Integrand size = 14, antiderivative size = 801 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (1+a^2 x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}} \]
1/2*x*arccot(a*x)/c/(d*x^2+c)+1/4*a*ln(a^2*x^2+1)/c/(a^2*c-d)-1/4*a*ln(d*x ^2+c)/c/(a^2*c-d)+1/2*arccot(a*x)*arctan(x*d^(1/2)/c^(1/2))/c^(3/2)/d^(1/2 )+1/8*I*a*ln(-(1+x*(-a^2)^(1/2))*d^(1/2)/(I*(-a^2)^(1/2)*c^(1/2)-d^(1/2))) *ln(1-I*x*d^(1/2)/c^(1/2))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)-1/8*I*a*ln((1-x*(- a^2)^(1/2))*d^(1/2)/(I*(-a^2)^(1/2)*c^(1/2)+d^(1/2)))*ln(1-I*x*d^(1/2)/c^( 1/2))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)+1/8*I*a*ln(-(1-x*(-a^2)^(1/2))*d^(1/2)/ (I*(-a^2)^(1/2)*c^(1/2)-d^(1/2)))*ln(1+I*x*d^(1/2)/c^(1/2))/c^(3/2)/(-a^2) ^(1/2)/d^(1/2)-1/8*I*a*ln((1+x*(-a^2)^(1/2))*d^(1/2)/(I*(-a^2)^(1/2)*c^(1/ 2)+d^(1/2)))*ln(1+I*x*d^(1/2)/c^(1/2))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)-1/8*I* a*polylog(2,(-a^2)^(1/2)*(c^(1/2)-I*x*d^(1/2))/((-a^2)^(1/2)*c^(1/2)-I*d^( 1/2)))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)+1/8*I*a*polylog(2,(-a^2)^(1/2)*(c^(1/2 )-I*x*d^(1/2))/((-a^2)^(1/2)*c^(1/2)+I*d^(1/2)))/c^(3/2)/(-a^2)^(1/2)/d^(1 /2)-1/8*I*a*polylog(2,(-a^2)^(1/2)*(c^(1/2)+I*x*d^(1/2))/((-a^2)^(1/2)*c^( 1/2)-I*d^(1/2)))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)+1/8*I*a*polylog(2,(-a^2)^(1/ 2)*(c^(1/2)+I*x*d^(1/2))/((-a^2)^(1/2)*c^(1/2)+I*d^(1/2)))/c^(3/2)/(-a^2)^ (1/2)/d^(1/2)
Time = 5.30 (sec) , antiderivative size = 806, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=-\frac {a \left (\frac {2 \log \left (\frac {a^2 c+d+\left (-a^2 c+d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}{a^2 c+d}\right )}{a^2 c-d}+\frac {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 )}{\sqrt {-a^2 c d}}-\frac {4 \cot ^{-1}(a x) \sin \left (2 \cot ^{-1}(a x)\right )}{a^2 c+d+\left (-a^2 c+d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}\right )}{8 c} \]
-1/8*(a*((2*Log[(a^2*c + d + (-(a^2*c) + d)*Cos[2*ArcCot[a*x]])/(a^2*c + d )])/(a^2*c - d) + (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)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] + (2*I)*ArcTanh[(a*d*x)/Sq rt[-(a^2*c*d)]])*Log[(Sqrt[2]*Sqrt[-(a^2*c*d)])/(Sqrt[a^2*c - d]*E^(I*ArcC ot[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*Ar cCot[a*x]))/(Sqrt[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))]))/Sqrt[-(a^2*c*d)] - (4*ArcCot[a*x]*Sin[2* ArcCot[a*x]])/(a^2*c + d + (-(a^2*c) + d)*Cos[2*ArcCot[a*x]])))/c
Time = 1.52 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5448, 27, 7276, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5448 |
| \(\displaystyle a \int \frac {\frac {x}{c \left (d x^2+c\right )}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d}}}{2 \left (a^2 x^2+1\right )}dx+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} a \int \frac {\frac {x}{c \left (d x^2+c\right )}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d}}}{a^2 x^2+1}dx+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {1}{2} a \int \left (\frac {x}{c \left (a^2 x^2+1\right ) \left (d x^2+c\right )}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d} \left (a^2 x^2+1\right )}\right )dx+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \cot ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {x \cot ^{-1}(a x)}{2 c \left (d x^2+c\right )}+\frac {1}{2} a \left (-\frac {i \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i \log \left (-\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i \log \left (\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {\log \left (a^2 x^2+1\right )}{2 c \left (a^2 c-d\right )}-\frac {\log \left (d x^2+c\right )}{2 c \left (a^2 c-d\right )}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{4 \sqrt {-a^2} c^{3/2} \sqrt {d}}\right )\) |
(x*ArcCot[a*x])/(2*c*(c + d*x^2)) + (ArcCot[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c ]])/(2*c^(3/2)*Sqrt[d]) + (a*(((-1/4*I)*Log[(Sqrt[d]*(1 - Sqrt[-a^2]*x))/( I*Sqrt[-a^2]*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^ 2]*c^(3/2)*Sqrt[d]) + ((I/4)*Log[-((Sqrt[d]*(1 + Sqrt[-a^2]*x))/(I*Sqrt[-a ^2]*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/ 2)*Sqrt[d]) + ((I/4)*Log[-((Sqrt[d]*(1 - Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt [c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[ d]) - ((I/4)*Log[(Sqrt[d]*(1 + Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] + Sqrt [d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + Log[1 + a^2*x^2]/(2*c*(a^2*c - d)) - Log[c + d*x^2]/(2*c*(a^2*c - d)) - ((I/4)* PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] - I*Sq rt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, (Sqrt[-a^2]*(Sqr t[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2 )*Sqrt[d]) - ((I/4)*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[ -a^2]*Sqrt[c] - I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/4)*PolyLog [2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])] )/(Sqrt[-a^2]*c^(3/2)*Sqrt[d])))/2
3.1.58.3.1 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[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcCot[c*x]) u, x] + Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ [{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2140 vs. \(2 (593 ) = 1186\).
