Integrand size = 21, antiderivative size = 1335 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} 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 )}{8 \sqrt {-c^2} \sqrt {d} 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 )}{8 \sqrt {-c^2} \sqrt {d} 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 )}{8 \sqrt {-c^2} \sqrt {d} 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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} 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 )}{8 \sqrt {-c^2} \sqrt {d} 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 )}{8 \sqrt {-c^2} \sqrt {d} 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 )}{8 \sqrt {-c^2} \sqrt {d} 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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}} \]
-1/2*x*(a+b*arctan(c*x))/e/(e*x^2+d)+1/4*b*c*ln(c^2*x^2+1)/(c^2*d-e)/e-1/4 *b*c*ln(e*x^2+d)/(c^2*d-e)/e-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^(3/2)/(-c^2)^(1/2 )/d^(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^(3/2)/(-c^2)^(1/2)/d^(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^(3/ 2)/(-c^2)^(1/2)/d^(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^(3/2)/(-c^2)^(1/2)/d^( 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^(3/2)/(-d)^(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^(3/2)/(-c^2)^(1/2)/d^(1/2)-1/4*I* b*polylog(2,(1-I*c*x)*e^(1/2)/(I*c*(-d)^(1/2)+e^(1/2)))/e^(3/2)/(-d)^(1/2) +1/4*I*b*polylog(2,(I-c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))/e^(3/2)/(-d)^ (1/2)+a*arctan(x*e^(1/2)/d^(1/2))/e^(3/2)/d^(1/2)-1/2*(a+b*arctan(c*x))*ar ctan(x*e^(1/2)/d^(1/2))/e^(3/2)/d^(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^(3/2)/(-d)^(1/2)+1/8*I*b*c*polyl og(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^(3/2)/(-c^2)^(1/2)/d^(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^(3/2)/(-c^2)^(1/ 2)/d^(1/2)-1/4*I*b*ln(1+I*c*x)*ln(c*((-d)^(1/2)-x*e^(1/2))/(c*(-d)^(1/2...
Time = 9.21 (sec) , antiderivative size = 877, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {a x}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {b c \left (-\frac {2 \log \left (\frac {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )}{c^2 d-e}+\frac {-4 \arctan (c x) \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c 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 (-\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (c^2 d-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 (\frac {2 i c^2 d \left (e+i \sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (c^2 d-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 {\sqrt {-c^2 d e}}{c e x}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\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 \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\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 (c^2 d+c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d-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 (c^2 d+c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}\right )\right )}{\sqrt {-c^2 d e}}-\frac {4 \arctan (c x) \sin (2 \arctan (c x))}{c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}\right )}{8 e} \]
-1/2*(a*x)/(e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^ (3/2)) + (b*c*((-2*Log[(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(c^2*d - 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)*A rcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[((2*I)*c^2*d*(e + I*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[Sqrt[-(c^2*d*e)]/(c*e*x)] + (2*I)*A rcTanh[(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*ArcT an[c*x]]])] - (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] + (2*I)*ArcTanh[Sqrt[-(c ^2*d*e)]/(c*e*x)] - (2*I)*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)]*x ))]))/Sqrt[-(c^2*d*e)] - (4*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e ...
Time = 2.06 (sec) , antiderivative size = 1335, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (\frac {a+b \arctan (c x)}{e \left (d+e x^2\right )}-\frac {d (a+b \arctan (c x))}{e \left (d+e x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 \sqrt {d} e^{3/2}}-\frac {x (a+b \arctan (c x))}{2 e \left (e x^2+d\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {i b \log (i c x+1) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (i c x+1) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} 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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {i b c \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} \sqrt {d} 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 (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {i b c \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} \sqrt {d} e^{3/2}}+\frac {b c \log \left (c^2 x^2+1\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (e x^2+d\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} 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 )}{8 \sqrt {-c^2} \sqrt {d} 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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {i b c \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} \sqrt {d} e^{3/2}}+\frac {i b c \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} \sqrt {d} e^{3/2}}\) |
-1/2*(x*(a + b*ArcTan[c*x]))/(e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[ d]])/(Sqrt[d]*e^(3/2)) - ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]]) /(2*Sqrt[d]*e^(3/2)) - ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]* x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 - I*c*x ]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3 /2)) - ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[- d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/8)*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]*Sqrt[d]*e^(3/2)) + ((I/8)*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]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*Log[-((Sq rt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sq rt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) - ((I/8)*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]*Sqrt[d]*e^(3/2)) + (b*c*Log[1 + c^2*x^2])/(4*(c^2* d - e)*e) - (b*c*Log[d + e*x^2])/(4*(c^2*d - e)*e) + ((I/4)*b*PolyLog[2, ( Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)* b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*e^ (3/2)) - ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt...
3.12.62.3.1 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. 2304 vs. \(2 (991 ) = 1982\).
Time = 1.86 (sec) , antiderivative size = 2305, normalized size of antiderivative = 1.73
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2305\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2344\) |
| default | \(\text {Expression too large to display}\) | \(2344\) |
| risch | \(\text {Expression too large to display}\) | \(2391\) |
3/4*I*b*c^3*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)*d/e/(c^2*d-e)/(c^4*d^2-2*c^2*d*e+e^2)*(c^2*d*e)^(1/2)-1 /4*I*b*c^5*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)/e^2/(c^2*d-e)/(c^4*d^2-2*c^2*d*e+e^2)*(c^2*d*e)^(1/2 )+1/4*I*b/c*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)/d/(c^2*d-e)/(c^4*d^2-2*c^2*d*e+e^2)*e*(c^2*d*e)^(1/2)-1 /4*I*b/c*(c^2*d*e)^(1/2)/d/e/(c^2*d-e)*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))+1/2*a/e/(e*d)^(1/2)*arctan(e *x/(e*d)^(1/2))-1/4*b*c^5*d^2*arctan(c*x)^2/e^2/(c^2*d-e)/(c^4*d^2-2*c^2*d *e+e^2)*(c^2*d*e)^(1/2)+1/4*b/c*arctan(c*x)^2/d/(c^2*d-e)/(c^4*d^2-2*c^2*d *e+e^2)*e*(c^2*d*e)^(1/2)+1/8*b/c*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))/d/(c^2*d-e)/(c^4*d^2-2*c^2*d*e+e^2)*e*(c ^2*d*e)^(1/2)+3/4*b*c^3*arctan(c*x)^2*d/e/(c^2*d-e)/(c^4*d^2-2*c^2*d*e+e^2 )*(c^2*d*e)^(1/2)+1/4*I*b*c*(c^2*d*e)^(1/2)/e^2/(c^2*d-e)*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))-1/2*I*b*c ^3*arctan(c*x)/(c^2*d-e)/e/(c^2*e*x^2+c^2*d)*d-3/4*I*b*c*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^2*d-e)/ (c^4*d^2-2*c^2*d*e+e^2)*(c^2*d*e)^(1/2)-1/2*b*c^4*arctan(c*x)/(c^2*d-e)/e/ (c^2*e*x^2+c^2*d)*x*d-1/8*b*c^5*d^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))/e^2/(c^2*d-e)/(c^4*d^2-2*c^2*d*e+e^...
\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]