Integrand size = 15, antiderivative size = 212 \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=-\frac {\cot ^{-1}(c x)}{x}-\frac {1}{2} i \arctan (x) \log \left (1-\frac {i}{c x}\right )+\frac {1}{2} i \arctan (x) \log \left (1+\frac {i}{c x}\right )-c \log (x)+\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac {1}{2} i \arctan (x) \log \left (-\frac {2 i (i+c x)}{(1+c) (1-i x)}\right )+\frac {1}{2} c \log \left (1+c^2 x^2\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,1+\frac {2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (2,1+\frac {2 i (i+c x)}{(1+c) (1-i x)}\right ) \] Output:
-arccot(c*x)/x-1/2*I*arctan(x)*ln(1-I/c/x)+1/2*I*arctan(x)*ln(1+I/c/x)-c*l n(x)+1/2*I*arctan(x)*ln(-2*I*(I-c*x)/(1-c)/(1-I*x))-1/2*I*arctan(x)*ln(-2* I*(I+c*x)/(1+c)/(1-I*x))+1/2*c*ln(c^2*x^2+1)+1/4*polylog(2,1+2*I*(I-c*x)/( 1-c)/(1-I*x))-1/4*polylog(2,1+2*I*(I+c*x)/(1+c)/(1-I*x))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(619\) vs. \(2(212)=424\).
Time = 1.31 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.92 \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=-\frac {\cot ^{-1}(c x)}{x}-c \log \left (\frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )-\frac {c \left (2 \arccos \left (\frac {1+c^2}{-1+c^2}\right ) \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )-4 \cot ^{-1}(c x) \text {arctanh}\left (\frac {c x}{\sqrt {-c^2}}\right )-\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )-2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )\right ) \log \left (-\frac {2 \left (c^2+i \sqrt {-c^2}\right ) (-i+c x)}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )-\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )+2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )\right ) \log \left (\frac {2 i \left (i c^2+\sqrt {-c^2}\right ) (i+c x)}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )+\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )-2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )+2 i \text {arctanh}\left (\frac {c x}{\sqrt {-c^2}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2} e^{-i \cot ^{-1}(c x)}}{\sqrt {-1+c^2} \sqrt {-1-c^2+\left (-1+c^2\right ) \cos \left (2 \cot ^{-1}(c x)\right )}}\right )+\left (\arccos \left (\frac {1+c^2}{-1+c^2}\right )+2 i \text {arctanh}\left (\frac {\sqrt {-c^2}}{c x}\right )-2 i \text {arctanh}\left (\frac {c x}{\sqrt {-c^2}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2} e^{i \cot ^{-1}(c x)}}{\sqrt {-1+c^2} \sqrt {-1-c^2+\left (-1+c^2\right ) \cos \left (2 \cot ^{-1}(c x)\right )}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (1+c^2-2 i \sqrt {-c^2}\right ) \left (\sqrt {-c^2}+c x\right )}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+c^2+2 i \sqrt {-c^2}\right ) \left (\sqrt {-c^2}+c x\right )}{\left (-1+c^2\right ) \left (\sqrt {-c^2}-c x\right )}\right )\right )\right )}{4 \sqrt {-c^2}} \] Input:
Integrate[ArcCot[c*x]/(x^2*(1 + x^2)),x]
Output:
