Integrand size = 21, antiderivative size = 114 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )^2 d}+\frac {a^3}{b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
-2*a*b*x/(a^2+b^2)^2+(a^2-b^2)*ln(cos(d*x+c))/(a^2+b^2)^2/d+a^2*(a^2+3*b^2 )*ln(a+b*tan(d*x+c))/b^2/(a^2+b^2)^2/d+a^3/b^2/(a^2+b^2)/d/(a+b*tan(d*x+c) )
Result contains complex when optimal does not.
Time = 3.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {\log (i-\tan (c+d x))}{(a+i b)^2}+\frac {\log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 a^2 \left (\left (3+\frac {a^2}{b^2}\right ) \log (a+b \tan (c+d x))+\frac {a \left (a^2+b^2\right )}{b^2 (a+b \tan (c+d x))}\right )}{\left (a^2+b^2\right )^2}}{2 d} \]
-1/2*(Log[I - Tan[c + d*x]]/(a + I*b)^2 + Log[I + Tan[c + d*x]]/(a - I*b)^ 2 - (2*a^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^2 + b^2)^2)/d
Time = 0.66 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4048, 3042, 4109, 3042, 3956, 4100, 16}
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.
| \(\displaystyle \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^3}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {a^2-b \tan (c+d x) a+\left (a^2+b^2\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a^2-b \tan (c+d x) a+\left (a^2+b^2\right ) \tan (c+d x)^2}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {\frac {a^2 \left (a^2+3 b^2\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b \left (a^2-b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}-\frac {2 a b^2 x}{a^2+b^2}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b \left (a^2-b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}-\frac {2 a b^2 x}{a^2+b^2}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {a^2 \left (a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {2 a b^2 x}{a^2+b^2}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {\frac {a^2 \left (a^2+3 b^2\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}+\frac {b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {2 a b^2 x}{a^2+b^2}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}+\frac {b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {2 a b^2 x}{a^2+b^2}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
((-2*a*b^2*x)/(a^2 + b^2) + (b*(a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2) *d) + (a^2*(a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)*d))/(b*(a ^2 + b^2)) - (a^2*Tan[c + d*x])/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))
3.5.70.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/(b*f) Subst[Int[(a + x)^m, x], x, b* Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 )/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*A + b*B - a *C)*(x/(a^2 + b^2)), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[( 1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(A*b - a*B - b*C)/( a^2 + b^2) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] & & NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C , 0]
Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} b^{2}}+\frac {a^{3}}{b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(114\) |
| default | \(\frac {\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} b^{2}}+\frac {a^{3}}{b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(114\) |
| norman | \(\frac {\frac {a^{3}}{d \left (a^{2}+b^{2}\right ) b^{2}}-\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{2} d}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(178\) |
| parallelrisch | \(-\frac {4 b^{4} a \tan \left (d x +c \right ) x d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{3}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) b^{5}-2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b -6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{3}+4 a^{2} b^{3} x d +\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{2}-a \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}-2 a^{5} \ln \left (a +b \tan \left (d x +c \right )\right )-6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{3} b^{2}-2 a^{5}-2 a^{3} b^{2}}{2 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{2} d}\) | \(240\) |
| risch | \(-\frac {i x}{2 i a b -a^{2}+b^{2}}-\frac {2 i a^{4} x}{b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i a^{4} c}{b^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {6 i a^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {6 i a^{2} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 i x}{b^{2}}+\frac {2 i c}{b^{2} d}-\frac {2 i a^{3}}{\left (i b +a \right ) d b \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}\) | \(337\) |
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)))+a^2*(a^2+3*b^2)/(a^2+b^2)^2/b^2*ln(a+b*tan(d*x+c))+a^3/b^2/(a^2+b^2) /(a+b*tan(d*x+c)))
Time = 0.27 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.98 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {4 \, a^{2} b^{3} d x - 2 \, a^{3} b^{2} - {\left (a^{5} + 3 \, a^{3} b^{2} + {\left (a^{4} b + 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, a b^{4} d x + a^{4} b\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d\right )}} \]
-1/2*(4*a^2*b^3*d*x - 2*a^3*b^2 - (a^5 + 3*a^3*b^2 + (a^4*b + 3*a^2*b^3)*t an(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)) + (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(1/(tan(d*x + c)^2 + 1)) + 2*(2*a*b^4*d*x + a^4*b)*tan(d*x + c) )/((a^4*b^3 + 2*a^2*b^5 + b^7)*d*tan(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b ^6)*d)
Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 1992, normalized size of antiderivative = 17.47 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*x*tan(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-log(tan(c + d *x)**2 + 1)/(2*d) + tan(c + d*x)**2/(2*d))/a**2, Eq(b, 0)), (3*I*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) + 6*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) - 3*I*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*log(tan(c + d*x)**2 + 1)*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) - 4*I*log(tan(c + d*x)**2 + 1)*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) - 2*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d *tan(c + d*x) - 4*b**2*d) - 5*I*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) - 4/(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)), (-3*I*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) + 6*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) + 3 *I*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 *log(tan(c + d*x)**2 + 1)*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) + 4*I*log(tan(c + d*x)**2 + 1)*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) - 2*log( tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 5*I*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*ta...
Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.36 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, a^{3}}{a^{3} b^{2} + a b^{4} + {\left (a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )} - \frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{2 \, d} \]
1/2*(2*a^3/(a^3*b^2 + a*b^4 + (a^2*b^3 + b^5)*tan(d*x + c)) - 4*(d*x + c)* a*b/(a^4 + 2*a^2*b^2 + b^4) + 2*(a^4 + 3*a^2*b^2)*log(b*tan(d*x + c) + a)/ (a^4*b^2 + 2*a^2*b^4 + b^6) - (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2 *a^2*b^2 + b^4))/d
Time = 0.66 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.59 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (a^{4} \tan \left (d x + c\right ) + 3 \, a^{2} b^{2} \tan \left (d x + c\right ) + 2 \, a^{3} b\right )}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]
-1/2*(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*(a^4 + 3*a^2*b^2)*log(abs(b*tan(d*x + c) + a))/(a^4*b^2 + 2*a^2*b^4 + b^6) + 2*(a^4*tan(d*x + c) + 3*a^2*b^2*t an(d*x + c) + 2*a^3*b)/((a^4*b + 2*a^2*b^3 + b^5)*(b*tan(d*x + c) + a)))/d
Time = 4.76 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.20 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {a^3}{b^2\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2+3\,b^2\right )}{b^2\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]