Integrand size = 29, antiderivative size = 115 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (2 a A b-a^2 B+b^2 B\right ) x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2 A-A b^2+2 a b B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a (A b-a B)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
(2*A*a*b-B*a^2+B*b^2)*x/(a^2+b^2)^2-(A*a^2-A*b^2+2*B*a*b)*ln(a*cos(d*x+c)+ b*sin(d*x+c))/(a^2+b^2)^2/d+a*(A*b-B*a)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.22 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {(A+i B) \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {(A-i B) \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {2 \left (\left (-a^2 A+A b^2-2 a b B\right ) \log (a+b \tan (c+d x))-\frac {a \left (a^2+b^2\right ) (-A b+a B)}{b (a+b \tan (c+d x))}\right )}{\left (a^2+b^2\right )^2}}{2 d} \]
(((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*((-(a^2*A) + A*b^2 - 2*a*b*B)*Log[a + b*Tan[c + d*x]] - (a*(a^2 + b^2)*(-(A*b) + a*B))/(b*(a + b*Tan[c + d*x]))))/(a^2 + b ^2)^2)/(2*d)
Time = 0.57 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3042, 4074, 3042, 4014, 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.
| \(\displaystyle \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4074 |
| \(\displaystyle \frac {\int \frac {A b-a B+(a A+b B) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a (A b-a B)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {A b-a B+(a A+b B) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a (A b-a B)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}-\frac {\left (a^2 A+2 a b B-A b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a (A b-a B)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}-\frac {\left (a^2 A+2 a b B-A b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a (A b-a B)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {a (A b-a B)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {x \left (a^2 (-B)+2 a A b+b^2 B\right )}{a^2+b^2}-\frac {\left (a^2 A+2 a b B-A b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}}{a^2+b^2}\) |
(((2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2) - ((a^2*A - A*b^2 + 2*a*b*B)*Lo g[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d))/(a^2 + b^2) + (a*(A*b - a*B))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))
3.3.77.3.1 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[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[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] && N eQ[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_)]), x_Symbol] :> Simp[(b *c - a*d)*(A*b - a*B)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2 ))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m , -1] && NeQ[a^2 + b^2, 0]
Time = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.26
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a \left (A b -B a \right )}{\left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(145\) |
| default | \(\frac {\frac {\frac {\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a \left (A b -B a \right )}{\left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(145\) |
| norman | \(\frac {\frac {a \left (2 A a b -B \,a^{2}+B \,b^{2}\right ) x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {b \left (2 A a b -B \,a^{2}+B \,b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {a \left (A b -B a \right )}{\left (a^{2}+b^{2}\right ) b d}}{a +b \tan \left (d x +c \right )}+\frac {\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(221\) |
| parallelrisch | \(\frac {2 A \,a^{3} b +2 A a \,b^{3}-2 B \,a^{2} b^{2}-2 B \,a^{4}-2 A \ln \left (a +b \tan \left (d x +c \right )\right ) a^{3} b +A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{2}-2 A \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{2}+2 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}-4 B \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}-A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) b^{4}+2 A \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) b^{4}+2 A \ln \left (a +b \tan \left (d x +c \right )\right ) a \,b^{3}-4 B \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2} b^{2}+2 B x \tan \left (d x +c \right ) b^{4} d +4 A \,a^{2} b^{2} d x -2 B \,a^{3} b d x +2 B a \,b^{3} d x +4 A x \tan \left (d x +c \right ) a \,b^{3} d -2 B x \tan \left (d x +c \right ) a^{2} b^{2} d +A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b -A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}+2 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}}{2 \left (a +b \tan \left (d x +c \right )\right ) d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b}\) | \(399\) |
