Integrand size = 19, antiderivative size = 66 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {b x}{a^2+b^2}+\frac {\log (\sin (c+d x))}{a d}-\frac {b^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a \left (a^2+b^2\right ) d} \] Output:
-b*x/(a^2+b^2)+ln(sin(d*x+c))/a/d-b^2*ln(a*cos(d*x+c)+b*sin(d*x+c))/a/(a^2 +b^2)/d
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {\log (i-\tan (c+d x))}{a+i b}-\frac {2 \log (\tan (c+d x))}{a}+\frac {\log (i+\tan (c+d x))}{a-i b}+\frac {2 b^2 \log (a+b \tan (c+d x))}{a^3+a b^2}}{2 d} \] Input:
Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x]),x]
Output:
-1/2*(Log[I - Tan[c + d*x]]/(a + I*b) - (2*Log[Tan[c + d*x]])/a + Log[I + Tan[c + d*x]]/(a - I*b) + (2*b^2*Log[a + b*Tan[c + d*x]])/(a^3 + a*b^2))/d
Time = 0.43 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 4054, 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.
\(\displaystyle \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x) (a+b \tan (c+d x))}dx\) |
\(\Big \downarrow \) 4054 |
\(\displaystyle -\frac {b^2 \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\int \cot (c+d x)dx}{a}-\frac {b x}{a^2+b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b^2 \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b x}{a^2+b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {b^2 \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {\int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b x}{a^2+b^2}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {b^2 \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {b x}{a^2+b^2}+\frac {\log (-\sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 4013 |
\(\displaystyle -\frac {b^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac {b x}{a^2+b^2}+\frac {\log (-\sin (c+d x))}{a d}\) |
Input:
Int[Cot[c + d*x]/(a + b*Tan[c + d*x]),x]
Output:
-((b*x)/(a^2 + b^2)) + Log[-Sin[c + d*x]]/(a*d) - (b^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d)
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[1/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f _.)*(x_)])), x_Symbol] :> Simp[(a*c - b*d)*(x/((a^2 + b^2)*(c^2 + d^2))), x ] + (Simp[b^2/((b*c - a*d)*(a^2 + b^2)) Int[(b - a*Tan[e + f*x])/(a + b*T an[e + f*x]), x], x] - Simp[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}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 0.96 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15
method | result | size |
parallelrisch | \(\frac {a^{2} \left (2 \ln \left (\tan \left (d x +c \right )\right )-\ln \left (\sec \left (d x +c \right )^{2}\right )\right )+2 b^{2} \left (\ln \left (\tan \left (d x +c \right )\right )-\ln \left (a +b \tan \left (d x +c \right )\right )\right )-2 a b d x}{2 \left (a^{2}+b^{2}\right ) a d}\) | \(76\) |
derivativedivides | \(\frac {\frac {-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{a \left (a^{2}+b^{2}\right )}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a}}{d}\) | \(80\) |
default | \(\frac {\frac {-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{a \left (a^{2}+b^{2}\right )}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a}}{d}\) | \(80\) |
norman | \(-\frac {b x}{a^{2}+b^{2}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a d}-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a d}\) | \(86\) |
risch | \(-\frac {i x}{i b -a}+\frac {2 i b^{2} x}{\left (a^{2}+b^{2}\right ) a}+\frac {2 i b^{2} c}{\left (a^{2}+b^{2}\right ) a d}-\frac {2 i x}{a}-\frac {2 i c}{a d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{2}+b^{2}\right ) a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}\) | \(142\) |
Input:
int(cot(d*x+c)/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/2*(a^2*(2*ln(tan(d*x+c))-ln(sec(d*x+c)^2))+2*b^2*(ln(tan(d*x+c))-ln(a+b* tan(d*x+c)))-2*a*b*d*x)/(a^2+b^2)/a/d
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.48 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {2 \, a b d x + b^{2} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \] Input:
