Integrand size = 14, antiderivative size = 403 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \] Output:
1/2*I*arctan(d^(1/2)*x/c^(1/2))*ln(1-I/a/x)/c^(1/2)/d^(1/2)-1/2*I*arctan(d ^(1/2)*x/c^(1/2))*ln(1+I/a/x)/c^(1/2)/d^(1/2)-1/2*I*arctan(d^(1/2)*x/c^(1/ 2))*ln(2*I*c^(1/2)*d^(1/2)*(I-a*x)/(a*c^(1/2)-d^(1/2))/(c^(1/2)-I*d^(1/2)* x))/c^(1/2)/d^(1/2)+1/2*I*arctan(d^(1/2)*x/c^(1/2))*ln(-2*I*c^(1/2)*d^(1/2 )*(I+a*x)/(a*c^(1/2)+d^(1/2))/(c^(1/2)-I*d^(1/2)*x))/c^(1/2)/d^(1/2)-1/4*p olylog(2,1-2*I*c^(1/2)*d^(1/2)*(I-a*x)/(a*c^(1/2)-d^(1/2))/(c^(1/2)-I*d^(1 /2)*x))/c^(1/2)/d^(1/2)+1/4*polylog(2,1+2*I*c^(1/2)*d^(1/2)*(I+a*x)/(a*c^( 1/2)+d^(1/2))/(c^(1/2)-I*d^(1/2)*x))/c^(1/2)/d^(1/2)
Time = 1.11 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.78 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {a \left (-2 \arccos \left (\frac {a^2 c+d}{a^2 c-d}\right ) \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )-4 \cot ^{-1}(a x) \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )-\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )-2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 i d \left (i a^2 c+\sqrt {-a^2 c d}\right ) (i+a x)}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )-\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )+2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c+i \sqrt {-a^2 c d}\right ) (-i+a x)}{\left (a^2 c-d\right ) \left (-\sqrt {-a^2 c d}+a d x\right )}\right )+\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )+2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )+2 i \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-a^2 c d} e^{-i \cot ^{-1}(a x)}}{\sqrt {a^2 c-d} \sqrt {-a^2 c-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )+\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )-2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )-2 i \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-a^2 c d} e^{i \cot ^{-1}(a x)}}{\sqrt {a^2 c-d} \sqrt {-a^2 c-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (a^2 c+d-2 i \sqrt {-a^2 c d}\right ) \left (\sqrt {-a^2 c d}+a d x\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (a^2 c+d+2 i \sqrt {-a^2 c d}\right ) \left (\sqrt {-a^2 c d}+a d x\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )\right )\right )}{4 \sqrt {-a^2 c d}} \] Input:
Integrate[ArcCot[a*x]/(c + d*x^2),x]
Output:
(a*(-2*ArcCos[(a^2*c + d)/(a^2*c - d)]*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] - 4*ArcCot[a*x]*ArcTanh[(a*d*x)/Sqrt[-(a^2*c*d)]] - (ArcCos[(a^2*c + d)/( a^2*c - d)] - (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)])*Log[((2*I)*d*(I*a ^2*c + Sqrt[-(a^2*c*d)])*(I + a*x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x ))] - (ArcCos[(a^2*c + d)/(a^2*c - d)] + (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c *d)]*x)])*Log[(2*d*(a^2*c + I*Sqrt[-(a^2*c*d)])*(-I + a*x))/((a^2*c - d)*( -Sqrt[-(a^2*c*d)] + a*d*x))] + (ArcCos[(a^2*c + d)/(a^2*c - d)] + (2*I)*Ar cTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] + (2*I)*ArcTanh[(a*d*x)/Sqrt[-(a^2*c*d)] ])*Log[(Sqrt[2]*Sqrt[-(a^2*c*d)])/(Sqrt[a^2*c - d]*E^(I*ArcCot[a*x])*Sqrt[ -(a^2*c) - d + (a^2*c - d)*Cos[2*ArcCot[a*x]]])] + (ArcCos[(a^2*c + d)/(a^ 2*c - d)] - (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] - (2*I)*ArcTanh[(a*d *x)/Sqrt[-(a^2*c*d)]])*Log[(Sqrt[2]*Sqrt[-(a^2*c*d)]*E^(I*ArcCot[a*x]))/(S qrt[a^2*c - d]*Sqrt[-(a^2*c) - d + (a^2*c - d)*Cos[2*ArcCot[a*x]]])] + I*( -PolyLog[2, ((a^2*c + d - (2*I)*Sqrt[-(a^2*c*d)])*(Sqrt[-(a^2*c*d)] + a*d* x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x))] + PolyLog[2, ((a^2*c + d + ( 2*I)*Sqrt[-(a^2*c*d)])*(Sqrt[-(a^2*c*d)] + a*d*x))/((a^2*c - d)*(Sqrt[-(a^ 2*c*d)] - a*d*x))])))/(4*Sqrt[-(a^2*c*d)])
Time = 1.05 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.56, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {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.
