Integrand size = 31, antiderivative size = 193 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {d^2 \left (12 c^2-16 c d+7 d^2\right ) \text {arctanh}(\sin (e+f x))}{2 a^2 f}+\frac {(c-d) (c+8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )+d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f} \]
1/2*d^2*(12*c^2-16*c*d+7*d^2)*arctanh(sin(f*x+e))/a^2/f+1/3*(c-d)*(c+8*d)* (c+d*sec(f*x+e))^2*tan(f*x+e)/f/(a^2+a^2*sec(f*x+e))+1/3*(c-d)*(c+d*sec(f* x+e))^3*tan(f*x+e)/f/(a+a*sec(f*x+e))^2-1/6*d*(4*c^3+32*c^2*d-80*c*d^2+32* d^3+d*(2*c^2+16*c*d-21*d^2)*sec(f*x+e))*tan(f*x+e)/a^2/f
Time = 5.10 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.61 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {-24 d^2 \left (12 c^2-16 c d+7 d^2\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (2 c^4+16 c^3 d-60 c^2 d^2+112 c d^3-37 d^4+6 \left (c^4+2 c^3 d-12 c^2 d^2+28 c d^3-10 d^4\right ) \cos (e+f x)+\left (2 c^4+16 c^3 d-60 c^2 d^2+112 c d^3-43 d^4\right ) \cos (2 (e+f x))+2 c^4 \cos (3 (e+f x))+4 c^3 d \cos (3 (e+f x))-24 c^2 d^2 \cos (3 (e+f x))+40 c d^3 \cos (3 (e+f x))-16 d^4 \cos (3 (e+f x))\right ) \sec ^2(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{12 a^2 f (1+\cos (e+f x))^2} \]
(-24*d^2*(12*c^2 - 16*c*d + 7*d^2)*Cos[(e + f*x)/2]^4*(Log[Cos[(e + f*x)/2 ] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 2*Cos[ (e + f*x)/2]*(2*c^4 + 16*c^3*d - 60*c^2*d^2 + 112*c*d^3 - 37*d^4 + 6*(c^4 + 2*c^3*d - 12*c^2*d^2 + 28*c*d^3 - 10*d^4)*Cos[e + f*x] + (2*c^4 + 16*c^3 *d - 60*c^2*d^2 + 112*c*d^3 - 43*d^4)*Cos[2*(e + f*x)] + 2*c^4*Cos[3*(e + f*x)] + 4*c^3*d*Cos[3*(e + f*x)] - 24*c^2*d^2*Cos[3*(e + f*x)] + 40*c*d^3* Cos[3*(e + f*x)] - 16*d^4*Cos[3*(e + f*x)])*Sec[e + f*x]^2*Sin[(e + f*x)/2 ])/(12*a^2*f*(1 + Cos[e + f*x])^2)
Time = 0.48 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.56, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3042, 4475, 109, 25, 27, 167, 27, 164, 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.
| \(\displaystyle \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a \sec (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 4475 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {(c+d \sec (e+f x))^4}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2}}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (-\frac {\int -\frac {a^2 (c+d \sec (e+f x))^2 \left (c^2+5 d c-3 d^2-(2 c-5 d) d \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2}}d\sec (e+f x)}{3 a^3}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {a^2 (c+d \sec (e+f x))^2 \left (c^2+5 d c-3 d^2-(2 c-5 d) d \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2}}d\sec (e+f x)}{3 a^3}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {(c+d \sec (e+f x))^2 \left (c^2+5 d c-3 d^2-(2 c-5 d) d \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2}}d\sec (e+f x)}{3 a}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {a^2 d (c+d \sec (e+f x)) \left ((19 c-16 d) d-\left (2 c^2+16 d c-21 d^2\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}d\sec (e+f x)}{a^3}-\frac {(c-d) (c+8 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{a^2 \sqrt {a \sec (e+f x)+a}}}{3 a}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {d \int \frac {(c+d \sec (e+f x)) \left ((19 c-16 d) d-\left (2 c^2+16 d c-21 d^2\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}d\sec (e+f x)}{a}-\frac {(c-d) (c+8 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{a^2 \sqrt {a \sec (e+f x)+a}}}{3 a}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {d \left (\frac {3}{2} d \left (12 c^2-16 c d+7 d^2\right ) \int \frac {1}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}d\sec (e+f x)+\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a} \left (d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)+4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )\right )}{2 a^2}\right )}{a}-\frac {(c-d) (c+8 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{a^2 \sqrt {a \sec (e+f x)+a}}}{3 a}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {d \left (3 d \left (12 c^2-16 c d+7 d^2\right ) \int \frac {1}{-\frac {(a-a \sec (e+f x)) a}{\sec (e+f x) a+a}-a}d\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {\sec (e+f x) a+a}}+\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a} \left (d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)+4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )\right )}{2 a^2}\right )}{a}-\frac {(c-d) (c+8 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{a^2 \sqrt {a \sec (e+f x)+a}}}{3 a}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {d \left (\frac {\sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a} \left (d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)+4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )\right )}{2 a^2}-\frac {3 d \left (12 c^2-16 c d+7 d^2\right ) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a \sec (e+f x)+a}}\right )}{a}\right )}{a}-\frac {(c-d) (c+8 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{a^2 \sqrt {a \sec (e+f x)+a}}}{3 a}-\frac {(c-d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{3 a^2 (a \sec (e+f x)+a)^{3/2}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^2*(-1/3*((c - d)*Sqrt[a - a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3)/(a^ 2*(a + a*Sec[e + f*x])^(3/2)) + (-(((c - d)*(c + 8*d)*Sqrt[a - a*Sec[e + f *x]]*(c + d*Sec[e + f*x])^2)/(a^2*Sqrt[a + a*Sec[e + f*x]])) + (d*((-3*d*( 12*c^2 - 16*c*d + 7*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a + a*Sec[e + f*x]]])/a + (Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]*(4*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3) + d*(2*c^2 + 16*c*d - 21*d^2)*Sec[e + f*x]))/ (2*a^2)))/a)/(3*a))*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*S ec[e + f*x]]))
3.3.18.3.1 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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.01 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.42
| method | result | size |
| parallelrisch | \(\frac {-12 \left (1+\cos \left (2 f x +2 e \right )\right ) \left (c^{2}-\frac {4}{3} c d +\frac {7}{12} d^{2}\right ) d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+12 \left (1+\cos \left (2 f x +2 e \right )\right ) \left (c^{2}-\frac {4}{3} c d +\frac {7}{12} d^{2}\right ) d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (\frac {1}{3} c^{4}+\frac {8}{3} c^{3} d -10 c^{2} d^{2}+\frac {56}{3} c \,d^{3}-\frac {43}{6} d^{4}\right ) \cos \left (2 f x +2 e \right )+\left (\frac {1}{3} c^{4}+\frac {2}{3} c^{3} d -4 c^{2} d^{2}+\frac {20}{3} c \,d^{3}-\frac {8}{3} d^{4}\right ) \cos \left (3 f x +3 e \right )+\left (c^{4}+2 c^{3} d -12 c^{2} d^{2}+28 c \,d^{3}-10 d^{4}\right ) \cos \left (f x +e \right )+\frac {c^{4}}{3}+\frac {8 c^{3} d}{3}-10 c^{2} d^{2}+\frac {56 c \,d^{3}}{3}-\frac {37 d^{4}}{6}\right )}{2 f \,a^{2} \left (1+\cos \left (2 f x +2 e \right )\right )}\) | \(275\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{4}}{3}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{3} d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{2} d^{2}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c \,d^{3}}{3}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{4}}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{4}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{3} d -18 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2} d^{2}+20 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c \,d^{3}-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{4}-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}+\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{2 f \,a^{2}}\) | \(317\) |
| default | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{4}}{3}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{3} d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{2} d^{2}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c \,d^{3}}{3}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{4}}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{4}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{3} d -18 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2} d^{2}+20 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c \,d^{3}-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{4}-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}+\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{2 f \,a^{2}}\) | \(317\) |
| norman | \(\frac {-\frac {\left (c^{4}-4 c^{3} d +6 c^{2} d^{2}-4 c \,d^{3}+d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{6 a f}+\frac {\left (c^{4}+4 c^{3} d -18 c^{2} d^{2}+36 c \,d^{3}-13 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a f}-\frac {\left (3 c^{4}+4 c^{3} d -30 c^{2} d^{2}+44 c \,d^{3}-18 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f}+\frac {\left (7 c^{4}-4 c^{3} d -30 c^{2} d^{2}+44 c \,d^{3}-17 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{6 a f}+\frac {\left (11 c^{4}+28 c^{3} d -150 c^{2} d^{2}+244 c \,d^{3}-100 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}-\frac {\left (13 c^{4}+44 c^{3} d -210 c^{2} d^{2}+380 c \,d^{3}-149 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{6 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4} a}-\frac {d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{2} f}+\frac {d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{2} f}\) | \(390\) |
| risch | \(\frac {i \left (-36 c^{2} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-120 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+256 c \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+16 c^{3} d \,{\mathrm e}^{2 i \left (f x +e \right )}+144 c \,d^{3} {\mathrm e}^{5 i \left (f x +e \right )}+224 c \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-108 c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+192 c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+48 c \,d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+8 c^{3} d \,{\mathrm e}^{4 i \left (f x +e \right )}-108 c^{2} d^{2} {\mathrm e}^{5 i \left (f x +e \right )}-216 c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+336 c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-132 c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-32 d^{4}+4 c^{4}+48 c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}+24 c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}+24 c^{3} d \,{\mathrm e}^{5 i \left (f x +e \right )}-63 d^{4} {\mathrm e}^{5 i \left (f x +e \right )}-75 d^{4} {\mathrm e}^{i \left (f x +e \right )}-97 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{6 i \left (f x +e \right )}-126 d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-98 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{i \left (f x +e \right )}-21 d^{4} {\mathrm e}^{6 i \left (f x +e \right )}+14 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+16 c^{4} {\mathrm e}^{4 i \left (f x +e \right )}+8 c^{3} d -48 c^{2} d^{2}+80 c \,d^{3}+12 c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{5 i \left (f x +e \right )}\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2}}+\frac {6 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2}}{a^{2} f}-\frac {8 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c}{a^{2} f}+\frac {7 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a^{2} f}-\frac {6 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2}}{a^{2} f}+\frac {8 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c}{a^{2} f}-\frac {7 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a^{2} f}\) | \(655\) |
1/2*(-12*(1+cos(2*f*x+2*e))*(c^2-4/3*c*d+7/12*d^2)*d^2*ln(tan(1/2*f*x+1/2* e)-1)+12*(1+cos(2*f*x+2*e))*(c^2-4/3*c*d+7/12*d^2)*d^2*ln(tan(1/2*f*x+1/2* e)+1)+sec(1/2*f*x+1/2*e)^2*tan(1/2*f*x+1/2*e)*((1/3*c^4+8/3*c^3*d-10*c^2*d ^2+56/3*c*d^3-43/6*d^4)*cos(2*f*x+2*e)+(1/3*c^4+2/3*c^3*d-4*c^2*d^2+20/3*c *d^3-8/3*d^4)*cos(3*f*x+3*e)+(c^4+2*c^3*d-12*c^2*d^2+28*c*d^3-10*d^4)*cos( f*x+e)+1/3*c^4+8/3*c^3*d-10*c^2*d^2+56/3*c*d^3-37/6*d^4))/f/a^2/(1+cos(2*f *x+2*e))
Time = 0.31 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.87 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {3 \, {\left ({\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left ({\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (3 \, d^{4} + 4 \, {\left (c^{4} + 2 \, c^{3} d - 12 \, c^{2} d^{2} + 20 \, c d^{3} - 8 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (2 \, c^{4} + 16 \, c^{3} d - 60 \, c^{2} d^{2} + 112 \, c d^{3} - 43 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (4 \, c d^{3} - d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2}\right )}} \]
1/12*(3*((12*c^2*d^2 - 16*c*d^3 + 7*d^4)*cos(f*x + e)^4 + 2*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*cos(f*x + e)^3 + (12*c^2*d^2 - 16*c*d^3 + 7*d^4)*cos(f*x + e)^2)*log(sin(f*x + e) + 1) - 3*((12*c^2*d^2 - 16*c*d^3 + 7*d^4)*cos(f* x + e)^4 + 2*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*cos(f*x + e)^3 + (12*c^2*d^2 - 16*c*d^3 + 7*d^4)*cos(f*x + e)^2)*log(-sin(f*x + e) + 1) + 2*(3*d^4 + 4* (c^4 + 2*c^3*d - 12*c^2*d^2 + 20*c*d^3 - 8*d^4)*cos(f*x + e)^3 + (2*c^4 + 16*c^3*d - 60*c^2*d^2 + 112*c*d^3 - 43*d^4)*cos(f*x + e)^2 + 6*(4*c*d^3 - d^4)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^4 + 2*a^2*f*cos(f*x + e)^3 + a^2*f*cos(f*x + e)^2)
\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {\int \frac {c^{4} \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{4} \sec ^{5}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {4 c d^{3} \sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {6 c^{2} d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {4 c^{3} d \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
(Integral(c**4*sec(e + f*x)/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + I ntegral(d**4*sec(e + f*x)**5/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + Integral(4*c*d**3*sec(e + f*x)**4/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + Integral(6*c**2*d**2*sec(e + f*x)**3/(sec(e + f*x)**2 + 2*sec(e + f*x ) + 1), x) + Integral(4*c**3*d*sec(e + f*x)**2/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x))/a**2
Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (184) = 368\).
