Integrand size = 25, antiderivative size = 290 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {(b (c-d)+a (c+d)) (a (c-d)-b (c+d)) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}+\frac {2 b^3 \left (a b c-2 a^2 d-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^3 f}-\frac {2 d^3 \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^2 f}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))} \]
(b*(c-d)+a*(c+d))*(a*(c-d)-b*(c+d))*x/(a^2+b^2)^2/(c^2+d^2)^2+2*b^3*(-2*a^ 2*d+a*b*c-b^2*d)*ln(a*cos(f*x+e)+b*sin(f*x+e))/(a^2+b^2)^2/(-a*d+b*c)^3/f- 2*d^3*(a*c*d-b*(2*c^2+d^2))*ln(c*cos(f*x+e)+d*sin(f*x+e))/(-a*d+b*c)^3/(c^ 2+d^2)^2/f-d*(a^2*d^2+b^2*(c^2+2*d^2))/(a^2+b^2)/(-a*d+b*c)^2/(c^2+d^2)/f/ (c+d*tan(f*x+e))-b^2/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))/(c+d*tan(f*x+ e))
Time = 7.03 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.92 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {-\frac {\frac {b (b c-a d)^2 \left (2 a b c^2+2 a^2 c d-2 b^2 c d-2 a b d^2+\frac {b \left (4 a b c d-a^2 \left (c^2-d^2\right )+b^2 \left (c^2-d^2\right )\right )}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b^4 \left (a b c-2 a^2 d-b^2 d\right ) \left (c^2+d^2\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}+\frac {b (b c-a d)^2 \left (2 a b c^2+2 a^2 c d-2 b^2 c d-2 a b d^2+\frac {\sqrt {-b^2} \left (4 a b c d-a^2 \left (c^2-d^2\right )+b^2 \left (c^2-d^2\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac {2 b \left (a^2+b^2\right ) d^3 \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c+d \tan (e+f x))}{(b c-a d) \left (c^2+d^2\right )}}{b (-b c+a d) \left (c^2+d^2\right ) f}-\frac {d^2 \left (-a b c+a^2 d+2 b^2 d\right )-c \left (-2 b^2 c d+b d (b c-a d)\right )}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)} \]
-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]) )) - (-(((b*(b*c - a*d)^2*(2*a*b*c^2 + 2*a^2*c*d - 2*b^2*c*d - 2*a*b*d^2 + (b*(4*a*b*c*d - a^2*(c^2 - d^2) + b^2*(c^2 - d^2)))/Sqrt[-b^2])*Log[Sqrt[ -b^2] - b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (2*b^4*(a*b*c - 2*a ^2*d - b^2*d)*(c^2 + d^2)*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d )) + (b*(b*c - a*d)^2*(2*a*b*c^2 + 2*a^2*c*d - 2*b^2*c*d - 2*a*b*d^2 + (Sq rt[-b^2]*(4*a*b*c*d - a^2*(c^2 - d^2) + b^2*(c^2 - d^2)))/b)*Log[Sqrt[-b^2 ] + b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) + (2*b*(a^2 + b^2)*d^3*(a *c*d - b*(2*c^2 + d^2))*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)) )/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (d^2*(-(a*b*c) + a^2*d + 2*b^2*d) - c*(-2*b^2*c*d + b*d*(b*c - a*d)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan [e + f*x])))/((a^2 + b^2)*(b*c - a*d))
Time = 1.80 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4052, 25, 3042, 4132, 3042, 4134, 3042, 4013}
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.