-(ArcCot[c*x]/x) - c*Log[1/Sqrt[1 + 1/(c^2*x^2)]] - (c*(2*ArcCos[(1 + c^2) /(-1 + c^2)]*ArcTanh[Sqrt[-c^2]/(c*x)] - 4*ArcCot[c*x]*ArcTanh[(c*x)/Sqrt[ -c^2]] - (ArcCos[(1 + c^2)/(-1 + c^2)] - (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)])* Log[(-2*(c^2 + I*Sqrt[-c^2])*(-I + c*x))/((-1 + c^2)*(Sqrt[-c^2] - c*x))] - (ArcCos[(1 + c^2)/(-1 + c^2)] + (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)])*Log[((2 *I)*(I*c^2 + Sqrt[-c^2])*(I + c*x))/((-1 + c^2)*(Sqrt[-c^2] - c*x))] + (Ar cCos[(1 + c^2)/(-1 + c^2)] - (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)] + (2*I)*ArcTa nh[(c*x)/Sqrt[-c^2]])*Log[(Sqrt[2]*Sqrt[-c^2])/(Sqrt[-1 + c^2]*E^(I*ArcCot [c*x])*Sqrt[-1 - c^2 + (-1 + c^2)*Cos[2*ArcCot[c*x]]])] + (ArcCos[(1 + c^2 )/(-1 + c^2)] + (2*I)*ArcTanh[Sqrt[-c^2]/(c*x)] - (2*I)*ArcTanh[(c*x)/Sqrt [-c^2]])*Log[(Sqrt[2]*Sqrt[-c^2]*E^(I*ArcCot[c*x]))/(Sqrt[-1 + c^2]*Sqrt[- 1 - c^2 + (-1 + c^2)*Cos[2*ArcCot[c*x]]])] + I*(-PolyLog[2, ((1 + c^2 - (2 *I)*Sqrt[-c^2])*(Sqrt[-c^2] + c*x))/((-1 + c^2)*(Sqrt[-c^2] - c*x))] + Pol yLog[2, ((1 + c^2 + (2*I)*Sqrt[-c^2])*(Sqrt[-c^2] + c*x))/((-1 + c^2)*(Sqr t[-c^2] - c*x))])))/(4*Sqrt[-c^2])
Time = 0.87 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.61, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5454, 5362, 243, 47, 14, 16, 5444, 2920, 27, 2005, 5411, 2009}
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.
| \(\displaystyle \int \frac {\cot ^{-1}(c x)}{x^2 \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 5454 |
| \(\displaystyle \int \frac {\cot ^{-1}(c x)}{x^2}dx-\int \frac {\cot ^{-1}(c x)}{x^2+1}dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -c \int \frac {1}{x \left (c^2 x^2+1\right )}dx-\int \frac {\cot ^{-1}(c x)}{x^2+1}dx-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{2} c \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx^2-\int \frac {\cot ^{-1}(c x)}{x^2+1}dx-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle -\frac {1}{2} c \left (\int \frac {1}{x^2}dx^2-c^2 \int \frac {1}{c^2 x^2+1}dx^2\right )-\int \frac {\cot ^{-1}(c x)}{x^2+1}dx-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle -\frac {1}{2} c \left (\log \left (x^2\right )-c^2 \int \frac {1}{c^2 x^2+1}dx^2\right )-\int \frac {\cot ^{-1}(c x)}{x^2+1}dx-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\int \frac {\cot ^{-1}(c x)}{x^2+1}dx-\frac {1}{2} c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 5444 |
| \(\displaystyle -\frac {1}{2} i \int \frac {\log \left (1-\frac {i}{c x}\right )}{x^2+1}dx+\frac {1}{2} i \int \frac {\log \left (1+\frac {i}{c x}\right )}{x^2+1}dx-\frac {1}{2} c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 2920 |
| \(\displaystyle -\frac {1}{2} i \left (\arctan (x) \log \left (1-\frac {i}{c x}\right )-\frac {i \int \frac {c \arctan (x)}{\left (c-\frac {i}{x}\right ) x^2}dx}{c}\right )+\frac {1}{2} i \left (\frac {i \int \frac {c \arctan (x)}{\left (c+\frac {i}{x}\right ) x^2}dx}{c}+\arctan (x) \log \left (1+\frac {i}{c x}\right )\right )-\frac {1}{2} c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} i \left (\arctan (x) \log \left (1-\frac {i}{c x}\right )-i \int \frac {\arctan (x)}{\left (c-\frac {i}{x}\right ) x^2}dx\right )+\frac {1}{2} i \left (i \int \frac {\arctan (x)}{\left (c+\frac {i}{x}\right ) x^2}dx+\arctan (x) \log \left (1+\frac {i}{c x}\right )\right )-\frac {1}{2} c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -\frac {1}{2} i \left (\arctan (x) \log \left (1-\frac {i}{c x}\right )-i \int \frac {\arctan (x)}{x (c x-i)}dx\right )+\frac {1}{2} i \left (i \int \frac {\arctan (x)}{x (c x+i)}dx+\arctan (x) \log \left (1+\frac {i}{c x}\right )\right )-\frac {1}{2} c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle -\frac {1}{2} i \left (\arctan (x) \log \left (1-\frac {i}{c x}\right )-i \int \left (\frac {i \arctan (x)}{x}-\frac {i c \arctan (x)}{c x-i}\right )dx\right )+\frac {1}{2} i \left (i \int \left (\frac {i c \arctan (x)}{c x+i}-\frac {i \arctan (x)}{x}\right )dx+\arctan (x) \log \left (1+\frac {i}{c x}\right )\right )-\frac {1}{2} c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )-\frac {\cot ^{-1}(c x)}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} i \left (\arctan (x) \log \left (1-\frac {i}{c x}\right )-i \left (-i \arctan (x) \log \left (-\frac {2 i (-c x+i)}{(1-c) (1-i x)}\right )+i \arctan (x) \log \left (\frac {2}{1-i x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2 i (i-c x)}{(1-c) (1-i x)}+1\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-i x}\right )-\frac {\operatorname {PolyLog}(2,-i x)}{2}+\frac {\operatorname {PolyLog}(2,i x)}{2}\right )\right )+\frac {1}{2} i \left (i \left (i \arctan (x) \log \left (-\frac {2 i (c x+i)}{(c+1) (1-i x)}\right )-i \arctan (x) \log \left (\frac {2}{1-i x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2 i (c x+i)}{(c+1) (1-i x)}+1\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-i x}\right )+\frac {\operatorname {PolyLog}(2,-i x)}{2}-\frac {\operatorname {PolyLog}(2,i x)}{2}\right )+\arctan (x) \log \left (1+\frac {i}{c x}\right )\right )-\frac {1}{2} c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )-\frac {\cot ^{-1}(c x)}{x}\) |
Input:
Int[ArcCot[c*x]/(x^2*(1 + x^2)),x]
Output:
-(ArcCot[c*x]/x) - (c*(Log[x^2] - Log[1 + c^2*x^2]))/2 - (I/2)*(ArcTan[x]* Log[1 - I/(c*x)] - I*(I*ArcTan[x]*Log[2/(1 - I*x)] - I*ArcTan[x]*Log[((-2* I)*(I - c*x))/((1 - c)*(1 - I*x))] + PolyLog[2, 1 - 2/(1 - I*x)]/2 - PolyL og[2, (-I)*x]/2 + PolyLog[2, I*x]/2 - PolyLog[2, 1 + ((2*I)*(I - c*x))/((1 - c)*(1 - I*x))]/2)) + (I/2)*(ArcTan[x]*Log[1 + I/(c*x)] + I*((-I)*ArcTan [x]*Log[2/(1 - I*x)] + I*ArcTan[x]*Log[((-2*I)*(I + c*x))/((1 + c)*(1 - I* x))] - PolyLog[2, 1 - 2/(1 - I*x)]/2 + PolyLog[2, (-I)*x]/2 - PolyLog[2, I *x]/2 + PolyLog[2, 1 + ((2*I)*(I + c*x))/((1 + c)*(1 - I*x))]/2))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.) *(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p Int[u*(x^(n - 1)/(d + e*x^n)), x ], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^n])^p/(m + 1)), x] + Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Int[ArcCot[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[I/2 Int[ Log[1 - I/(c*x)]/(d + e*x^2), x], x] - Simp[I/2 Int[Log[1 + I/(c*x)]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcCot[c*x])^p/(d + e*x^2) ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Time = 1.34 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(-\frac {\pi \arctan \left (x \right )}{2}-\frac {\pi }{2 x}+\frac {\ln \left (-i c x +1\right ) \ln \left (\frac {-i c x -c}{-c -1}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {-i c x -c}{-c -1}\right )}{4}-\frac {c \ln \left (-i c x \right )}{2}+\frac {c \ln \left (-i c x +1\right )}{2}+\frac {i \ln \left (-i c x +1\right )}{2 x}-\frac {\ln \left (-i c x +1\right ) \ln \left (\frac {-i c x +c}{c -1}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {-i c x +c}{c -1}\right )}{4}+\frac {\ln \left (i c x +1\right ) \ln \left (\frac {i c x -c}{-c -1}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {i c x -c}{-c -1}\right )}{4}-\frac {c \ln \left (i c x \right )}{2}+\frac {c \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (i c x +1\right )}{2 x}-\frac {\ln \left (i c x +1\right ) \ln \left (\frac {i c x +c}{c -1}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {i c x +c}{c -1}\right )}{4}\) | \(257\) |
