Integrand size = 14, antiderivative size = 152 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx=-\frac {\cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (a+b x)}\right )}{2 d}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{2 d} \]
-arccot(b*x+a)*ln(2/(1-I*(b*x+a)))/d+arccot(b*x+a)*ln(2*b*(d*x+c)/(b*c+I*d -a*d)/(1-I*(b*x+a)))/d-1/2*I*polylog(2,1-2/(1-I*(b*x+a)))/d+1/2*I*polylog( 2,1-2*b*(d*x+c)/(b*c+I*d-a*d)/(1-I*(b*x+a)))/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(325\) vs. \(2(152)=304\).
Time = 0.20 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.14 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx=\frac {\left (\cot ^{-1}(a+b x)+\arctan (a+b x)\right ) \log (c+d x)+\arctan (a+b x) \left (\log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )-\log \left (\sin \left (\arctan \left (\frac {b c-a d}{d}\right )+\arctan (a+b x)\right )\right )\right )+\frac {1}{2} \left (\frac {1}{4} i (\pi -2 \arctan (a+b x))^2+i \left (\arctan \left (\frac {b c-a d}{d}\right )+\arctan (a+b x)\right )^2-(\pi -2 \arctan (a+b x)) \log \left (1+e^{-2 i \arctan (a+b x)}\right )-2 \left (\arctan \left (\frac {b c-a d}{d}\right )+\arctan (a+b x)\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {b c-a d}{d}\right )+\arctan (a+b x)\right )}\right )+(\pi -2 \arctan (a+b x)) \log \left (\frac {2}{\sqrt {1+(a+b x)^2}}\right )+2 \left (\arctan \left (\frac {b c-a d}{d}\right )+\arctan (a+b x)\right ) \log \left (2 \sin \left (\arctan \left (\frac {b c-a d}{d}\right )+\arctan (a+b x)\right )\right )+i \operatorname {PolyLog}\left (2,-e^{-2 i \arctan (a+b x)}\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {b c-a d}{d}\right )+\arctan (a+b x)\right )}\right )\right )}{d} \]
((ArcCot[a + b*x] + ArcTan[a + b*x])*Log[c + d*x] + ArcTan[a + b*x]*(Log[1 /Sqrt[1 + (a + b*x)^2]] - Log[Sin[ArcTan[(b*c - a*d)/d] + ArcTan[a + b*x]] ]) + ((I/4)*(Pi - 2*ArcTan[a + b*x])^2 + I*(ArcTan[(b*c - a*d)/d] + ArcTan [a + b*x])^2 - (Pi - 2*ArcTan[a + b*x])*Log[1 + E^((-2*I)*ArcTan[a + b*x]) ] - 2*(ArcTan[(b*c - a*d)/d] + ArcTan[a + b*x])*Log[1 - E^((2*I)*(ArcTan[( b*c - a*d)/d] + ArcTan[a + b*x]))] + (Pi - 2*ArcTan[a + b*x])*Log[2/Sqrt[1 + (a + b*x)^2]] + 2*(ArcTan[(b*c - a*d)/d] + ArcTan[a + b*x])*Log[2*Sin[A rcTan[(b*c - a*d)/d] + ArcTan[a + b*x]]] + I*PolyLog[2, -E^((-2*I)*ArcTan[ a + b*x])] + I*PolyLog[2, E^((2*I)*(ArcTan[(b*c - a*d)/d] + ArcTan[a + b*x ]))])/2)/d
Time = 0.54 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5571, 27, 5382, 2849, 2752, 2897}
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.
| \(\displaystyle \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx\) |
| \(\Big \downarrow \) 5571 |
| \(\displaystyle \frac {\int \frac {b \cot ^{-1}(a+b x)}{b \left (c-\frac {a d}{b}\right )+d (a+b x)}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\cot ^{-1}(a+b x)}{d (a+b x)-a d+b c}d(a+b x)\) |
| \(\Big \downarrow \) 5382 |
| \(\displaystyle \frac {\int \frac {\log \left (\frac {2 (b c-a d+d (a+b x))}{(b c-a d+i d) (1-i (a+b x))}\right )}{(a+b x)^2+1}d(a+b x)}{d}-\frac {\int \frac {\log \left (\frac {2}{1-i (a+b x)}\right )}{(a+b x)^2+1}d(a+b x)}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{d}-\frac {\log \left (\frac {2}{1-i (a+b x)}\right ) \cot ^{-1}(a+b x)}{d}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {\int \frac {\log \left (\frac {2 (b c-a d+d (a+b x))}{(b c-a d+i d) (1-i (a+b x))}\right )}{(a+b x)^2+1}d(a+b x)}{d}-\frac {i \int \frac {\log \left (\frac {2}{1-i (a+b x)}\right )}{1-\frac {2}{1-i (a+b x)}}d\frac {1}{1-i (a+b x)}}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{d}-\frac {\log \left (\frac {2}{1-i (a+b x)}\right ) \cot ^{-1}(a+b x)}{d}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {\int \frac {\log \left (\frac {2 (b c-a d+d (a+b x))}{(b c-a d+i d) (1-i (a+b x))}\right )}{(a+b x)^2+1}d(a+b x)}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{d}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (a+b x)}\right )}{2 d}-\frac {\log \left (\frac {2}{1-i (a+b x)}\right ) \cot ^{-1}(a+b x)}{d}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {i \operatorname {PolyLog}\left (2,1-\frac {2 (b c-a d+d (a+b x))}{(b c-a d+i d) (1-i (a+b x))}\right )}{2 d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 (d (a+b x)-a d+b c)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{d}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (a+b x)}\right )}{2 d}-\frac {\log \left (\frac {2}{1-i (a+b x)}\right ) \cot ^{-1}(a+b x)}{d}\) |
-((ArcCot[a + b*x]*Log[2/(1 - I*(a + b*x))])/d) + (ArcCot[a + b*x]*Log[(2* (b*c - a*d + d*(a + b*x)))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/d - ((I /2)*PolyLog[2, 1 - 2/(1 - I*(a + b*x))])/d + ((I/2)*PolyLog[2, 1 - (2*(b*c - a*d + d*(a + b*x)))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/d
