Integrand size = 31, antiderivative size = 211 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx=\frac {2 d^2 (3 c+2 d) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^2 (c-d)^{7/2} (c+d)^{3/2} f}+\frac {d \left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sec (e+f x))}+\frac {(c-6 d) \tan (e+f x)}{3 a^2 (c-d)^2 f (1+\sec (e+f x)) (c+d \sec (e+f x))}+\frac {\tan (e+f x)}{3 (c-d) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))} \]
2*d^2*(3*c+2*d)*arctanh((c-d)^(1/2)*tan(1/2*f*x+1/2*e)/(c+d)^(1/2))/a^2/(c -d)^(7/2)/(c+d)^(3/2)/f+1/3*d*(c^2-6*c*d-10*d^2)*tan(f*x+e)/a^2/(c-d)^3/(c +d)/f/(c+d*sec(f*x+e))+1/3*(c-6*d)*tan(f*x+e)/a^2/(c-d)^2/f/(1+sec(f*x+e)) /(c+d*sec(f*x+e))+1/3*tan(f*x+e)/(c-d)/f/(a+a*sec(f*x+e))^2/(c+d*sec(f*x+e ))
Result contains complex when optimal does not.
Time = 3.96 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.78 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx=\frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \sec ^4(e+f x) \left (\frac {12 d^2 (3 c+2 d) \arctan \left (\frac {(i \cos (e)+\sin (e)) \left (c \sin (e)+(-d+c \cos (e)) \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) (i \cos (e)+\sin (e))}{(c+d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+(c-d) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )-4 (c-4 d) \cos ^2\left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+\frac {6 d^3 \cos ^3\left (\frac {1}{2} (e+f x)\right ) (-d \sin (e)+c \sin (f x))}{c (c+d) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )}+(c-d) \cos \left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \tan \left (\frac {e}{2}\right )\right )}{3 a^2 (-c+d)^3 f (1+\sec (e+f x))^2 (c+d \sec (e+f x))^2} \]
(2*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*Sec[e + f*x]^4*((12*d^2*(3*c + 2* d)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/ (Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*Cos[(e + f*x)/2]^3*(d + c*C os[e + f*x])*(I*Cos[e] + Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (c - d)*(d + c*Cos[e + f*x])*Sec[e/2]*Sin[(f*x)/2] - 4*(c - 4*d)*Cos[(e + f*x)/2]^2*(d + c*Cos[e + f*x])*Sec[e/2]*Sin[(f*x)/2] + (6* d^3*Cos[(e + f*x)/2]^3*(-(d*Sin[e]) + c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])) + (c - d)*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*Tan[e/2]))/(3*a^2*(-c + d)^3*f*(1 + Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2)
Time = 0.48 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.46, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 4475, 114, 27, 169, 25, 27, 169, 27, 104, 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.
| \(\displaystyle \int \frac {\sec (e+f x)}{(a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^2 \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4475 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {1}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))^2}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {a^2 (c+2 d-2 d \sec (e+f x))}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{a^2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {c+2 d-2 d \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {-\frac {\int -\frac {a^2 ((c-6 d) (c+d)+d (c+4 d) \sec (e+f x))}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a^3 (c-d)}-\frac {(c+4 d) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {a^2 ((c-6 d) (c+d)+d (c+4 d) \sec (e+f x))}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a^3 (c-d)}-\frac {(c+4 d) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {(c-6 d) (c+d)+d (c+4 d) \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a (c-d)}-\frac {(c+4 d) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {-\frac {\int -\frac {3 a^2 d^2 (3 c+2 d)}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}d\sec (e+f x)}{a^3 (c-d)}-\frac {\left (c^2-6 c d-10 d^2\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {(c+4 d) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {3 d^2 (3 c+2 d) \int \frac {1}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}d\sec (e+f x)}{a (c-d)}-\frac {\left (c^2-6 c d-10 d^2\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {(c+4 d) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {6 d^2 (3 c+2 d) \int \frac {1}{a (c-d)+\frac {a (c+d) (\sec (e+f x) a+a)}{a-a \sec (e+f x)}}d\frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {a-a \sec (e+f x)}}}{a (c-d)}-\frac {\left (c^2-6 c d-10 d^2\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {(c+4 d) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {6 d^2 (3 c+2 d) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a^2 (c-d)^{3/2} \sqrt {c+d}}-\frac {\left (c^2-6 c d-10 d^2\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {(c+4 d) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^2*((d*Sqrt[a - a*Sec[e + f*x]])/(a^2*(c^2 - d^2)*(a + a*Sec[e + f*x]) ^(3/2)*(c + d*Sec[e + f*x])) + (-1/3*((c + 4*d)*Sqrt[a - a*Sec[e + f*x]])/ (a^2*(c - d)*(a + a*Sec[e + f*x])^(3/2)) + ((6*d^2*(3*c + 2*d)*ArcTan[(Sqr t[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])] )/(a^2*(c - d)^(3/2)*Sqrt[c + d]) - ((c^2 - 6*c*d - 10*d^2)*Sqrt[a - a*Sec [e + f*x]])/(a^2*(c - d)*Sqrt[a + a*Sec[e + f*x]]))/(3*a*(c - d)))/(c^2 - d^2))*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))
