Integrand size = 29, antiderivative size = 137 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (2 a A b-a^2 B+b^2 B\right ) x}{\left (a^2+b^2\right )^2}+\frac {A \log (\sin (c+d x))}{a^2 d}-\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
-(2*A*a*b-B*a^2+B*b^2)*x/(a^2+b^2)^2+A*ln(sin(d*x+c))/a^2/d-b*(3*A*a^2*b+A *b^3-2*B*a^3)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^2/(a^2+b^2)^2/d+b*(A*b-B*a)/ a/(a^2+b^2)/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.34 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {-\frac {a (a-i b) (A+i B) \log (i-\tan (c+d x))}{2 (a+i b)}+\frac {A \left (a^2+b^2\right ) \log (\tan (c+d x))}{a}-\frac {a (a+i b) (A-i B) \log (i+\tan (c+d x))}{2 (a-i b)}+\frac {b \left (-3 a^2 A b-A b^3+2 a^3 B\right ) \log (a+b \tan (c+d x))}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a+b \tan (c+d x)}}{a \left (a^2+b^2\right ) d} \]
(-1/2*(a*(a - I*b)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) + (A*(a^2 + b^2)*Log[Tan[c + d*x]])/a - (a*(a + I*b)*(A - I*B)*Log[I + Tan[c + d*x]])/ (2*(a - I*b)) + (b*(-3*a^2*A*b - A*b^3 + 2*a^3*B)*Log[a + b*Tan[c + d*x]]) /(a*(a^2 + b^2)) + (b*(A*b - a*B))/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)
Time = 0.82 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3042, 4092, 3042, 4134, 3042, 25, 3956, 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.
| \(\displaystyle \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x) (a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle \frac {\int \frac {\cot (c+d x) \left (b (A b-a B) \tan ^2(c+d x)-a (A b-a B) \tan (c+d x)+A \left (a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b (A b-a B) \tan (c+d x)^2-a (A b-a B) \tan (c+d x)+A \left (a^2+b^2\right )}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4134 |
| \(\displaystyle \frac {\frac {A \left (a^2+b^2\right ) \int \cot (c+d x)dx}{a}-\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {A \left (a^2+b^2\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {A \left (a^2+b^2\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {-\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}+\frac {A \left (a^2+b^2\right ) \log (-\sin (c+d x))}{a d}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {-\frac {a x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}+\frac {A \left (a^2+b^2\right ) \log (-\sin (c+d x))}{a d}-\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}\) |
(-((a*(2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2)) + (A*(a^2 + b^2)*Log[-Sin[ c + d*x]])/(a*d) - (b*(3*a^2*A*b + A*b^3 - 2*a^3*B)*Log[a*Cos[c + d*x] + b *Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 + b^2)) + (b*(A*b - a*B))/(a*(a ^2 + b^2)*d*(a + b*Tan[c + d*x]))
3.3.79.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*(A*b - a*B)*(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[b* B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 )) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(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 = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 A a b +B \,a^{2}-B \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}-\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}+\frac {\left (A b -B a \right ) b}{\left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(163\) |
| default | \(\frac {\frac {\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 A a b +B \,a^{2}-B \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}-\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}+\frac {\left (A b -B a \right ) b}{\left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(163\) |
| parallelrisch | \(\frac {-6 \left (A \,a^{2} b +\frac {1}{3} A \,b^{3}-\frac {2}{3} B \,a^{3}\right ) b \left (a +b \tan \left (d x +c \right )\right ) \ln \left (a +b \tan \left (d x +c \right )\right )-a^{2} \left (a +b \tan \left (d x +c \right )\right ) \left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+2 A \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\tan \left (d x +c \right )\right )-4 \left (\frac {A \,b^{4}}{2}-\frac {B a \,b^{3}}{2}+\frac {a^{2} \left (B d x +A \right ) b^{2}}{2}+a^{3} \left (A d x -\frac {B}{2}\right ) b -\frac {B x \,a^{4} d}{2}\right ) b \tan \left (d x +c \right )-4 d \,a^{3} x \left (A a b -\frac {1}{2} B \,a^{2}+\frac {1}{2} B \,b^{2}\right )}{2 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )^{2} a^{2} d}\) | \(226\) |
| norman | \(\frac {-\frac {a \left (2 A a b -B \,a^{2}+B \,b^{2}\right ) x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (2 A a b -B \,a^{2}+B \,b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\left (A \,b^{2}-B a b \right ) b \tan \left (d x +c \right )}{d \,a^{2} \left (a^{2}+b^{2}\right )}}{a +b \tan \left (d x +c \right )}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}-\frac {\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}\) | \(254\) |
| risch | \(-\frac {x B}{2 i b a -a^{2}+b^{2}}+\frac {2 i b^{4} A c}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {6 i A \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {i x A}{2 i b a -a^{2}+b^{2}}+\frac {6 i A \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i x A}{a^{2}}+\frac {2 i b^{4} A x}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i b^{2} B}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}-\frac {4 i B a b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i A c}{a^{2} d}-\frac {4 i B a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b^{3} A}{\left (-i a +b \right ) d a \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B a b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {A \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}\) | \(535\) |
1/d*(1/(a^2+b^2)^2*(1/2*(-A*a^2+A*b^2-2*B*a*b)*ln(1+tan(d*x+c)^2)+(-2*A*a* b+B*a^2-B*b^2)*arctan(tan(d*x+c)))+1/a^2*A*ln(tan(d*x+c))-b*(3*A*a^2*b+A*b ^3-2*B*a^3)/(a^2+b^2)^2/a^2*ln(a+b*tan(d*x+c))+(A*b-B*a)*b/(a^2+b^2)/a/(a+ b*tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (137) = 274\).