\(\displaystyle \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2}dx\) |
\(\Big \downarrow \) 4052 |
\(\displaystyle -\frac {\int -\frac {-d a^2+b c a-2 b^2 d \tan ^2(e+f x)-2 b^2 d-b (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {-d a^2+b c a-2 b^2 d \tan ^2(e+f x)-2 b^2 d-b (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {-d a^2+b c a-2 b^2 d \tan (e+f x)^2-2 b^2 d-b (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {\frac {\int \frac {c d^2 a^3-2 b d \left (c^2+d^2\right ) a^2+b^2 c \left (c^2+2 d^2\right ) a-b d \left (\left (c^2+2 d^2\right ) b^2+a^2 d^2\right ) \tan ^2(e+f x)-2 b^3 d \left (c^2+d^2\right )-(b c-a d)^2 (b c+a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {c d^2 a^3-2 b d \left (c^2+d^2\right ) a^2+b^2 c \left (c^2+2 d^2\right ) a-b d \left (\left (c^2+2 d^2\right ) b^2+a^2 d^2\right ) \tan (e+f x)^2-2 b^3 d \left (c^2+d^2\right )-(b c-a d)^2 (b c+a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
\(\Big \downarrow \) 4134 |
\(\displaystyle \frac {\frac {-\frac {2 d^3 \left (a^2+b^2\right ) \left (a c d-b \left (2 c^2+d^2\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {2 b^3 \left (c^2+d^2\right ) \left (-2 a^2 d+a b c-b^2 d\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {x (b c-a d)^2 (a c-a d-b c-b d) (a c+a d+b c-b d)}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {2 d^3 \left (a^2+b^2\right ) \left (a c d-b \left (2 c^2+d^2\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {2 b^3 \left (c^2+d^2\right ) \left (-2 a^2 d+a b c-b^2 d\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {x (b c-a d)^2 (a c-a d-b c-b d) (a c+a d+b c-b d)}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
\(\Big \downarrow \) 4013 |
\(\displaystyle \frac {\frac {\frac {x (b c-a d)^2 (a c-a d-b c-b d) (a c+a d+b c-b d)}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 d^3 \left (a^2+b^2\right ) \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)}+\frac {2 b^3 \left (c^2+d^2\right ) \left (-2 a^2 d+a b c-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}\) |
-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]) )) + ((((b*c - a*d)^2*(a*c - b*c - a*d - b*d)*(a*c + b*c + a*d - b*d)*x)/( (a^2 + b^2)*(c^2 + d^2)) + (2*b^3*(a*b*c - 2*a^2*d - b^2*d)*(c^2 + d^2)*Lo g[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f) - (2*(a^2 + b^2)*d^3*(a*c*d - b*(2*c^2 + d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]]) /((b*c - a*d)*(c^2 + d^2)*f))/((b*c - a*d)*(c^2 + d^2)) - (d*(a^2*d^2 + b^ 2*(c^2 + 2*d^2)))/((b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d))
3.13.21.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^ 2)/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))*(x/ ((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d) *(a^2 + b^2)) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Sim p[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)) Int[(d - c*Tan[e + f* x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 1.72 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {d^{3}}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {b^{3}}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}+\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-2 a^{2} c d -2 a b \,c^{2}+2 a b \,d^{2}+2 b^{2} c d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}}{f}\) | \(289\) |
default | \(\frac {-\frac {d^{3}}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {b^{3}}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}+\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-2 a^{2} c d -2 a b \,c^{2}+2 a b \,d^{2}+2 b^{2} c d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}}{f}\) | \(289\) |
norman | \(\frac {\frac {\left (a d +b c \right ) \left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) x \tan \left (f x +e \right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) a c x}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b d \left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (a^{4} d^{4}+a^{2} b^{2} d^{4}+b^{4} c^{4}+b^{4} c^{2} d^{2}\right ) \tan \left (f x +e \right )}{f c a \left (c^{2}+d^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a^{2}+b^{2}\right )}+\frac {d b \left (a^{3} d^{3}+a \,b^{2} d^{3}+b^{3} c^{3}+b^{3} c \,d^{2}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{f c a \left (c^{2}+d^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}-\frac {\left (a^{2} c d +a b \,c^{2}-a b \,d^{2}-b^{2} c d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f \left (a^{4} c^{4}+2 a^{4} c^{2} d^{2}+a^{4} d^{4}+2 a^{2} b^{2} c^{4}+4 a^{2} b^{2} c^{2} d^{2}+2 a^{2} b^{2} d^{4}+b^{4} c^{4}+2 b^{4} c^{2} d^{2}+b^{4} d^{4}\right )}+\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) f}+\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}\) | \(770\) |
risch | \(\text {Expression too large to display}\) | \(3046\) |
parallelrisch | \(\text {Expression too large to display}\) | \(4238\) |
1/f*(-d^3/(a*d-b*c)^2/(c^2+d^2)/(c+d*tan(f*x+e))+2*d^3*(a*c*d-2*b*c^2-b*d^ 2)/(a*d-b*c)^3/(c^2+d^2)^2*ln(c+d*tan(f*x+e))-b^3/(a*d-b*c)^2/(a^2+b^2)/(a +b*tan(f*x+e))+2*b^3*(2*a^2*d-a*b*c+b^2*d)/(a*d-b*c)^3/(a^2+b^2)^2*ln(a+b* tan(f*x+e))+1/(a^2+b^2)^2/(c^2+d^2)^2*(1/2*(-2*a^2*c*d-2*a*b*c^2+2*a*b*d^2 +2*b^2*c*d)*ln(1+tan(f*x+e)^2)+(a^2*c^2-a^2*d^2-4*a*b*c*d-b^2*c^2+b^2*d^2) *arctan(tan(f*x+e))))
Leaf count of result is larger than twice the leaf count of optimal. 2217 vs. \(2 (291) = 582\).