3.3.23.3.1 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[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 = 0.79 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(\frac {-\frac {\frac {c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )}-\frac {4 d^{2} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (3 c +2 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3}}}{2 f \,a^{2}}\) | \(203\) |
| default | \(\frac {-\frac {\frac {c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )}-\frac {4 d^{2} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (3 c +2 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3}}}{2 f \,a^{2}}\) | \(203\) |
| risch | \(\frac {2 i \left (-3 c^{4} {\mathrm e}^{4 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{4 i \left (f x +e \right )}+9 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+3 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}-3 c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}+27 c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+21 c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+9 d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-5 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{2 i \left (f x +e \right )}+41 c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+39 c \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+9 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-3 c^{4} {\mathrm e}^{i \left (f x +e \right )}+8 c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}+27 c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+25 c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+3 d^{4} {\mathrm e}^{i \left (f x +e \right )}-2 c^{4}+6 c^{3} d +8 c^{2} d^{2}+3 c \,d^{3}\right )}{3 \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right ) \left (-c^{2}+d^{2}\right ) c f \,a^{2} \left (-c +d \right )^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}+\frac {2 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}-\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}-\frac {2 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}\) | \(733\) |
1/2/f/a^2*(-1/(c^2-2*c*d+d^2)/(c-d)*(1/3*c*tan(1/2*f*x+1/2*e)^3-1/3*d*tan( 1/2*f*x+1/2*e)^3-c*tan(1/2*f*x+1/2*e)+5*d*tan(1/2*f*x+1/2*e))-4*d^2/(c-d)^ 3*(-d/(c+d)*tan(1/2*f*x+1/2*e)/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^ 2*d-c-d)-(3*c+2*d)/(c+d)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2 *e)/((c+d)*(c-d))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (196) = 392\).
Time = 0.33 (sec) , antiderivative size = 1242, normalized size of antiderivative = 5.89 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx=\text {Too large to display} \]
[-1/6*(3*(3*c*d^3 + 2*d^4 + (3*c^2*d^2 + 2*c*d^3)*cos(f*x + e)^3 + (6*c^2* d^2 + 7*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (3*c^2*d^2 + 8*c*d^3 + 4*d^4)*cos( f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2 )/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) - 2*(c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*d^5 + (2*c^5 - 6*c^4*d - 10*c^3*d^2 + 3*c^2*d^ 3 + 8*c*d^4 + 3*d^5)*cos(f*x + e)^2 + (c^5 - 4*c^4*d - 14*c^3*d^2 - 10*c^2 *d^3 + 13*c*d^4 + 14*d^5)*cos(f*x + e))*sin(f*x + e))/((a^2*c^7 - 2*a^2*c^ 6*d - a^2*c^5*d^2 + 4*a^2*c^4*d^3 - a^2*c^3*d^4 - 2*a^2*c^2*d^5 + a^2*c*d^ 6)*f*cos(f*x + e)^3 + (2*a^2*c^7 - 3*a^2*c^6*d - 4*a^2*c^5*d^2 + 7*a^2*c^4 *d^3 + 2*a^2*c^3*d^4 - 5*a^2*c^2*d^5 + a^2*d^7)*f*cos(f*x + e)^2 + (a^2*c^ 7 - 5*a^2*c^5*d^2 + 2*a^2*c^4*d^3 + 7*a^2*c^3*d^4 - 4*a^2*c^2*d^5 - 3*a^2* c*d^6 + 2*a^2*d^7)*f*cos(f*x + e) + (a^2*c^6*d - 2*a^2*c^5*d^2 - a^2*c^4*d ^3 + 4*a^2*c^3*d^4 - a^2*c^2*d^5 - 2*a^2*c*d^6 + a^2*d^7)*f), 1/3*(3*(3*c* d^3 + 2*d^4 + (3*c^2*d^2 + 2*c*d^3)*cos(f*x + e)^3 + (6*c^2*d^2 + 7*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (3*c^2*d^2 + 8*c*d^3 + 4*d^4)*cos(f*x + e))*sqrt (-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*si n(f*x + e))) + (c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*d^5 + (2*c^5 - 6*c^4*d - 10*c^3*d^2 + 3*c^2*d^3 + 8*c*d^4 + 3*d^5)*cos(f*x + e)^2 + (c ^5 - 4*c^4*d - 14*c^3*d^2 - 10*c^2*d^3 + 13*c*d^4 + 14*d^5)*cos(f*x + e...
