Integrand size = 19, antiderivative size = 107 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\log (\sin (c+d x))}{a^2 d}-\frac {b^2 \left (3 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^2}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
-2*a*b*x/(a^2+b^2)^2+ln(sin(d*x+c))/a^2/d-b^2*(3*a^2+b^2)*ln(a*cos(d*x+c)+ b*sin(d*x+c))/a^2/(a^2+b^2)^2/d+b^2/a/(a^2+b^2)/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.44 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {-\frac {a (a-i b) \log (i-\tan (c+d x))}{2 (a+i b)}+\frac {\left (a^2+b^2\right ) \log (\tan (c+d x))}{a}-\frac {a (a+i b) \log (i+\tan (c+d x))}{2 (a-i b)}-\frac {b^2 \left (3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{a \left (a^2+b^2\right )}+\frac {b^2}{a+b \tan (c+d x)}}{a \left (a^2+b^2\right ) d} \]
(-1/2*(a*(a - I*b)*Log[I - Tan[c + d*x]])/(a + I*b) + ((a^2 + b^2)*Log[Tan [c + d*x]])/a - (a*(a + I*b)*Log[I + Tan[c + d*x]])/(2*(a - I*b)) - (b^2*( 3*a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)) + b^2/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)
Time = 0.72 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 4052, 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.
\(\displaystyle \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x) (a+b \tan (c+d x))^2}dx\) |
\(\Big \downarrow \) 4052 |
\(\displaystyle \frac {\int \frac {\cot (c+d x) \left (a^2-b \tan (c+d x) a+b^2+b^2 \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a^2-b \tan (c+d x) a+b^2+b^2 \tan (c+d x)^2}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
\(\Big \downarrow \) 4134 |
\(\displaystyle \frac {-\frac {b^2 \left (3 a^2+b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right ) \int \cot (c+d x)dx}{a}-\frac {2 a^2 b x}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {b^2 \left (3 a^2+b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 a^2 b x}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {b^2 \left (3 a^2+b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {\left (a^2+b^2\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {2 a^2 b x}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {-\frac {b^2 \left (3 a^2+b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right ) \log (-\sin (c+d x))}{a d}-\frac {2 a^2 b x}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
\(\Big \downarrow \) 4013 |
\(\displaystyle \frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {\left (a^2+b^2\right ) \log (-\sin (c+d x))}{a d}-\frac {b^2 \left (3 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac {2 a^2 b x}{a^2+b^2}}{a \left (a^2+b^2\right )}\) |
((-2*a^2*b*x)/(a^2 + b^2) + ((a^2 + b^2)*Log[-Sin[c + d*x]])/(a*d) - (b^2* (3*a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a* (a^2 + b^2)) + b^2/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))
3.5.74.3.1 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_)*((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_)] + (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 = 0.49 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {b^{2}}{a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(126\) |
default | \(\frac {\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {b^{2}}{a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(126\) |
parallelrisch | \(\frac {-6 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{2}+\frac {b^{2}}{3}\right ) b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )-a^{2} \left (a -b \right ) \left (a +b \right ) \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\tan \left (d x +c \right )\right )+\left (-4 b^{2} a^{3} x d -2 a^{2} b^{3}-2 b^{5}\right ) \tan \left (d x +c \right )-4 a^{4} b x d}{2 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )^{2} d \,a^{2}}\) | \(165\) |
norman | \(\frac {-\frac {b^{3} \tan \left (d x +c \right )}{d \,a^{2} \left (a^{2}+b^{2}\right )}-\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{2} \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}\) | \(200\) |
risch | \(-\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {6 i x \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {6 i b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b^{4} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {2 i b^{4} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}-\frac {2 i x}{a^{2}}-\frac {2 i c}{a^{2} d}+\frac {2 i b^{3}}{\left (-i a +b \right ) d a \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}\) | \(339\) |
1/d*(1/(a^2+b^2)^2*(1/2*(-a^2+b^2)*ln(1+tan(d*x+c)^2)-2*a*b*arctan(tan(d*x +c)))+b^2/a/(a^2+b^2)/(a+b*tan(d*x+c))-b^2*(3*a^2+b^2)/a^2/(a^2+b^2)^2*ln( a+b*tan(d*x+c))+1/a^2*ln(tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (107) = 214\).
Time = 0.27 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.20 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {4 \, a^{4} b d x - 2 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (3 \, a^{3} b^{2} + a b^{4} + {\left (3 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, a^{3} b^{2} d x + a^{2} b^{3}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d\right )}} \]
-1/2*(4*a^4*b*d*x - 2*a*b^4 - (a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^ 3 + b^5)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + (3*a^3*b ^2 + a*b^4 + (3*a^2*b^3 + 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*(2*a^3*b^2*d*x + a^2*b^3) *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 = 1.62 (sec) , antiderivative size = 2961, normalized size of antiderivative = 27.67 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*x*cot(c)/tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-log (tan(c + d*x)**2 + 1)/(2*d) + log(tan(c + d*x))/d)/a**2, Eq(b, 0)), ((log( tan(c + d*x)**2 + 1)/(2*d) - log(tan(c + d*x))/d - 1/(2*d*tan(c + d*x)**2) )/b**2, Eq(a, 0)), (3*I*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*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*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*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*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*log(tan(c + d*x)**2 + 1)/(4*a* *2*d*tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d) + 4*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*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*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*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/(4*a**2*d *tan(c + d*x)**2 + 8*I*a**2*d*tan(c + d*x) - 4*a**2*d), Eq(b, -I*a)), (-3* I*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*d*x*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**2 - 8*I*a**2*d*ta n(c + d*x) - 4*a**2*d) + 3*I*d*x/(4*a**2*d*tan(c + d*x)**2 - 8*I*a**2*d...
Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.53 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, b^{2}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )} + \frac {2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \]
-1/2*(4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) - 2*b^2/(a^4 + a^2*b^2 + (a^ 3*b + a*b^3)*tan(d*x + c)) + 2*(3*a^2*b^2 + b^4)*log(b*tan(d*x + c) + a)/( a^6 + 2*a^4*b^2 + a^2*b^4) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2* a^2*b^2 + b^4) - 2*log(tan(d*x + c))/a^2)/d
Time = 0.60 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.93 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (3 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac {2 \, {\left (3 \, a^{2} b^{3} \tan \left (d x + c\right ) + b^{5} \tan \left (d x + c\right ) + 4 \, a^{3} b^{2} + 2 \, a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} - \frac {2 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}}}{2 \, d} \]
-1/2*(4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 2*(3*a^2*b^3 + b^5)*log(abs(b*tan(d*x + c) + a))/(a^6*b + 2*a^4*b^3 + a^2*b^5) - 2*(3*a^2*b^3*tan(d*x + c) + b^5 *tan(d*x + c) + 4*a^3*b^2 + 2*a*b^4)/((a^6 + 2*a^4*b^2 + a^2*b^4)*(b*tan(d *x + c) + a)) - 2*log(abs(tan(d*x + c)))/a^2)/d
Time = 4.92 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.42 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {b^2}{a\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^2+b^2\right )}{a^2\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]