Time = 1.00 (sec) , antiderivative size = 2217, normalized size of antiderivative = 7.64 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \]
-(b^6*c^6 - a*b^5*c^5*d + 2*b^6*c^4*d^2 - 2*a*b^5*c^3*d^3 + b^6*c^2*d^4 + (a^5*b + 2*a^3*b^3)*c*d^5 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^6 - ((a^3*b^3 - a*b^5)*c^6 - (3*a^4*b^2 + a^2*b^4)*c^5*d + (3*a^5*b + 8*a^3*b^3 + a*b^5)*c ^4*d^2 - (a^6 + 8*a^4*b^2 + 3*a^2*b^4)*c^3*d^3 + (a^5*b + 3*a^3*b^3)*c^2*d ^4 + (a^6 - a^4*b^2)*c*d^5)*f*x - (a*b^5*c^5*d - a^2*b^4*c^4*d^2 + 2*a*b^5 *c^3*d^3 - a^2*b^4*d^6 + (a^4*b^2 + b^6)*c^2*d^4 - (a^5*b + 2*a^3*b^3)*c*d ^5 + ((a^2*b^4 - b^6)*c^5*d - (3*a^3*b^3 + a*b^5)*c^4*d^2 + (3*a^4*b^2 + 8 *a^2*b^4 + b^6)*c^3*d^3 - (a^5*b + 8*a^3*b^3 + 3*a*b^5)*c^2*d^4 + (a^4*b^2 + 3*a^2*b^4)*c*d^5 + (a^5*b - a^3*b^3)*d^6)*f*x)*tan(f*x + e)^2 - (a^2*b^ 4*c^6 + 2*a^2*b^4*c^4*d^2 + a^2*b^4*c^2*d^4 - (2*a^3*b^3 + a*b^5)*c^5*d - 2*(2*a^3*b^3 + a*b^5)*c^3*d^3 - (2*a^3*b^3 + a*b^5)*c*d^5 + (a*b^5*c^5*d + 2*a*b^5*c^3*d^3 + a*b^5*c*d^5 - (2*a^2*b^4 + b^6)*c^4*d^2 - 2*(2*a^2*b^4 + b^6)*c^2*d^4 - (2*a^2*b^4 + b^6)*d^6)*tan(f*x + e)^2 + (a*b^5*c^6 - (a^2 *b^4 + b^6)*c^5*d - (2*a^3*b^3 - a*b^5)*c^4*d^2 - 2*(a^2*b^4 + b^6)*c^3*d^ 3 - (4*a^3*b^3 + a*b^5)*c^2*d^4 - (a^2*b^4 + b^6)*c*d^5 - (2*a^3*b^3 + a*b ^5)*d^6)*tan(f*x + e))*log((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) + a^2) /(tan(f*x + e)^2 + 1)) - (2*(a^5*b + 2*a^3*b^3 + a*b^5)*c^3*d^3 - (a^6 + 2 *a^4*b^2 + a^2*b^4)*c^2*d^4 + (a^5*b + 2*a^3*b^3 + a*b^5)*c*d^5 + (2*(a^4* b^2 + 2*a^2*b^4 + b^6)*c^2*d^4 - (a^5*b + 2*a^3*b^3 + a*b^5)*c*d^5 + (a^4* b^2 + 2*a^2*b^4 + b^6)*d^6)*tan(f*x + e)^2 + (2*(a^4*b^2 + 2*a^2*b^4 + ...
Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\text {Exception raised: NotImplementedError} \]
Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (291) = 582\).
