Integrand size = 21, antiderivative size = 1382 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {b c \log (x)}{d^2}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {b c e \log \left (d+e x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}} \]
(-a-b*arctan(c*x))/d^2/x-1/2*e*x*(a+b*arctan(c*x))/d^2/(e*x^2+d)+b*c*ln(x) /d^2-1/2*b*c*ln(c^2*x^2+1)/d^2+1/4*b*c*e*ln(c^2*x^2+1)/d^2/(c^2*d-e)-1/4*b *c*e*ln(e*x^2+d)/d^2/(c^2*d-e)-a*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(5/2) -1/2*(a+b*arctan(c*x))*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(5/2)+1/8*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))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)+1/8*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)))*e^(1/2)/d^(5 /2)/(-c^2)^(1/2)-1/4*I*b*polylog(2,(I-c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2) ))*e^(1/2)/(-d)^(5/2)+1/4*I*b*polylog(2,(1+I*c*x)*e^(1/2)/(I*c*(-d)^(1/2)+ e^(1/2)))*e^(1/2)/(-d)^(5/2)-1/4*I*b*polylog(2,(c*x+I)*e^(1/2)/(c*(-d)^(1/ 2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+1/4*I*b*polylog(2,(1-I*c*x)*e^(1/2)/(I*c *(-d)^(1/2)+e^(1/2)))*e^(1/2)/(-d)^(5/2)+1/8*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))*e^(1/2 )/d^(5/2)/(-c^2)^(1/2)-1/8*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)))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)-1/8*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))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)+1/8*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)))*e^(1/2)/d^( 5/2)/(-c^2)^(1/2)+1/4*I*b*ln(1-I*c*x)*ln(c*((-d)^(1/2)+x*e^(1/2))/(c*(-d)^ (1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(5/2)-1/4*I*b*ln(1-I*c*x)*ln(c*((-d)^(1/...
Time = 12.75 (sec) , antiderivative size = 992, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {a}{d^2 x}-\frac {a e x}{2 d^2 \left (d+e x^2\right )}-\frac {3 a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+b c^5 \left (-\frac {\arctan (c x)}{c^5 d^2 x}+\frac {\log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )}{c^4 d^2}-\frac {e \log \left (1-\frac {\left (-c^2 d+e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )}{4 c^4 d^2 \left (c^2 d-e\right )}-\frac {3 e \left (4 \arctan (c x) \text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+2 \arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (1-\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )+\left (-\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (1-\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )\right )\right )}{8 c^4 d^2 \sqrt {-c^2 d e}}-\frac {e \arctan (c x) \sin (2 \arctan (c x))}{2 c^4 d^2 \left (c^2 d+e+c^2 d \cos (2 \arctan (c x))-e \cos (2 \arctan (c x))\right )}\right ) \]
-(a/(d^2*x)) - (a*e*x)/(2*d^2*(d + e*x^2)) - (3*a*Sqrt[e]*ArcTan[(Sqrt[e]* x)/Sqrt[d]])/(2*d^(5/2)) + b*c^5*(-(ArcTan[c*x]/(c^5*d^2*x)) + Log[(c*x)/S qrt[1 + c^2*x^2]]/(c^4*d^2) - (e*Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x] ])/(c^2*d + e)])/(4*c^4*d^2*(c^2*d - e)) - (3*e*(4*ArcTan[c*x]*ArcTanh[(c* d)/(Sqrt[-(c^2*d*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)*ArcT anh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e) ])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*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[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^ 2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*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)*(ArcT anh[(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)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2 *d + 2*c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(...
Time = 1.89 (sec) , antiderivative size = 1382, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (-\frac {e (a+b \arctan (c x))}{d^2 \left (d+e x^2\right )}+\frac {a+b \arctan (c x)}{d^2 x^2}-\frac {e (a+b \arctan (c x))}{d \left (d+e x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 d^{5/2}}-\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (e x^2+d\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b c \log (x)}{d^2}+\frac {i b \sqrt {e} \log (i c x+1) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (i c x+1) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {b c e \log \left (c^2 x^2+1\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d^2}-\frac {b c e \log \left (e x^2+d\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \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 )}{8 \sqrt {-c^2} d^{5/2}}\) |
-((a + b*ArcTan[c*x])/(d^2*x)) - (e*x*(a + b*ArcTan[c*x]))/(2*d^2*(d + e*x ^2)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(5/2) - (Sqrt[e]*(a + b*A rcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)) + (b*c*Log[x])/d^2 + ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(5/2) - ((I/4)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[ -d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) + ((I/4)*b*Sqrt[e] *Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/ (-d)^(5/2) - ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x) )/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) - ((I/8)*b*c*Sqrt[e]*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^(5/2)) + ((I/8)*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sq rt[d]])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*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^(5/2)) - ((I/8)*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 + Sqrt[-c^2] *x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sq rt[-c^2]*d^(5/2)) - (b*c*Log[1 + c^2*x^2])/(2*d^2) + (b*c*e*Log[1 + c^2*x^ 2])/(4*d^2*(c^2*d - e)) - (b*c*e*Log[d + e*x^2])/(4*d^2*(c^2*d - e)) - ((I /4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(- d)^(5/2) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[...
3.12.64.3.1 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. 2567 vs. \(2 (1038 ) = 2076\).
Time = 1.48 (sec) , antiderivative size = 2568, normalized size of antiderivative = 1.86
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2568\) |
| parts | \(\text {Expression too large to display}\) | \(3558\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3591\) |
| default | \(\text {Expression too large to display}\) | \(3591\) |
-1/4*b*c^2/d*e/(c^2*d-e)/(e*d)^(1/2)*arctanh(1/2*(2*(1+I*c*x)*e-2*e)/c/(e* d)^(1/2))-1/4*b*c^3/d*e*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)+1/8*b*c^4 /d*e^2*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1 /2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*x^2-1/8*b*c^4/d*e^2*ln(1+I*c*x)/(c^2 *d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*( e*d)^(1/2)-e))*x^2-1/8*b*c^2/d^2*e^3*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2 *d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*x^2+1/ 8*b*c^2/d^2*e^3*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c *(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))*x^2-1/8*b*c^2/d*e^2*ln(1+I* c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)* e+e)/(c*(e*d)^(1/2)+e))+1/8*b*c^2/d*e^2*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2- c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))-1/4 *I*b*c^4/d*e*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x+1/4*I*b*c^2/d^2*e^ 2*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x+1/8*c^2*b/d*e^2*ln(1-I*c*x)/( c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1-I*c*x)*e-e)/( c*(e*d)^(1/2)-e))-1/8*c^2*b/d*e^2*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d) /(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))+1/4*I*c^4 *b*ln(1-I*c*x)/d/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*e*x-1/4*I*c^2*b/d^2*e^2*ln(1 -I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x-a/x/d^2-1/4*c^2*b/d*e/(c^2*d-e)/(e* d)^(1/2)*arctanh(1/2*(2*(1-I*c*x)*e-2*e)/c/(e*d)^(1/2))-1/4*c^3*b/d*e*l...
\[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \]