Integrand size = 21, antiderivative size = 403 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {b c^2 d \arctan (c x)}{2 \left (c^2 d-e\right ) e^2}+\frac {d (a+b \arctan (c x))}{2 e^2 \left (d+e x^2\right )}+\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}-\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^2}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^2}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2} \] Output:
-1/2*b*c^2*d*arctan(c*x)/(c^2*d-e)/e^2+1/2*d*(a+b*arctan(c*x))/e^2/(e*x^2+ d)+1/2*b*c*d^(1/2)*arctan(e^(1/2)*x/d^(1/2))/(c^2*d-e)/e^(3/2)-(a+b*arctan (c*x))*ln(2/(1-I*c*x))/e^2+1/2*(a+b*arctan(c*x))*ln(2*c*((-d)^(1/2)-e^(1/2 )*x)/(c*(-d)^(1/2)-I*e^(1/2))/(1-I*c*x))/e^2+1/2*(a+b*arctan(c*x))*ln(2*c* ((-d)^(1/2)+e^(1/2)*x)/(c*(-d)^(1/2)+I*e^(1/2))/(1-I*c*x))/e^2+1/2*I*b*pol ylog(2,1-2/(1-I*c*x))/e^2-1/4*I*b*polylog(2,1-2*c*((-d)^(1/2)-e^(1/2)*x)/( c*(-d)^(1/2)-I*e^(1/2))/(1-I*c*x))/e^2-1/4*I*b*polylog(2,1-2*c*((-d)^(1/2) +e^(1/2)*x)/(c*(-d)^(1/2)+I*e^(1/2))/(1-I*c*x))/e^2
Time = 6.75 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.30 \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {2 a \left (\frac {d}{d+e x^2}+\log \left (d+e x^2\right )\right )+b \left (-\frac {2 c^2 d \arctan (c x)}{c^2 d-e}+\frac {2 d \arctan (c x)}{d+e x^2}+\frac {2 c \sqrt {d} \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{c^2 d-e}+2 \arctan (c x) \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+2 \arctan (c x) \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1-i c x)}{c \sqrt {d}-\sqrt {e}}\right )-i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1-i c x)}{c \sqrt {d}+\sqrt {e}}\right )-i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1+i c x)}{c \sqrt {d}-\sqrt {e}}\right )+i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1+i c x)}{c \sqrt {d}+\sqrt {e}}\right )-i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )+i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )+i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )-i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )\right )}{4 e^2} \] Input:
Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]
Output:
(2*a*(d/(d + e*x^2) + Log[d + e*x^2]) + b*((-2*c^2*d*ArcTan[c*x])/(c^2*d - e) + (2*d*ArcTan[c*x])/(d + e*x^2) + (2*c*Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt[e] *x)/Sqrt[d]])/(c^2*d - e) + 2*ArcTan[c*x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + 2*ArcTan[c*x]*Log[(I*Sqrt[d])/Sqrt[e] + x] + I*Log[((-I)*Sqrt[d])/Sqrt[e ] + x]*Log[(Sqrt[e]*(-1 - I*c*x))/(c*Sqrt[d] - Sqrt[e])] - I*Log[((-I)*Sqr t[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 - I*c*x))/(c*Sqrt[d] + Sqrt[e])] - I*Lo g[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 + I*c*x))/(c*Sqrt[d] - Sqrt[e] )] + I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 + I*c*x))/(c*Sqrt[d] + Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e] )] + I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])] + I*P olyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] - I*PolyLog[2 , (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])]))/(4*e^2)
Time = 0.79 (sec) , antiderivative size = 403, 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.
