Integrand size = 21, antiderivative size = 966 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+\frac {i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d \left (c^2 d-e\right )^2 e}-\frac {b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right ) e}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d \left (c^2 d-e\right )^2 e}+\frac {b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right ) e}+\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}} \]
1/8*b*c/(c^2*d-e)/e/(e*x^2+d)-1/4*x*(a+b*arctan(c*x))/e/(e*x^2+d)^2+1/8*x* (a+b*arctan(c*x))/d/e/(e*x^2+d)+1/8*(a+b*arctan(c*x))*arctan(x*e^(1/2)/d^( 1/2))/d^(3/2)/e^(3/2)+1/16*b*c*(5*c^2*d-3*e)*ln(c^2*x^2+1)/d/(c^2*d-e)^2/e -1/4*b*c*ln(c^2*x^2+1)/d/(c^2*d-e)/e-1/16*b*c*(5*c^2*d-3*e)*ln(e*x^2+d)/d/ (c^2*d-e)^2/e+1/4*b*c*ln(e*x^2+d)/d/(c^2*d-e)/e+1/32*I*b*c*polylog(2,(-c^2 )^(1/2)*(d^(1/2)-I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)-I*e^(1/2)))/d^(3/2)/e^ (3/2)/(-c^2)^(1/2)-1/32*I*b*c*ln(-(1-x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/ 2)*d^(1/2)-e^(1/2)))*ln(1+I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)/(-c^2)^(1/2 )-1/32*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)-I*x*e^(1/2))/((-c^2)^(1/2)*d^ (1/2)+I*e^(1/2)))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)-1/32*I*b*c*ln(-(1+x*(-c^2)^ (1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)-e^(1/2)))*ln(1-I*x*e^(1/2)/d^(1/2)) /d^(3/2)/e^(3/2)/(-c^2)^(1/2)+1/32*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)+I *x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)-I*e^(1/2)))/d^(3/2)/e^(3/2)/(-c^2)^(1/2) +1/32*I*b*c*ln((1-x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1/2)) )*ln(1-I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)-1/32*I*b*c*polylo g(2,(-c^2)^(1/2)*(d^(1/2)+I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)+I*e^(1/2)))/d ^(3/2)/e^(3/2)/(-c^2)^(1/2)+1/32*I*b*c*ln((1+x*(-c^2)^(1/2))*e^(1/2)/(I*(- c^2)^(1/2)*d^(1/2)+e^(1/2)))*ln(1+I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)/(-c ^2)^(1/2)
Time = 12.54 (sec) , antiderivative size = 1744, normalized size of antiderivative = 1.81 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \]
-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt [e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + (b*c^3*((-2*Log[1 + ((c^2*d - e)*Co s[2*ArcTan[c*x]])/(c^2*d + e)])/(c^2*d) - (2*Log[1 + ((c^2*d - e)*Cos[2*Ar cTan[c*x]])/(c^2*d + e)])/e + ((c^2*d - e)*e*(-4*ArcTan[c*x]*ArcTanh[Sqrt[ -(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x )/Sqrt[-(c^2*d*e)]] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[ (c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e )/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I *e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)] *x))] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[- (c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c ^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)* Cos[2*ArcTan[c*x]]])] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTa nh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[( Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)* Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c* Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*( c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]...
Time = 2.12 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5515, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (\frac {a+b \arctan (c x)}{e \left (d+e x^2\right )^2}-\frac {d (a+b \arctan (c x))}{e \left (d+e x^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i b \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \log \left (-\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \log \left (\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {b \log \left (c^2 x^2+1\right ) c}{4 d \left (c^2 d-e\right ) e}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {b \log \left (e x^2+d\right ) c}{4 d \left (c^2 d-e\right ) e}-\frac {b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {b c}{8 \left (c^2 d-e\right ) e \left (e x^2+d\right )}+\frac {x (a+b \arctan (c x))}{8 d e \left (e x^2+d\right )}-\frac {x (a+b \arctan (c x))}{4 e \left (e x^2+d\right )^2}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}\) |
(b*c)/(8*(c^2*d - e)*e*(d + e*x^2)) - (x*(a + b*ArcTan[c*x]))/(4*e*(d + e* x^2)^2) + (x*(a + b*ArcTan[c*x]))/(8*d*e*(d + e*x^2)) + ((a + b*ArcTan[c*x ])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Log[(Sqr t[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt [e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*Log[-((Sqrt[e] *(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e] *x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x) /Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Log[(Sqrt[e]*(1 + Sq rt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[ d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2]) /(16*d*(c^2*d - e)^2*e) - (b*c*Log[1 + c^2*x^2])/(4*d*(c^2*d - e)*e) - (b* c*(5*c^2*d - 3*e)*Log[d + e*x^2])/(16*d*(c^2*d - e)^2*e) + (b*c*Log[d + e* x^2])/(4*d*(c^2*d - e)*e) + ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/ 2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c ^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Poly Log[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e ])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sq rt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^...
3.12.70.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*ArcTan[c*x] )^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d , e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3773 vs. \(2 (750 ) = 1500\).
Time = 2.10 (sec) , antiderivative size = 3774, normalized size of antiderivative = 3.91
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3774\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3815\) |
| default | \(\text {Expression too large to display}\) | \(3815\) |
| risch | \(\text {Expression too large to display}\) | \(6982\) |
a*((1/8/d*x^3-1/8/e*x)/(e*x^2+d)^2+1/8/e/d/(e*d)^(1/2)*arctan(e*x/(e*d)^(1 /2)))-1/16*I*b*c^7*d^2*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c ^2*d*e)^(1/2)-e))*arctan(c*x)/(c^4*d^2-2*c^2*d*e+e^2)^2/e^2*(c^2*d*e)^(1/2 )+1/4*I*b*c^5*d*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e) ^(1/2)-e))*arctan(c*x)/(c^4*d^2-2*c^2*d*e+e^2)^2/e*(c^2*d*e)^(1/2)-1/8*I*b *c^5/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*e^2*arctan(c*x)*x^4-1/8 *I*b*c*(c^2*d*e)^(1/2)/d/e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)*ln(1-(c^2*d -e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))+1/4*I*b*c*e*ln(1 -(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))*arctan(c* x)/(c^4*d^2-2*c^2*d*e+e^2)^2/d*(c^2*d*e)^(1/2)-1/16*I*b/c*e^2*ln(1-(c^2*d- e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))*arctan(c*x)/(c^4* d^2-2*c^2*d*e+e^2)^2/d^2*(c^2*d*e)^(1/2)-3/16*b*c^3*polylog(2,(c^2*d-e)*(1 +I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/(c^4*d^2-2*c^2*d*e+e^2 )^2*(c^2*d*e)^(1/2)-1/8*b*c^5/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2* d-1/16*b*(e*d)^(1/2)/d^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+ 2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)+1/32*b/c*(c^2*d*e)^(1/ 2)/d^2/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1) /(-c^2*d-2*(c^2*d*e)^(1/2)-e))+1/16*b/c*(c^2*d*e)^(1/2)/d^2/(c^4*d^2-2*c^2 *d*e+e^2)*arctan(c*x)^2+1/32*b*c^3*(c^2*d*e)^(1/2)/e^2/(c^4*d^2-2*c^2*d*e+ e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1...
\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]