| \(\displaystyle \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 5444 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log \left (1-\frac {i}{a x}\right )}{d x^2+c}dx-\frac {1}{2} i \int \frac {\log \left (1+\frac {i}{a x}\right )}{d x^2+c}dx\) |
| \(\Big \downarrow \) 2920 |
| \(\displaystyle \frac {1}{2} i \left (\frac {\log \left (1-\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}-\frac {i \int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} \left (a-\frac {i}{x}\right ) x^2}dx}{a}\right )-\frac {1}{2} i \left (\frac {i \int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} \left (a+\frac {i}{x}\right ) x^2}dx}{a}+\frac {\log \left (1+\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} i \left (\frac {\log \left (1-\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}-\frac {i \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (a-\frac {i}{x}\right ) x^2}dx}{\sqrt {c} \sqrt {d}}\right )-\frac {1}{2} i \left (\frac {i \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (a+\frac {i}{x}\right ) x^2}dx}{\sqrt {c} \sqrt {d}}+\frac {\log \left (1+\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}\right )\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \frac {1}{2} i \left (\frac {\log \left (1-\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}-\frac {i \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (a x-i)}dx}{\sqrt {c} \sqrt {d}}\right )-\frac {1}{2} i \left (\frac {i \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (a x+i)}dx}{\sqrt {c} \sqrt {d}}+\frac {\log \left (1+\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}\right )\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \frac {1}{2} i \left (\frac {\log \left (1-\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}-\frac {i \int \left (\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}-\frac {i a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{a x-i}\right )dx}{\sqrt {c} \sqrt {d}}\right )-\frac {1}{2} i \left (\frac {i \int \left (\frac {i a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{a x+i}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}\right )dx}{\sqrt {c} \sqrt {d}}+\frac {\log \left (1+\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} i \left (\frac {\log \left (1-\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}-\frac {i \left (-i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (-a x+i)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )+i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i \sqrt {d} x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {d} x}{\sqrt {c}}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {d} x}{\sqrt {c}}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}-i \sqrt {d} x}\right )\right )}{\sqrt {c} \sqrt {d}}\right )-\frac {1}{2} i \left (\frac {i \left (i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (\sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )-i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i \sqrt {d} x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {d} x}{\sqrt {c}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {d} x}{\sqrt {c}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}-i \sqrt {d} x}\right )\right )}{\sqrt {c} \sqrt {d}}+\frac {\log \left (1+\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}\right )\) |
Input:
Int[ArcCot[a*x]/(c + d*x^2),x]
Output:
(I/2)*((ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - I/(a*x)])/(Sqrt[c]*Sqrt[d]) - (I*(I*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*Sqrt[d]*x)] - I*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((2*I)*Sqrt[c]*Sqrt[d]*(I - a*x))/((a *Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))] - PolyLog[2, ((-I)*Sqrt[d]*x )/Sqrt[c]]/2 + PolyLog[2, (I*Sqrt[d]*x)/Sqrt[c]]/2 + PolyLog[2, 1 - (2*Sqr t[c])/(Sqrt[c] - I*Sqrt[d]*x)]/2 - PolyLog[2, 1 - ((2*I)*Sqrt[c]*Sqrt[d]*( I - a*x))/((a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/2))/(Sqrt[c]*Sq rt[d])) - (I/2)*((ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 + I/(a*x)])/(Sqrt[c]*S qrt[d]) + (I*((-I)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*Sqrt[d]*x)] + I*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((-2*I)*Sqrt[c]*Sqrt[d]* (I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))] + PolyLog[2, ( (-I)*Sqrt[d]*x)/Sqrt[c]]/2 - PolyLog[2, (I*Sqrt[d]*x)/Sqrt[c]]/2 - PolyLog [2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*Sqrt[d]*x)]/2 + PolyLog[2, 1 + ((2*I)*Sqr t[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/2 ))/(Sqrt[c]*Sqrt[d]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + 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]