| parts | \(-\operatorname {arccot}\left (c x \right ) \arctan \left (x \right )-\frac {\operatorname {arccot}\left (c x \right )}{x}+c \left (-\ln \left (x \right )+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c -2}-\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c \left (c -1\right )}+\frac {\arctan \left (x \right )^{2}}{2 c -2}+\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c -4}-\frac {\arctan \left (x \right )^{2}}{2 c \left (c -1\right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c \left (c -1\right )}-\frac {i \arctan \left (x \right ) \ln \left (1-\frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{2 c}-\frac {\arctan \left (x \right )^{2}}{2 c}-\frac {\operatorname {polylog}\left (2, \frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{4 c}\right )\) | \(287\) |
| derivativedivides | \(c \left (-\frac {\operatorname {arccot}\left (c x \right )}{c x}-\frac {\operatorname {arccot}\left (c x \right ) \arctan \left (x \right )}{c}+c^{3} \left (-\frac {\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}}{c^{3}}+\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{3} \left (c -1\right )}-\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{4} \left (c -1\right )}+\frac {\arctan \left (x \right )^{2}}{2 c^{3} \left (c -1\right )}+\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{3} \left (c -1\right )}-\frac {\arctan \left (x \right )^{2}}{2 c^{4} \left (c -1\right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{4} \left (c -1\right )}-\frac {i \arctan \left (x \right ) \ln \left (1-\frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{2 c^{4}}-\frac {\arctan \left (x \right )^{2}}{2 c^{4}}-\frac {\operatorname {polylog}\left (2, \frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{4 c^{4}}\right )\right )\) | \(310\) |
| default | \(c \left (-\frac {\operatorname {arccot}\left (c x \right )}{c x}-\frac {\operatorname {arccot}\left (c x \right ) \arctan \left (x \right )}{c}+c^{3} \left (-\frac {\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}}{c^{3}}+\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{3} \left (c -1\right )}-\frac {i \ln \left (1-\frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right ) \arctan \left (x \right )}{2 c^{4} \left (c -1\right )}+\frac {\arctan \left (x \right )^{2}}{2 c^{3} \left (c -1\right )}+\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{3} \left (c -1\right )}-\frac {\arctan \left (x \right )^{2}}{2 c^{4} \left (c -1\right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (1+c \right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (c -1\right )}\right )}{4 c^{4} \left (c -1\right )}-\frac {i \arctan \left (x \right ) \ln \left (1-\frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{2 c^{4}}-\frac {\arctan \left (x \right )^{2}}{2 c^{4}}-\frac {\operatorname {polylog}\left (2, \frac {\left (c -1\right ) \left (i x +1\right )^{2}}{\left (x^{2}+1\right ) \left (1+c \right )}\right )}{4 c^{4}}\right )\right )\) | \(310\) |
Input:
int(arccot(c*x)/x^2/(x^2+1),x,method=_RETURNVERBOSE)
Output:
-1/2*Pi*arctan(x)-1/2*Pi/x+1/4*ln(1-I*c*x)*ln((-c-I*c*x)/(-c-1))+1/4*dilog ((-c-I*c*x)/(-c-1))-1/2*c*ln(-I*c*x)+1/2*c*ln(1-I*c*x)+1/2*I*ln(1-I*c*x)/x -1/4*ln(1-I*c*x)*ln((c-I*c*x)/(c-1))-1/4*dilog((c-I*c*x)/(c-1))+1/4*ln(1+I *c*x)*ln((-c+I*c*x)/(-c-1))+1/4*dilog((-c+I*c*x)/(-c-1))-1/2*c*ln(I*c*x)+1 /2*c*ln(1+I*c*x)-1/2*I*ln(1+I*c*x)/x-1/4*ln(1+I*c*x)*ln((c+I*c*x)/(c-1))-1 /4*dilog((c+I*c*x)/(c-1))