Time = 0.37 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.03 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=-\frac {\frac {{\left (4 \, a b c d - {\left (a^{2} - b^{2}\right )} c^{2} + {\left (a^{2} - b^{2}\right )} d^{2}\right )} {\left (f x + e\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} d^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}} - \frac {2 \, {\left (a b^{4} c - {\left (2 \, a^{2} b^{3} + b^{5}\right )} d\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} c^{3} - 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} c^{2} d + 3 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} c d^{2} - {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d^{3}} - \frac {2 \, {\left (2 \, b c^{2} d^{3} - a c d^{4} + b d^{5}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c d^{6} - a^{3} d^{7} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{5} d^{2} - {\left (a^{3} + 6 \, a b^{2}\right )} c^{4} d^{3} + {\left (6 \, a^{2} b + b^{3}\right )} c^{3} d^{4} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5}} + \frac {{\left (a b c^{2} - a b d^{2} + {\left (a^{2} - b^{2}\right )} c d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} d^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}} + \frac {b^{3} c^{3} + b^{3} c d^{2} + {\left (a^{3} + a b^{2}\right )} d^{3} + {\left (b^{3} c^{2} d + {\left (a^{2} b + 2 \, b^{3}\right )} d^{3}\right )} \tan \left (f x + e\right )}{{\left (a^{3} b^{2} + a b^{4}\right )} c^{5} - 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} c^{4} d + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} c^{3} d^{2} - 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} c^{2} d^{3} + {\left (a^{5} + a^{3} b^{2}\right )} c d^{4} + {\left ({\left (a^{2} b^{3} + b^{5}\right )} c^{4} d - 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} c^{3} d^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} c^{2} d^{3} - 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} c d^{4} + {\left (a^{4} b + a^{2} b^{3}\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} + {\left ({\left (a^{2} b^{3} + b^{5}\right )} c^{5} - {\left (a^{3} b^{2} + a b^{4}\right )} c^{4} d - {\left (a^{4} b - b^{5}\right )} c^{3} d^{2} + {\left (a^{5} - a b^{4}\right )} c^{2} d^{3} - {\left (a^{4} b + a^{2} b^{3}\right )} c d^{4} + {\left (a^{5} + a^{3} b^{2}\right )} d^{5}\right )} \tan \left (f x + e\right )}}{f} \]
-((4*a*b*c*d - (a^2 - b^2)*c^2 + (a^2 - b^2)*d^2)*(f*x + e)/((a^4 + 2*a^2* b^2 + b^4)*c^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^2 + (a^4 + 2*a^2*b^2 + b^ 4)*d^4) - 2*(a*b^4*c - (2*a^2*b^3 + b^5)*d)*log(b*tan(f*x + e) + a)/((a^4* b^3 + 2*a^2*b^5 + b^7)*c^3 - 3*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*c^2*d + 3*(a^ 6*b + 2*a^4*b^3 + a^2*b^5)*c*d^2 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d^3) - 2*(2 *b*c^2*d^3 - a*c*d^4 + b*d^5)*log(d*tan(f*x + e) + c)/(b^3*c^7 - 3*a*b^2*c ^6*d + 3*a^2*b*c*d^6 - a^3*d^7 + (3*a^2*b + 2*b^3)*c^5*d^2 - (a^3 + 6*a*b^ 2)*c^4*d^3 + (6*a^2*b + b^3)*c^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5) + (a*b*c ^2 - a*b*d^2 + (a^2 - b^2)*c*d)*log(tan(f*x + e)^2 + 1)/((a^4 + 2*a^2*b^2 + b^4)*c^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^2 + (a^4 + 2*a^2*b^2 + b^4)*d ^4) + (b^3*c^3 + b^3*c*d^2 + (a^3 + a*b^2)*d^3 + (b^3*c^2*d + (a^2*b + 2*b ^3)*d^3)*tan(f*x + e))/((a^3*b^2 + a*b^4)*c^5 - 2*(a^4*b + a^2*b^3)*c^4*d + (a^5 + 2*a^3*b^2 + a*b^4)*c^3*d^2 - 2*(a^4*b + a^2*b^3)*c^2*d^3 + (a^5 + a^3*b^2)*c*d^4 + ((a^2*b^3 + b^5)*c^4*d - 2*(a^3*b^2 + a*b^4)*c^3*d^2 + ( a^4*b + 2*a^2*b^3 + b^5)*c^2*d^3 - 2*(a^3*b^2 + a*b^4)*c*d^4 + (a^4*b + a^ 2*b^3)*d^5)*tan(f*x + e)^2 + ((a^2*b^3 + b^5)*c^5 - (a^3*b^2 + a*b^4)*c^4* d - (a^4*b - b^5)*c^3*d^2 + (a^5 - a*b^4)*c^2*d^3 - (a^4*b + a^2*b^3)*c*d^ 4 + (a^5 + a^3*b^2)*d^5)*tan(f*x + e)))/f
Leaf count of result is larger than twice the leaf count of optimal. 1371 vs. \(2 (291) = 582\).