Time = 2.10 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {i \pi \,\operatorname {arctanh}\left (\frac {2 \left (-i a x +1\right ) d -2 d}{2 a \sqrt {c d}}\right )}{2 \sqrt {c d}}-\frac {\ln \left (-i a x +1\right ) \ln \left (\frac {a \sqrt {c d}-\left (-i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (-i a x +1\right ) \ln \left (\frac {a \sqrt {c d}+\left (-i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}-\left (-i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}+\left (-i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\ln \left (i a x +1\right ) \ln \left (\frac {a \sqrt {c d}-\left (i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (i a x +1\right ) \ln \left (\frac {a \sqrt {c d}+\left (i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}-\left (i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}+\left (i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}\) | \(392\) |
| derivativedivides | \(\frac {-\frac {i \sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{2 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right )^{2}}{2 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{4 c d}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right )}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {i \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right ) a^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right )}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right )}{4 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {arccot}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) a^{2}}{2 \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right )}{4 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}}{a}\) | \(977\) |
| default | \(\frac {-\frac {i \sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{2 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right )^{2}}{2 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{4 c d}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right )}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {i \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right ) a^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right )}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right )}{4 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {arccot}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) a^{2}}{2 \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right )}{4 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}}{a}\) | \(977\) |
Input:
int(arccot(a*x)/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
1/2*I*Pi/(c*d)^(1/2)*arctanh(1/2*(2*(1-I*a*x)*d-2*d)/a/(c*d)^(1/2))-1/4*ln (1-I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)-(1-I*a*x)*d+d)/(a*(c*d)^(1/2)+d))+ 1/4*ln(1-I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1-I*a*x)*d-d)/(a*(c*d)^(1/2 )-d))-1/4/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)-(1-I*a*x)*d+d)/(a*(c*d)^(1/2)+d ))+1/4/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)+(1-I*a*x)*d-d)/(a*(c*d)^(1/2)-d))- 1/4*ln(1+I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)-(1+I*a*x)*d+d)/(a*(c*d)^(1/2 )+d))+1/4*ln(1+I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1+I*a*x)*d-d)/(a*(c*d )^(1/2)-d))-1/4/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)-(1+I*a*x)*d+d)/(a*(c*d)^( 1/2)+d))+1/4/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)+(1+I*a*x)*d-d)/(a*(c*d)^(1/2 )-d))
\[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{d x^{2} + c} \,d x } \] Input:
integrate(arccot(a*x)/(d*x^2+c),x, algorithm="fricas")
Output:
integral(arccot(a*x)/(d*x^2 + c), x)
\[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{c + d x^{2}}\, dx \] Input:
integrate(acot(a*x)/(d*x**2+c),x)
Output:
Integral(acot(a*x)/(c + d*x**2), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (263) = 526\).