Time = 0.30 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.36 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 \, B a^{2} b^{3} - 2 \, A a b^{4} - 2 \, {\left (B a^{5} - 2 \, A a^{4} b - B a^{3} b^{2}\right )} d x - {\left (A a^{5} + 2 \, A a^{3} b^{2} + A a b^{4} + {\left (A a^{4} b + 2 \, A a^{2} b^{3} + A b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2} - A a b^{4} + {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (B a^{3} b^{2} - A a^{2} b^{3} + {\left (B a^{4} b - 2 \, A a^{3} b^{2} - B a^{2} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d\right )}} \]
-1/2*(2*B*a^2*b^3 - 2*A*a*b^4 - 2*(B*a^5 - 2*A*a^4*b - B*a^3*b^2)*d*x - (A *a^5 + 2*A*a^3*b^2 + A*a*b^4 + (A*a^4*b + 2*A*a^2*b^3 + A*b^5)*tan(d*x + c ))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - (2*B*a^4*b - 3*A*a^3*b^2 - A *a*b^4 + (2*B*a^3*b^2 - 3*A*a^2*b^3 - A*b^5)*tan(d*x + c))*log((b^2*tan(d* x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - 2*(B*a^3*b^2 - A*a^2*b^3 + (B*a^4*b - 2*A*a^3*b^2 - B*a^2*b^3)*d*x)*tan(d*x + c))/((a^6 *b + 2*a^4*b^3 + a^2*b^5)*d*tan(d*x + c) + (a^7 + 2*a^5*b^2 + a^3*b^4)*d)
Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 4488, normalized size of antiderivative = 32.76 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*x*(A + B*tan(c))*cot(c)/tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq (d, 0)), ((-A*log(tan(c + d*x)**2 + 1)/(2*d) + A*log(tan(c + d*x))/d + B*x )/a**2, Eq(b, 0)), ((A*log(tan(c + d*x)**2 + 1)/(2*d) - A*log(tan(c + d*x) )/d - A/(2*d*tan(c + d*x)**2) - B*x - B/(d*tan(c + d*x)))/b**2, Eq(a, 0)), (3*I*A*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) - 6*A*d*x*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**2 + 8*I*a **2*d*tan(c + d*x) - 4*a**2*d) - 3*I*A*d*x/(4*a**2*d*tan(c + d*x)**2 + 8*I *a**2*d*tan(c + d*x) - 4*a**2*d) - 2*A*log(tan(c + d*x)**2 + 1)*tan(c + d* x)**2/(4*a**2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) - 4* I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**2 + 8*I* a**2*d*tan(c + d*x) - 4*a**2*d) + 2*A*log(tan(c + d*x)**2 + 1)/(4*a**2*d*t an(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) + 4*A*log(tan(c + d*x ))*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4 *a**2*d) + 8*I*A*log(tan(c + d*x))*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) - 4*A*log(tan(c + d*x))/(4*a**2*d*ta n(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) + 3*I*A*tan(c + d*x)/( 4*a**2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) - 4*A/(4*a* *2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) + B*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) + 2*I*B*d*x*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + ...
Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.52 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2} - A b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (B a b - A b^{2}\right )}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )} + \frac {2 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \]
1/2*(2*(B*a^2 - 2*A*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(2* B*a^3*b - 3*A*a^2*b^2 - A*b^4)*log(b*tan(d*x + c) + a)/(a^6 + 2*a^4*b^2 + a^2*b^4) - (A*a^2 + 2*B*a*b - A*b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2* b^2 + b^4) - 2*(B*a*b - A*b^2)/(a^4 + a^2*b^2 + (a^3*b + a*b^3)*tan(d*x + c)) + 2*A*log(tan(d*x + c))/a^2)/d
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (137) = 274\).
Time = 0.66 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.04 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} + \frac {2 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac {2 \, {\left (2 \, B a^{3} b^{2} \tan \left (d x + c\right ) - 3 \, A a^{2} b^{3} \tan \left (d x + c\right ) - A b^{5} \tan \left (d x + c\right ) + 3 \, B a^{4} b - 4 \, A a^{3} b^{2} + B a^{2} b^{3} - 2 \, A a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]
1/2*(2*(B*a^2 - 2*A*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (A*a^ 2 + 2*B*a*b - A*b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 2*( 2*B*a^3*b^2 - 3*A*a^2*b^3 - A*b^5)*log(abs(b*tan(d*x + c) + a))/(a^6*b + 2 *a^4*b^3 + a^2*b^5) + 2*A*log(abs(tan(d*x + c)))/a^2 - 2*(2*B*a^3*b^2*tan( d*x + c) - 3*A*a^2*b^3*tan(d*x + c) - A*b^5*tan(d*x + c) + 3*B*a^4*b - 4*A *a^3*b^2 + B*a^2*b^3 - 2*A*a*b^4)/((a^6 + 2*a^4*b^2 + a^2*b^4)*(b*tan(d*x + c) + a)))/d
Time = 9.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.31 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )}+\frac {A\,b^2-B\,a\,b}{a\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-2\,B\,a^3+3\,A\,a^2\,b+A\,b^3\right )}{a^2\,d\,{\left (a^2+b^2\right )}^2} \]
(A*log(tan(c + d*x)))/(a^2*d) - (log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*( a*b*2i + a^2 - b^2)) - (log(tan(c + d*x) + 1i)*(A*1i + B))/(2*d*(2*a*b + a ^2*1i - b^2*1i)) + (A*b^2 - B*a*b)/(a*d*(a^2 + b^2)*(a + b*tan(c + d*x))) - (b*log(a + b*tan(c + d*x))*(A*b^3 - 2*B*a^3 + 3*A*a^2*b))/(a^2*d*(a^2 + b^2)^2)