Time = 1.59 (sec) , antiderivative size = 2141, normalized size of antiderivative = 2.67
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2141\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2275\) |
| default | \(\text {Expression too large to display}\) | \(2275\) |
1/4*I*a^4*ln(1-I*a*x)/(a^2*c-d)/(-a^2*d*x^2-a^2*c)*x-1/4*I*a^4*ln(1+I*a*x) /(a^2*c-d)/(-a^2*d*x^2-a^2*c)*x-1/8*a^4*ln(1-I*a*x)*c/(a^2*c-d)/(-a^2*d*x^ 2-a^2*c)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1-I*a*x)*d-d)/(a*(c*d)^(1/2)-d))+1 /8*a^4*ln(1-I*a*x)*c/(a^2*c-d)/(-a^2*d*x^2-a^2*c)/(c*d)^(1/2)*ln((a*(c*d)^ (1/2)-(1-I*a*x)*d+d)/(a*(c*d)^(1/2)+d))+1/8*a^2*ln(1-I*a*x)/(a^2*c-d)/(-a^ 2*d*x^2-a^2*c)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1-I*a*x)*d-d)/(a*(c*d)^(1/2) -d))*d-1/4*a^3*ln(1-I*a*x)/c/(a^2*c-d)/(-a^2*d*x^2-a^2*c)*d*x^2-1/8*a^2*ln (1-I*a*x)/(a^2*c-d)/(-a^2*d*x^2-a^2*c)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)-(1-I* a*x)*d+d)/(a*(c*d)^(1/2)+d))*d-1/4*a^3*ln(1+I*a*x)/c/(a^2*c-d)/(-a^2*d*x^2 -a^2*c)*d*x^2+1/8*a^4*ln(1+I*a*x)*c/(a^2*c-d)/(-a^2*d*x^2-a^2*c)/(c*d)^(1/ 2)*ln((a*(c*d)^(1/2)-(1+I*a*x)*d+d)/(a*(c*d)^(1/2)+d))-1/8*a^4*ln(1+I*a*x) *c/(a^2*c-d)/(-a^2*d*x^2-a^2*c)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1+I*a*x)*d- d)/(a*(c*d)^(1/2)-d))-1/8*a^2*ln(1+I*a*x)/(a^2*c-d)/(-a^2*d*x^2-a^2*c)/(c* d)^(1/2)*ln((a*(c*d)^(1/2)-(1+I*a*x)*d+d)/(a*(c*d)^(1/2)+d))*d+1/8*a^2*ln( 1+I*a*x)/(a^2*c-d)/(-a^2*d*x^2-a^2*c)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1+I*a *x)*d-d)/(a*(c*d)^(1/2)-d))*d+1/8*a^4*ln(1-I*a*x)/(a^2*c-d)/(-a^2*d*x^2-a^ 2*c)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)-(1-I*a*x)*d+d)/(a*(c*d)^(1/2)+d))*d*x^2 -1/8/c/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)-(1-I*a*x)*d+d)/(a*(c*d)^(1/2)+d))+ 1/8/c/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)+(1-I*a*x)*d-d)/(a*(c*d)^(1/2)-d))-1 /8/c/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)-(1+I*a*x)*d+d)/(a*(c*d)^(1/2)+d))...
\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.38 (sec) , antiderivative size = 628, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{c d x^{2} + c^{2}} + \frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c}\right )} \operatorname {arccot}\left (a x\right ) + \frac {{\left (4 \, a c d \log \left (a^{2} x^{2} + 1\right ) - 4 \, a c d \log \left (d x^{2} + c\right ) + {\left (4 \, {\left (a^{2} c - d\right )} \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, {\left (a^{2} c - d\right )} \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 ) + {\left (a^{2} c - 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 ) - {\left (a^{2} c - 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 ) + 2 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 )\right )} \sqrt {c} \sqrt {d}\right )} a}{16 \, {\left (a^{3} c^{3} d - a c^{2} d^{2}\right )}} \]
1/2*(x/(c*d*x^2 + c^2) + arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c))*arccot(a*x) + 1/16*(4*a*c*d*log(a^2*x^2 + 1) - 4*a*c*d*log(d*x^2 + c) + (4*(a^2*c - d) *arctan(a*x)*arctan(sqrt(d)*x/sqrt(c)) + 4*(a^2*c - d)*arctan(sqrt(d)*x/sq rt(c))*arctan2(-a*sqrt(d)*x/(a*sqrt(c) - sqrt(d)), -sqrt(d)/(a*sqrt(c) - s qrt(d))) + (a^2*c - 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)) - (a^2*c - 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)) + 2*(a^2* c - d)*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*(a^2*c - d)*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*(a^2*c - d)*d ilog((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*(a^2*c - d)*dilog((a^2*c - I*a*d*x + (I*a^2*x - a)*sqr t(c)*sqrt(d))/(a^2*c - 2*a*sqrt(c)*sqrt(d) + d)))*sqrt(c)*sqrt(d))*a/(a^3* c^3*d - a*c^2*d^2)
\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]