Time = 0.59 (sec) , antiderivative size = 1371, normalized size of antiderivative = 4.73 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \]
((a^2*c^2 - b^2*c^2 - 4*a*b*c*d - a^2*d^2 + b^2*d^2)*(f*x + e)/(a^4*c^4 + 2*a^2*b^2*c^4 + b^4*c^4 + 2*a^4*c^2*d^2 + 4*a^2*b^2*c^2*d^2 + 2*b^4*c^2*d^ 2 + a^4*d^4 + 2*a^2*b^2*d^4 + b^4*d^4) - (a*b*c^2 + a^2*c*d - b^2*c*d - a* b*d^2)*log(tan(f*x + e)^2 + 1)/(a^4*c^4 + 2*a^2*b^2*c^4 + b^4*c^4 + 2*a^4* c^2*d^2 + 4*a^2*b^2*c^2*d^2 + 2*b^4*c^2*d^2 + a^4*d^4 + 2*a^2*b^2*d^4 + b^ 4*d^4) + 2*(a*b^5*c - 2*a^2*b^4*d - b^6*d)*log(abs(b*tan(f*x + e) + a))/(a ^4*b^4*c^3 + 2*a^2*b^6*c^3 + b^8*c^3 - 3*a^5*b^3*c^2*d - 6*a^3*b^5*c^2*d - 3*a*b^7*c^2*d + 3*a^6*b^2*c*d^2 + 6*a^4*b^4*c*d^2 + 3*a^2*b^6*c*d^2 - a^7 *b*d^3 - 2*a^5*b^3*d^3 - a^3*b^5*d^3) + 2*(2*b*c^2*d^4 - a*c*d^5 + b*d^6)* log(abs(d*tan(f*x + e) + c))/(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^ 3 + 2*b^3*c^5*d^3 - a^3*c^4*d^4 - 6*a*b^2*c^4*d^4 + 6*a^2*b*c^3*d^5 + b^3* c^3*d^5 - 2*a^3*c^2*d^6 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8) - (a* b^4*c^4*d*tan(f*x + e)^2 - a^2*b^3*c^3*d^2*tan(f*x + e)^2 - b^5*c^3*d^2*ta n(f*x + e)^2 - a^3*b^2*c^2*d^3*tan(f*x + e)^2 + a*b^4*c^2*d^3*tan(f*x + e) ^2 + a^4*b*c*d^4*tan(f*x + e)^2 + a^2*b^3*c*d^4*tan(f*x + e)^2 - a^3*b^2*d ^5*tan(f*x + e)^2 + a*b^4*c^5*tan(f*x + e) + a^2*b^3*c^4*d*tan(f*x + e) - 2*a^3*b^2*c^3*d^2*tan(f*x + e) + a^4*b*c^2*d^3*tan(f*x + e) + 6*a^2*b^3*c^ 2*d^3*tan(f*x + e) + 3*b^5*c^2*d^3*tan(f*x + e) + a^5*c*d^4*tan(f*x + e) + 3*a^2*b^3*d^5*tan(f*x + e) + 2*b^5*d^5*tan(f*x + e) + 2*a^2*b^3*c^5 + b^5 *c^5 - a^3*b^2*c^4*d - a*b^4*c^4*d - a^4*b*c^3*d^2 + 3*a^2*b^3*c^3*d^2 ...