3.2.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Si mp[(-(a + b*ArcCot[c*x]))*(Log[2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcCot[ c*x])*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - Simp[b*(c/e) Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Simp[b*(c/e) Int[Log[2* c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x]) /; FreeQ[{a , b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && I GtQ[p, 0]
Time = 0.79 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \operatorname {arccot}\left (b x +a \right )}{d}-b \left (-\frac {i \ln \left (a d -b c -d \left (b x +a \right )\right ) \left (\ln \left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\ln \left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}-\frac {i \left (\operatorname {dilog}\left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\operatorname {dilog}\left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}\right )}{b}\) | \(187\) |
| default | \(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \operatorname {arccot}\left (b x +a \right )}{d}-b \left (-\frac {i \ln \left (a d -b c -d \left (b x +a \right )\right ) \left (\ln \left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\ln \left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}-\frac {i \left (\operatorname {dilog}\left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\operatorname {dilog}\left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}\right )}{b}\) | \(187\) |
| parts | \(\frac {\ln \left (d x +c \right ) \operatorname {arccot}\left (b x +a \right )}{d}+b \left (-\frac {i \ln \left (d x +c \right ) \left (\ln \left (\frac {i d -a d +b c -b \left (d x +c \right )}{-a d +b c +i d}\right )-\ln \left (\frac {i d +a d -b c +b \left (d x +c \right )}{a d -b c +i d}\right )\right )}{2 d b}-\frac {i \left (\operatorname {dilog}\left (\frac {i d -a d +b c -b \left (d x +c \right )}{-a d +b c +i d}\right )-\operatorname {dilog}\left (\frac {i d +a d -b c +b \left (d x +c \right )}{a d -b c +i d}\right )\right )}{2 d b}\right )\) | \(193\) |
| risch | \(-\frac {i \operatorname {dilog}\left (\frac {i a d -i b c +\left (-i b x -i a +1\right ) d -d}{i a d -i b c -d}\right )}{2 d}-\frac {i \ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d -i b c +\left (-i b x -i a +1\right ) d -d}{i a d -i b c -d}\right )}{2 d}+\frac {\pi \ln \left (i a d -i b c +\left (-i b x -i a +1\right ) d -d \right )}{2 d}+\frac {i \operatorname {dilog}\left (\frac {-i a d +i b c +\left (i b x +i a +1\right ) d -d}{-i a d +i b c -d}\right )}{2 d}+\frac {i \ln \left (i b x +i a +1\right ) \ln \left (\frac {-i a d +i b c +\left (i b x +i a +1\right ) d -d}{-i a d +i b c -d}\right )}{2 d}\) | \(264\) |
1/b*(b*ln(a*d-b*c-d*(b*x+a))/d*arccot(b*x+a)-b*(-1/2*I*ln(a*d-b*c-d*(b*x+a ))*(ln((I*d+d*(b*x+a))/(a*d-b*c+I*d))-ln((I*d-d*(b*x+a))/(b*c+I*d-a*d)))/d -1/2*I*(dilog((I*d+d*(b*x+a))/(a*d-b*c+I*d))-dilog((I*d-d*(b*x+a))/(b*c+I* d-a*d)))/d))
\[ \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{d x + c} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (130) = 260\).
Time = 0.36 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx=\frac {\operatorname {arccot}\left (b x + a\right ) \log \left (d x + c\right )}{d} + \frac {\arctan \left (\frac {b^{2} x + a b}{b}\right ) \log \left (d x + c\right )}{d} + \frac {\arctan \left (\frac {b d^{2} x + b c d}{b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 1\right )} d^{2}}, \frac {b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x}{b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 1\right )} d^{2}}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - \arctan \left (b x + a\right ) \log \left (\frac {b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}}{b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 1\right )} d^{2}}\right ) + i \, {\rm Li}_2\left (\frac {i \, b d x + {\left (i \, a + 1\right )} d}{-i \, b c + {\left (i \, a + 1\right )} d}\right ) - i \, {\rm Li}_2\left (\frac {i \, b d x + {\left (i \, a - 1\right )} d}{-i \, b c + {\left (i \, a - 1\right )} d}\right )}{2 \, d} \]
arccot(b*x + a)*log(d*x + c)/d + arctan((b^2*x + a*b)/b)*log(d*x + c)/d + 1/2*(arctan2((b*d^2*x + b*c*d)/(b^2*c^2 - 2*a*b*c*d + (a^2 + 1)*d^2), (b^2 *c^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x)/(b^2*c^2 - 2*a*b*c*d + (a^2 + 1)*d ^2))*log(b^2*x^2 + 2*a*b*x + a^2 + 1) - arctan(b*x + a)*log((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)/(b^2*c^2 - 2*a*b*c*d + (a^2 + 1)*d^2)) + I*dilog(( I*b*d*x + (I*a + 1)*d)/(-I*b*c + (I*a + 1)*d)) - I*dilog((I*b*d*x + (I*a - 1)*d)/(-I*b*c + (I*a - 1)*d)))/d
\[ \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{d x + c} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+d\,x} \,d x \]