Time = 0.23 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.78 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=-\frac {d^{4} {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - 4 \, c d^{3} {\left (\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (f x + e\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + 6 \, c^{2} d^{2} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - \frac {4 \, c^{3} d {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \]
-1/6*(d^4*(6*(3*sin(f*x + e)/(cos(f*x + e) + 1) - 5*sin(f*x + e)^3/(cos(f* x + e) + 1)^3)/(a^2 - 2*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin( f*x + e)^4/(cos(f*x + e) + 1)^4) + (21*sin(f*x + e)/(cos(f*x + e) + 1) + s in(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 - 21*log(sin(f*x + e)/(cos(f*x + e ) + 1) + 1)/a^2 + 21*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^2) - 4*c*d ^3*((15*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1 )^3)/a^2 - 12*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^2 + 12*log(sin(f* x + e)/(cos(f*x + e) + 1) - 1)/a^2 + 12*sin(f*x + e)/((a^2 - a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1))) + 6*c^2*d^2*((9*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 - 6*log( sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^2 + 6*log(sin(f*x + e)/(cos(f*x + e ) + 1) - 1)/a^2) - 4*c^3*d*(3*sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 - c^4*(3*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2)/f
Time = 0.36 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.86 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {\frac {3 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (8 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} - \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 54 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 60 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 21 \, a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6}}}{6 \, f} \]
1/6*(3*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*log(abs(tan(1/2*f*x + 1/2*e) + 1))/ a^2 - 3*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*log(abs(tan(1/2*f*x + 1/2*e) - 1)) /a^2 - 6*(8*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 5*d^4*tan(1/2*f*x + 1/2*e)^3 - 8*c*d^3*tan(1/2*f*x + 1/2*e) + 3*d^4*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*a^2) - (a^4*c^4*tan(1/2*f*x + 1/2*e)^3 - 4*a^4*c^3*d*tan( 1/2*f*x + 1/2*e)^3 + 6*a^4*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 - 4*a^4*c*d^3*ta n(1/2*f*x + 1/2*e)^3 + a^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 3*a^4*c^4*tan(1/2* f*x + 1/2*e) - 12*a^4*c^3*d*tan(1/2*f*x + 1/2*e) + 54*a^4*c^2*d^2*tan(1/2* f*x + 1/2*e) - 60*a^4*c*d^3*tan(1/2*f*x + 1/2*e) + 21*a^4*d^4*tan(1/2*f*x + 1/2*e))/a^6)/f
Time = 13.70 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,c\,d^3-3\,d^4\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (8\,c\,d^3-5\,d^4\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^2\right )}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,{\left (c-d\right )}^4}{2\,a^2}-\frac {2\,\left (c+d\right )\,{\left (c-d\right )}^3}{a^2}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\left (c-d\right )}^4}{6\,a^2\,f}+\frac {d^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (12\,c^2-16\,c\,d+7\,d^2\right )}{a^2\,f} \]
(tan(e/2 + (f*x)/2)*(8*c*d^3 - 3*d^4) - tan(e/2 + (f*x)/2)^3*(8*c*d^3 - 5* d^4))/(f*(a^2*tan(e/2 + (f*x)/2)^4 - 2*a^2*tan(e/2 + (f*x)/2)^2 + a^2)) - (tan(e/2 + (f*x)/2)*((3*(c - d)^4)/(2*a^2) - (2*(c + d)*(c - d)^3)/a^2))/f - (tan(e/2 + (f*x)/2)^3*(c - d)^4)/(6*a^2*f) + (d^2*atanh(tan(e/2 + (f*x) /2))*(12*c^2 - 16*c*d + 7*d^2))/(a^2*f)