integrate(cot(d*x+c)/(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
-1/2*(2*a*b*d*x + b^2*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/ (tan(d*x + c)^2 + 1)) - (a^2 + b^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1 )))/((a^3 + a*b^2)*d)
Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 626, normalized size of antiderivative = 9.48 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)/(a+b*tan(d*x+c)),x)
Output:
Piecewise((zoo*x*cot(c)/tan(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-log(ta n(c + d*x)**2 + 1)/(2*d) + log(tan(c + d*x))/d)/a, Eq(b, 0)), ((-x - 1/(d* tan(c + d*x)))/b, Eq(a, 0)), (d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b *d) - I*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) - I*log(tan(c + d*x)**2 + 1)*ta n(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) - log(tan(c + d*x)**2 + 1)/(2*b* d*tan(c + d*x) - 2*I*b*d) + 2*I*log(tan(c + d*x))*tan(c + d*x)/(2*b*d*tan( c + d*x) - 2*I*b*d) + 2*log(tan(c + d*x))/(2*b*d*tan(c + d*x) - 2*I*b*d) + 1/(2*b*d*tan(c + d*x) - 2*I*b*d), Eq(a, -I*b)), (d*x*tan(c + d*x)/(2*b*d* tan(c + d*x) + 2*I*b*d) + I*d*x/(2*b*d*tan(c + d*x) + 2*I*b*d) + I*log(tan (c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) - log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) + 2*I*b*d) - 2*I*log(tan(c + d*x))*tan( c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) + 2*log(tan(c + d*x))/(2*b*d*tan(c + d*x) + 2*I*b*d) + 1/(2*b*d*tan(c + d*x) + 2*I*b*d), Eq(a, I*b)), (x*cot (c)/(a + b*tan(c)), Eq(d, 0)), (-a**2*log(tan(c + d*x)**2 + 1)/(2*a**3*d + 2*a*b**2*d) + 2*a**2*log(tan(c + d*x))/(2*a**3*d + 2*a*b**2*d) - 2*a*b*d* x/(2*a**3*d + 2*a*b**2*d) - 2*b**2*log(a/b + tan(c + d*x))/(2*a**3*d + 2*a *b**2*d) + 2*b**2*log(tan(c + d*x))/(2*a**3*d + 2*a*b**2*d), True))
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.27 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {2 \, b^{2} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3} + a b^{2}} + \frac {2 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} + \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{a}}{2 \, d} \] Input:
integrate(cot(d*x+c)/(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
-1/2*(2*b^2*log(b*tan(d*x + c) + a)/(a^3 + a*b^2) + 2*(d*x + c)*b/(a^2 + b ^2) + a*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) - 2*log(tan(d*x + c))/a)/d
Time = 0.17 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.45 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {b^{3} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3} b d + a b^{3} d} - \frac {{\left (d x + c\right )} b}{a^{2} d + b^{2} d} - \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} + \frac {\log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a d} \] Input:
integrate(cot(d*x+c)/(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
-b^3*log(abs(b*tan(d*x + c) + a))/(a^3*b*d + a*b^3*d) - (d*x + c)*b/(a^2*d + b^2*d) - 1/2*a*log(tan(d*x + c)^2 + 1)/(a^2*d + b^2*d) + log(abs(tan(d* x + c)))/(a*d)
Time = 1.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.44 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,d\,\left (a^2+b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )} \] Input:
int(cot(c + d*x)/(a + b*tan(c + d*x)),x)
Output:
log(tan(c + d*x))/(a*d) - (log(tan(c + d*x) - 1i)*1i)/(2*d*(a*1i - b)) - l og(tan(c + d*x) + 1i)/(2*d*(a - b*1i)) - (b^2*log(a + b*tan(c + d*x)))/(a* d*(a^2 + b^2))
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.59 \[ \int \frac {\cot (c+d x)}{a+b \tan (c+d x)} \, dx=\frac {-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{2}-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right ) b^{2}+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-a b d x}{a d \left (a^{2}+b^{2}\right )} \] Input:
int(cot(d*x+c)/(a+b*tan(d*x+c)),x)
Output:
( - log(tan((c + d*x)/2)**2 + 1)*a**2 - log(tan((c + d*x)/2)**2*a - 2*tan( (c + d*x)/2)*b - a)*b**2 + log(tan((c + d*x)/2))*a**2 + log(tan((c + d*x)/ 2))*b**2 - a*b*d*x)/(a*d*(a**2 + b**2))