Time = 11.82 (sec) , antiderivative size = 725, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d\,\left (4\,a^2\,b^3+2\,b^5\right )-2\,a\,b^4\,c\right )}{f\,\left (a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d+2\,a^5\,b^2\,d^3-a^4\,b^3\,c^3-6\,a^4\,b^3\,c\,d^2+6\,a^3\,b^4\,c^2\,d+a^3\,b^4\,d^3-2\,a^2\,b^5\,c^3-3\,a^2\,b^5\,c\,d^2+3\,a\,b^6\,c^2\,d-b^7\,c^3\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}{2\,f\,\left (-a^2\,c^2\,1{}\mathrm {i}+2\,a^2\,c\,d+a^2\,d^2\,1{}\mathrm {i}+2\,a\,b\,c^2+a\,b\,c\,d\,4{}\mathrm {i}-2\,a\,b\,d^2+b^2\,c^2\,1{}\mathrm {i}-2\,b^2\,c\,d-b^2\,d^2\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{2\,f\,\left (-a^2\,c^2\,1{}\mathrm {i}-2\,a^2\,c\,d+a^2\,d^2\,1{}\mathrm {i}-2\,a\,b\,c^2+a\,b\,c\,d\,4{}\mathrm {i}+2\,a\,b\,d^2+b^2\,c^2\,1{}\mathrm {i}+2\,b^2\,c\,d-b^2\,d^2\,1{}\mathrm {i}\right )}-\frac {\frac {a^3\,d^3+a\,b^2\,d^3+b^3\,c^3+b^3\,c\,d^2}{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,c^2+a^2\,d^2+b^2\,c^2+b^2\,d^2\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b\,d^3+b^3\,c^2\,d+2\,b^3\,d^3\right )}{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,c^2+a^2\,d^2+b^2\,c^2+b^2\,d^2\right )}}{f\,\left (b\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2+\left (a\,d+b\,c\right )\,\mathrm {tan}\left (e+f\,x\right )+a\,c\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b\,\left (4\,c^2\,d^3+2\,d^5\right )-2\,a\,c\,d^4\right )}{f\,\left (a^3\,c^4\,d^3+2\,a^3\,c^2\,d^5+a^3\,d^7-3\,a^2\,b\,c^5\,d^2-6\,a^2\,b\,c^3\,d^4-3\,a^2\,b\,c\,d^6+3\,a\,b^2\,c^6\,d+6\,a\,b^2\,c^4\,d^3+3\,a\,b^2\,c^2\,d^5-b^3\,c^7-2\,b^3\,c^5\,d^2-b^3\,c^3\,d^4\right )} \]
log(tan(e + f*x) + 1i)/(2*f*(a^2*d^2*1i - a^2*c^2*1i + b^2*c^2*1i - b^2*d^ 2*1i - 2*a*b*c^2 + 2*a*b*d^2 - 2*a^2*c*d + 2*b^2*c*d + a*b*c*d*4i)) - log( tan(e + f*x) - 1i)/(2*f*(a^2*d^2*1i - a^2*c^2*1i + b^2*c^2*1i - b^2*d^2*1i + 2*a*b*c^2 - 2*a*b*d^2 + 2*a^2*c*d - 2*b^2*c*d + a*b*c*d*4i)) - ((a^3*d^ 3 + b^3*c^3 + a*b^2*d^3 + b^3*c*d^2)/((a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2 *c^2 + a^2*d^2 + b^2*c^2 + b^2*d^2)) + (tan(e + f*x)*(2*b^3*d^3 + a^2*b*d^ 3 + b^3*c^2*d))/((a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*c^2 + a^2*d^2 + b^2* c^2 + b^2*d^2)))/(f*(a*c + tan(e + f*x)*(a*d + b*c) + b*d*tan(e + f*x)^2)) + (log(a + b*tan(e + f*x))*(d*(2*b^5 + 4*a^2*b^3) - 2*a*b^4*c))/(f*(a^7*d ^3 - b^7*c^3 - 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 + 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 + 6*a^3*b^4*c^2*d - 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3 *a*b^6*c^2*d - 3*a^6*b*c*d^2)) - (log(c + d*tan(e + f*x))*(b*(2*d^5 + 4*c^ 2*d^3) - 2*a*c*d^4))/(f*(a^3*d^7 - b^3*c^7 + 2*a^3*c^2*d^5 + a^3*c^4*d^3 - b^3*c^3*d^4 - 2*b^3*c^5*d^2 + 3*a*b^2*c^2*d^5 + 6*a*b^2*c^4*d^3 - 6*a^2*b *c^3*d^4 - 3*a^2*b*c^5*d^2 + 3*a*b^2*c^6*d - 3*a^2*b*c*d^6))