\[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \] Input:
integrate(arccot(c*x)/x^2/(x^2+1),x, algorithm="fricas")
Output:
integral(arccot(c*x)/(x^4 + x^2), x)
\[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\operatorname {acot}{\left (c x \right )}}{x^{2} \left (x^{2} + 1\right )}\, dx \] Input:
integrate(acot(c*x)/x**2/(x**2+1),x)
Output:
Integral(acot(c*x)/(x**2*(x**2 + 1)), x)
Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=-{\left (\frac {1}{x} + \arctan \left (x\right )\right )} \operatorname {arccot}\left (c x\right ) - \frac {1}{2} \, \arctan \left (c x\right ) \arctan \left (x\right ) + \frac {1}{2} \, \arctan \left (x\right ) \arctan \left (\frac {c x}{c - 1}, -\frac {1}{c - 1}\right ) + \frac {1}{2} \, c \log \left (c^{2} x^{2} + 1\right ) - c \log \left (x\right ) - \frac {1}{8} \, \log \left (x^{2} + 1\right ) \log \left (\frac {c^{2} x^{2} + 1}{c^{2} + 2 \, c + 1}\right ) + \frac {1}{8} \, \log \left (x^{2} + 1\right ) \log \left (\frac {c^{2} x^{2} + 1}{c^{2} - 2 \, c + 1}\right ) - \frac {1}{4} \, {\rm Li}_2\left (\frac {i \, c x + c}{c + 1}\right ) - \frac {1}{4} \, {\rm Li}_2\left (-\frac {i \, c x - c}{c + 1}\right ) + \frac {1}{4} \, {\rm Li}_2\left (\frac {i \, c x + c}{c - 1}\right ) + \frac {1}{4} \, {\rm Li}_2\left (-\frac {i \, c x - c}{c - 1}\right ) \] Input:
integrate(arccot(c*x)/x^2/(x^2+1),x, algorithm="maxima")
Output:
-(1/x + arctan(x))*arccot(c*x) - 1/2*arctan(c*x)*arctan(x) + 1/2*arctan(x) *arctan2(c*x/(c - 1), -1/(c - 1)) + 1/2*c*log(c^2*x^2 + 1) - c*log(x) - 1/ 8*log(x^2 + 1)*log((c^2*x^2 + 1)/(c^2 + 2*c + 1)) + 1/8*log(x^2 + 1)*log(( c^2*x^2 + 1)/(c^2 - 2*c + 1)) - 1/4*dilog((I*c*x + c)/(c + 1)) - 1/4*dilog (-(I*c*x - c)/(c + 1)) + 1/4*dilog((I*c*x + c)/(c - 1)) + 1/4*dilog(-(I*c* x - c)/(c - 1))
\[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \] Input:
integrate(arccot(c*x)/x^2/(x^2+1),x, algorithm="giac")
Output:
integrate(arccot(c*x)/((x^2 + 1)*x^2), x)
Timed out. \[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\mathrm {acot}\left (c\,x\right )}{x^2\,\left (x^2+1\right )} \,d x \] Input:
int(acot(c*x)/(x^2*(x^2 + 1)),x)
Output:
int(acot(c*x)/(x^2*(x^2 + 1)), x)
\[ \int \frac {\cot ^{-1}(c x)}{x^2 \left (1+x^2\right )} \, dx=\frac {\mathit {acot} \left (c x \right )^{2} c^{3} x -2 \mathit {acot} \left (c x \right ) c^{2}-\mathit {atan} \left (\frac {1}{c x}\right )^{2} c^{3} x +\mathit {atan} \left (\frac {1}{c x}\right )^{2} c x +2 \mathit {atan} \left (\frac {1}{c x}\right ) c^{2}-2 \mathit {atan} \left (\frac {1}{c x}\right )+2 \left (\int \frac {\mathit {atan} \left (\frac {1}{c x}\right )}{c^{2} x^{4}+c^{2} x^{2}+x^{2}+1}d x \right ) c^{2} x -2 \left (\int \frac {\mathit {atan} \left (\frac {1}{c x}\right )}{c^{2} x^{4}+c^{2} x^{2}+x^{2}+1}d x \right ) x +\mathrm {log}\left (c^{2} x^{2}+1\right ) c x -2 \,\mathrm {log}\left (x \right ) c x}{2 x} \] Input:
int(acot(c*x)/x^2/(x^2+1),x)
Output:
(acot(c*x)**2*c**3*x - 2*acot(c*x)*c**2 - atan(1/(c*x))**2*c**3*x + atan(1 /(c*x))**2*c*x + 2*atan(1/(c*x))*c**2 - 2*atan(1/(c*x)) + 2*int(atan(1/(c* x))/(c**2*x**4 + c**2*x**2 + x**2 + 1),x)*c**2*x - 2*int(atan(1/(c*x))/(c* *2*x**4 + c**2*x**2 + x**2 + 1),x)*x + log(c**2*x**2 + 1)*c*x - 2*log(x)*c *x)/(2*x)