| \(\displaystyle \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (\frac {x (a+b \arctan (c x))}{e \left (d+e x^2\right )}-\frac {d x (a+b \arctan (c x))}{e \left (d+e x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d (a+b \arctan (c x))}{2 e^2 \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 e^2}+\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 e^2}-\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e^2}+\frac {b c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{3/2} \left (c^2 d-e\right )}-\frac {b c^2 d \arctan (c x)}{2 e^2 \left (c^2 d-e\right )}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^2}\) |
Input:
Int[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]
Output:
-1/2*(b*c^2*d*ArcTan[c*x])/((c^2*d - e)*e^2) + (d*(a + b*ArcTan[c*x]))/(2* e^2*(d + e*x^2)) + (b*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*(c^2*d - e )*e^(3/2)) - ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^2 + ((a + b*ArcTan [c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c *x))])/(2*e^2) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c *Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + ((I/2)*b*PolyLog[2, 1 - 2/ (1 - I*c*x)])/e^2 - ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/( (c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e^2 - ((I/4)*b*PolyLog[2, 1 - (2*c *(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^2
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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.59 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.78
| method | result | size |
| parts | \(\frac {a d}{2 e^{2} \left (e \,x^{2}+d \right )}+\frac {a \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {b \left (\frac {\arctan \left (c x \right ) c^{4} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e^{2}}+\frac {\arctan \left (c x \right ) c^{6} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c^{4} \left (\frac {-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}}{e^{2}}+\frac {c^{2} d \arctan \left (c x \right )}{e^{2} \left (c^{2} d -e \right )}-\frac {c d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \left (c^{2} d -e \right ) \sqrt {d e}}\right )}{2}\right )}{c^{4}}\) | \(716\) |
| derivativedivides | \(\frac {\frac {a \,c^{6} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{4} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e^{2}}+b \,c^{4} \left (\frac {\arctan \left (c x \right ) c^{2} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e^{2}}-\frac {-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}}{2 e^{2}}+\frac {c d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \left (c^{2} d -e \right ) \sqrt {d e}}-\frac {c^{2} d \arctan \left (c x \right )}{2 e^{2} \left (c^{2} d -e \right )}\right )}{c^{4}}\) | \(733\) |
| default | \(\frac {\frac {a \,c^{6} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{4} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e^{2}}+b \,c^{4} \left (\frac {\arctan \left (c x \right ) c^{2} d}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e^{2}}-\frac {-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}}{2 e^{2}}+\frac {c d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \left (c^{2} d -e \right ) \sqrt {d e}}-\frac {c^{2} d \arctan \left (c x \right )}{2 e^{2} \left (c^{2} d -e \right )}\right )}{c^{4}}\) | \(733\) |
| risch | \(-\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {d e}-\left (i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e^{2}}-\frac {i b c d \,\operatorname {arctanh}\left (\frac {2 \left (i c x +1\right ) e -2 e}{2 c \sqrt {d e}}\right )}{4 e \left (c^{2} d -e \right ) \sqrt {d e}}-\frac {i b \,c^{2} d \ln \left (i c x +1\right )}{4 e \left (c^{2} d -e \right ) \left (-c^{2} e \,x^{2}-c^{2} d \right )}-\frac {i b \,c^{2} d \ln \left (\left (i c x +1\right )^{2} e -c^{2} d -2 \left (i c x +1\right ) e +e \right )}{8 e^{2} \left (c^{2} d -e \right )}-\frac {i b \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {d e}+\left (i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e^{2}}+\frac {i c^{2} b d \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{8 e^{2} \left (c^{2} d -e \right )}+\frac {i c^{2} b d \ln \left (-i c x +1\right )}{4 e \left (c^{2} d -e \right ) \left (-c^{2} e \,x^{2}-c^{2} d \right )}+\frac {i b \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {d e}+\left (-i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e^{2}}-\frac {c^{2} a d}{2 e^{2} \left (-c^{2} e \,x^{2}-c^{2} d \right )}+\frac {a \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{2 e^{2}}+\frac {i b \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {d e}-\left (-i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e^{2}}+\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {d e}+\left (-i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e^{2}}+\frac {i c^{4} b d \ln \left (-i c x +1\right ) x^{2}}{4 e \left (c^{2} d -e \right ) \left (-c^{2} e \,x^{2}-c^{2} d \right )}+\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {d e}-\left (-i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e^{2}}-\frac {i b \,c^{4} d \ln \left (i c x +1\right ) x^{2}}{4 e \left (c^{2} d -e \right ) \left (-c^{2} e \,x^{2}-c^{2} d \right )}-\frac {i b \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {d e}-\left (i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 e^{2}}-\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {d e}+\left (i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 e^{2}}+\frac {i c b d \,\operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {d e}}\right )}{4 e \left (c^{2} d -e \right ) \sqrt {d e}}\) | \(824\) |
Input:
int(x^3*(a+b*arctan(c*x))/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/2*a/e^2*d/(e*x^2+d)+1/2*a/e^2*ln(e*x^2+d)+b/c^4*(1/2*arctan(c*x)*c^4/e^2 *ln(c^2*e*x^2+c^2*d)+1/2*arctan(c*x)*c^6*d/e^2/(c^2*e*x^2+c^2*d)-1/2*c^4*( 1/e^2*(-1/2*I*(ln(c*x-I)*ln(c^2*e*x^2+c^2*d)-2*e*(1/2*ln(c*x-I)*(ln((RootO f(e*_Z^2+2*I*e*_Z+c^2*d-e,index=1)-c*x+I)/RootOf(e*_Z^2+2*I*e*_Z+c^2*d-e,i ndex=1))+ln((RootOf(e*_Z^2+2*I*e*_Z+c^2*d-e,index=2)-c*x+I)/RootOf(e*_Z^2+ 2*I*e*_Z+c^2*d-e,index=2)))/e+1/2*(dilog((RootOf(e*_Z^2+2*I*e*_Z+c^2*d-e,i ndex=1)-c*x+I)/RootOf(e*_Z^2+2*I*e*_Z+c^2*d-e,index=1))+dilog((RootOf(e*_Z ^2+2*I*e*_Z+c^2*d-e,index=2)-c*x+I)/RootOf(e*_Z^2+2*I*e*_Z+c^2*d-e,index=2 )))/e))+1/2*I*(ln(I+c*x)*ln(c^2*e*x^2+c^2*d)-2*e*(1/2*ln(I+c*x)*(ln((RootO f(e*_Z^2-2*I*e*_Z+c^2*d-e,index=1)-c*x-I)/RootOf(e*_Z^2-2*I*e*_Z+c^2*d-e,i ndex=1))+ln((RootOf(e*_Z^2-2*I*e*_Z+c^2*d-e,index=2)-c*x-I)/RootOf(e*_Z^2- 2*I*e*_Z+c^2*d-e,index=2)))/e+1/2*(dilog((RootOf(e*_Z^2-2*I*e*_Z+c^2*d-e,i ndex=1)-c*x-I)/RootOf(e*_Z^2-2*I*e*_Z+c^2*d-e,index=1))+dilog((RootOf(e*_Z ^2-2*I*e*_Z+c^2*d-e,index=2)-c*x-I)/RootOf(e*_Z^2-2*I*e*_Z+c^2*d-e,index=2 )))/e)))+c^2*d/e^2/(c^2*d-e)*arctan(c*x)-c*d/e/(c^2*d-e)/(d*e)^(1/2)*arcta n(e*x/(d*e)^(1/2))))
\[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^3*(a+b*arctan(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*x^3*arctan(c*x) + a*x^3)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*atan(c*x))/(e*x**2+d)**2,x)
Output:
Timed out
\[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^3*(a+b*arctan(c*x))/(e*x^2+d)^2,x, algorithm="maxima")
Output:
1/2*a*(d/(e^3*x^2 + d*e^2) + log(e*x^2 + d)/e^2) + 2*b*integrate(1/2*x^3*a rctan(c*x)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^3*(a+b*arctan(c*x))/(e*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arctan(c*x) + a)*x^3/(e*x^2 + d)^2, x)
Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((x^3*(a + b*atan(c*x)))/(d + e*x^2)^2,x)
Output:
int((x^3*(a + b*atan(c*x)))/(d + e*x^2)^2, x)
\[ \int \frac {x^3 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {atan} \left (c x \right ) x^{3}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b d \,e^{2}+2 \left (\int \frac {\mathit {atan} \left (c x \right ) x^{3}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,e^{3} x^{2}+\mathrm {log}\left (e \,x^{2}+d \right ) a d +\mathrm {log}\left (e \,x^{2}+d \right ) a e \,x^{2}-a e \,x^{2}}{2 e^{2} \left (e \,x^{2}+d \right )} \] Input:
int(x^3*(a+b*atan(c*x))/(e*x^2+d)^2,x)
Output:
(2*int((atan(c*x)*x**3)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d*e**2 + 2*in t((atan(c*x)*x**3)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*e**3*x**2 + log(d + e*x**2)*a*d + log(d + e*x**2)*a*e*x**2 - a*e*x**2)/(2*e**2*(d + e*x**2))