frac {75\,a^2\,c\,d^2}{4}+\frac {13\,a^2\,d^3}{2}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
(3*a^2*atanh((3*tan(e/2 + (f*x)/2)*(2*c + d)*(3*c*d + 2*c^2 + 2*d^2))/(2*( (21*c*d^2)/2 + 12*c^2*d + 6*c^3 + 3*d^3)))*(2*c + d)*(3*c*d + 2*c^2 + 2*d^ 2))/(4*f) - (tan(e/2 + (f*x)/2)^9*(3*a^2*c^3 + (3*a^2*d^3)/2 + (21*a^2*c*d ^2)/4 + 6*a^2*c^2*d) - tan(e/2 + (f*x)/2)^7*(14*a^2*c^3 + 7*a^2*d^3 + (49* a^2*c*d^2)/2 + 28*a^2*c^2*d) - tan(e/2 + (f*x)/2)^3*(18*a^2*c^3 + 9*a^2*d^ 3 + (79*a^2*c*d^2)/2 + 52*a^2*c^2*d) + tan(e/2 + (f*x)/2)^5*(24*a^2*c^3 + (72*a^2*d^3)/5 + 40*a^2*c*d^2 + 56*a^2*c^2*d) + tan(e/2 + (f*x)/2)*(5*a^2* c^3 + (13*a^2*d^3)/2 + (75*a^2*c*d^2)/4 + 18*a^2*c^2*d))/(f*(5*tan(e/2 + ( f*x)/2)^2 - 10*tan(e/2 + (f*x)/2)^4 + 10*tan(e/2 + (f*x)/2)^6 - 5*tan(e/2 + (f*x)/2)^8 + tan(e/2 + (f*x)/2)^10 - 1))