Time = 0.18 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.31 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=-\frac {a {\left (\frac {8 \, \arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{a} - \frac {4 \, \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \arctan \left (-\frac {a \sqrt {d} x}{a \sqrt {c} - \sqrt {d}}, -\frac {\sqrt {d}}{a \sqrt {c} - \sqrt {d}}\right ) + \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} + 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d + 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) - \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} - 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d - 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x + {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x - {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x - {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x + {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right )}{a}\right )}}{8 \, \sqrt {c d}} + \frac {\operatorname {arccot}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {\arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} \] Input:
integrate(arccot(a*x)/(d*x^2+c),x, algorithm="maxima")
Output:
-1/8*a*(8*arctan(a*x)*arctan(d*x/sqrt(c*d))/a - (4*arctan(a*x)*arctan(sqrt (d)*x/sqrt(c)) + 4*arctan(sqrt(d)*x/sqrt(c))*arctan2(-a*sqrt(d)*x/(a*sqrt( c) - sqrt(d)), -sqrt(d)/(a*sqrt(c) - sqrt(d))) + log(d*x^2 + c)*log((a^2*c *d + (a^4*c*d + a^2*d^2)*x^2 + 2*(a^3*d*x^2 + a*d)*sqrt(c)*sqrt(d) + d^2)/ (a^4*c^2 + 6*a^2*c*d + 4*(a^3*c + a*d)*sqrt(c)*sqrt(d) + d^2)) - log(d*x^2 + c)*log((a^2*c*d + (a^4*c*d + a^2*d^2)*x^2 - 2*(a^3*d*x^2 + a*d)*sqrt(c) *sqrt(d) + d^2)/(a^4*c^2 + 6*a^2*c*d - 4*(a^3*c + a*d)*sqrt(c)*sqrt(d) + d ^2)) + 2*dilog((a^2*c + I*a*d*x + (I*a^2*x + a)*sqrt(c)*sqrt(d))/(a^2*c + 2*a*sqrt(c)*sqrt(d) + d)) + 2*dilog((a^2*c - I*a*d*x - (I*a^2*x - a)*sqrt( c)*sqrt(d))/(a^2*c + 2*a*sqrt(c)*sqrt(d) + d)) - 2*dilog((a^2*c + I*a*d*x - (I*a^2*x + a)*sqrt(c)*sqrt(d))/(a^2*c - 2*a*sqrt(c)*sqrt(d) + d)) - 2*di log((a^2*c - I*a*d*x + (I*a^2*x - a)*sqrt(c)*sqrt(d))/(a^2*c - 2*a*sqrt(c) *sqrt(d) + d)))/a)/sqrt(c*d) + arccot(a*x)*arctan(d*x/sqrt(c*d))/sqrt(c*d) + arctan(a*x)*arctan(d*x/sqrt(c*d))/sqrt(c*d)
\[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{d x^{2} + c} \,d x } \] Input:
integrate(arccot(a*x)/(d*x^2+c),x, algorithm="giac")
Output:
integrate(arccot(a*x)/(d*x^2 + c), x)
Timed out. \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{d\,x^2+c} \,d x \] Input:
int(acot(a*x)/(c + d*x^2),x)
Output:
int(acot(a*x)/(c + d*x^2), x)
\[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {-\mathit {acot} \left (a x \right )^{2} a -2 \left (\int \frac {\mathit {acot} \left (a x \right )}{a^{2} d \,x^{4}+a^{2} c \,x^{2}+d \,x^{2}+c}d x \right ) a^{2} c +2 \left (\int \frac {\mathit {acot} \left (a x \right )}{a^{2} d \,x^{4}+a^{2} c \,x^{2}+d \,x^{2}+c}d x \right ) d}{2 d} \] Input:
int(acot(a*x)/(d*x^2+c),x)
Output:
( - acot(a*x)**2*a - 2*int(acot(a*x)/(a**2*c*x**2 + a**2*d*x**4 + c + d*x* *2),x)*a**2*c + 2*int(acot(a*x)/(a**2*c*x**2 + a**2*d*x**4 + c + d*x**2),x )*d)/(2*d)