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sec ^{2}{\left (e + f x \right )} + 2 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{3}{\left (e + f x \right )} + 4 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{4}{\left (e + f x \right )} + 2 d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx}{a^{2}} \]
Integral(sec(e + f*x)/(c**2*sec(e + f*x)**2 + 2*c**2*sec(e + f*x) + c**2 + 2*c*d*sec(e + f*x)**3 + 4*c*d*sec(e + f*x)**2 + 2*c*d*sec(e + f*x) + d**2 *sec(e + f*x)**4 + 2*d**2*sec(e + f*x)**3 + d**2*sec(e + f*x)**2), x)/a**2
Exception generated. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (196) = 392\).
Time = 0.33 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.25 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx=\frac {\frac {12 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}} + \frac {12 \, {\left (3 \, c d^{2} + 2 \, d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 54 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 48 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6} c^{6} - 6 \, a^{6} c^{5} d + 15 \, a^{6} c^{4} d^{2} - 20 \, a^{6} c^{3} d^{3} + 15 \, a^{6} c^{2} d^{4} - 6 \, a^{6} c d^{5} + a^{6} d^{6}}}{6 \, f} \]
1/6*(12*d^3*tan(1/2*f*x + 1/2*e)/((a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a ^2*d^4)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)) + 1 2*(3*c*d^2 + 2*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + ar ctan(-(c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2))) /((a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*sqrt(-c^2 + d^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*tan(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) + 24*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) + 48*a^4*c*d^3* tan(1/2*f*x + 1/2*e) - 15*a^4*d^4*tan(1/2*f*x + 1/2*e))/(a^6*c^6 - 6*a^6*c ^5*d + 15*a^6*c^4*d^2 - 20*a^6*c^3*d^3 + 15*a^6*c^2*d^4 - 6*a^6*c*d^5 + a^ 6*d^6))/f
Time = 14.29 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.49 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3}{2\,a^2\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{a^2\,{\left (c-d\right )}^4}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6\,a^2\,f\,{\left (c-d\right )}^2}+\frac {2\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left (a^2\,d^4-a^2\,c^4+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^2\,c^4-4\,a^2\,c^3\,d+6\,a^2\,c^2\,d^2-4\,a^2\,c\,d^3+a^2\,d^4\right )-2\,a^2\,c\,d^3+2\,a^2\,c^3\,d\right )}-\frac {d^2\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d+6{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^2-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^3+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^4}{\sqrt {c+d}\,{\left (c-d\right )}^{7/2}}\right )\,\left (3\,c+2\,d\right )\,2{}\mathrm {i}}{a^2\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{7/2}} \]
(tan(e/2 + (f*x)/2)*(3/(2*a^2*(c - d)^2) - (c^2 - d^2)/(a^2*(c - d)^4)))/f - tan(e/2 + (f*x)/2)^3/(6*a^2*f*(c - d)^2) + (2*d^3*tan(e/2 + (f*x)/2))/( f*(c + d)*(a^2*d^4 - a^2*c^4 + tan(e/2 + (f*x)/2)^2*(a^2*c^4 + a^2*d^4 - 4 *a^2*c*d^3 - 4*a^2*c^3*d + 6*a^2*c^2*d^2) - 2*a^2*c*d^3 + 2*a^2*c^3*d)) - (d^2*atan((c^4*tan(e/2 + (f*x)/2)*1i + d^4*tan(e/2 + (f*x)/2)*1i - c*d^3*t an(e/2 + (f*x)/2)*4i - c^3*d*tan(e/2 + (f*x)/2)*4i + c^2*d^2*tan(e/2 + (f* x)/2)*6i)/((c + d)^(1/2)*(c - d)^(7/2)))*(3*c + 2*d)*2i)/(a^2*f*(c + d)^(3 /2)*(c - d)^(7/2))