| risch | \(\frac {x B}{2 i b a -a^{2}+b^{2}}+\frac {i x A}{2 i b a -a^{2}+b^{2}}+\frac {2 i a^{2} A x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i A \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {4 i B a b x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i a^{2} A c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i A \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {4 i B a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i a A b}{\left (i b +a \right ) d \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}+\frac {2 i a^{2} B}{\left (i b +a \right ) d \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B a b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(477\) |
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)))+a*(A*b-B*a)/(a^2+b^2)/b/(a+b*tan(d*x+c))- (A*a^2-A*b^2+2*B*a*b)/(a^2+b^2)^2*ln(a+b*tan(d*x+c)))
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.92 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 \, B a^{2} b - 2 \, A a b^{2} + 2 \, {\left (B a^{3} - 2 \, A a^{2} b - B a b^{2}\right )} d x + {\left (A a^{3} + 2 \, B a^{2} b - A a b^{2} + {\left (A a^{2} b + 2 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (B a^{3} - A a^{2} b - {\left (B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]
-1/2*(2*B*a^2*b - 2*A*a*b^2 + 2*(B*a^3 - 2*A*a^2*b - B*a*b^2)*d*x + (A*a^3 + 2*B*a^2*b - A*a*b^2 + (A*a^2*b + 2*B*a*b^2 - A*b^3)*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 - A*a^2*b - (B*a^2*b - 2*A*a*b^2 - B*b^3)*d*x)*tan(d*x + c))/((a^4*b + 2*a^2*b^3 + b^5)*d*tan(d*x + c) + (a^5 + 2*a^3*b^2 + a*b^4)*d)
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 2987, normalized size of antiderivative = 25.97 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*x*(A + B*tan(c))/tan(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ( (A*log(tan(c + d*x)**2 + 1)/(2*d) - B*x + B*tan(c + d*x)/d)/a**2, Eq(b, 0) ), (I*A*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*A*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b **2*d*tan(c + d*x) - 4*b**2*d) - I*A*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b **2*d*tan(c + d*x) - 4*b**2*d) + I*A*tan(c + d*x)/(4*b**2*d*tan(c + d*x)** 2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + B*d*x*tan(c + d*x)**2/(4*b**2*d* tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 2*I*B*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - B*d *x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 3*B*t an(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d ) + 2*I*B/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I*b)), (-I*A*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b **2*d*tan(c + d*x) - 4*b**2*d) + 2*A*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d* x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + I*A*d*x/(4*b**2*d*tan(c + d* x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - I*A*tan(c + d*x)/(4*b**2*d*t an(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + B*d*x*tan(c + d*x)* *2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*I*B *d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4* b**2*d) - B*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4...
Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.61 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (B a^{2} - A a b\right )}}{a^{3} b + a b^{3} + {\left (a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )}}{2 \, d} \]
-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) + 2*(A *a^2 + 2*B*a*b - A*b^2)*log(b*tan(d*x + c) + a)/(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*(B*a^2 - A*a*b)/(a^3*b + a*b^3 + (a^2*b^2 + b^4)*tan(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (117) = 234\).
Time = 0.43 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.10 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (A a^{2} b + 2 \, B a b^{2} - A b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {2 \, {\left (A a^{2} b^{2} \tan \left (d x + c\right ) + 2 \, B a b^{3} \tan \left (d x + c\right ) - A b^{4} \tan \left (d x + c\right ) - B a^{4} + 2 \, A a^{3} b + B a^{2} b^{2}\right )}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]
-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* (A*a^2*b + 2*B*a*b^2 - A*b^3)*log(abs(b*tan(d*x + c) + a))/(a^4*b + 2*a^2* b^3 + b^5) - 2*(A*a^2*b^2*tan(d*x + c) + 2*B*a*b^3*tan(d*x + c) - A*b^4*ta n(d*x + c) - B*a^4 + 2*A*a^3*b + B*a^2*b^2)/((a^4*b + 2*a^2*b^3 + b^5)*(b* tan(d*x + c) + a)))/d
Time = 8.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.42 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {a\,\left (A\,b-B\,a\right )}{b\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {A}{a^2+b^2}-\frac {2\,b\,\left (A\,b-B\,a\right )}{{\left (a^2+b^2\right )}^2}\right )}{d} \]