Integrand size = 21, antiderivative size = 409 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2} \] Output:
-1/2*b*c/d/x-1/2*b*c^2*arctan(c*x)/d-1/2*(a+b*arctan(c*x))/d/x^2-a*e*ln(x) /d^2-e*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/d^2+1/2*e*(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))/d^2+1/2*e*(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))/d^2-1/2*I*b*e*polylog(2,-I*c*x)/d^2+1/2*I*b*e*polylog(2,I*c*x)/d^2+ 1/2*I*b*e*polylog(2,1-2/(1-I*c*x))/d^2-1/4*I*b*e*polylog(2,1-2*c*((-d)^(1/ 2)-e^(1/2)*x)/(c*(-d)^(1/2)-I*e^(1/2))/(1-I*c*x))/d^2-1/4*I*b*e*polylog(2, 1-2*c*((-d)^(1/2)+e^(1/2)*x)/(c*(-d)^(1/2)+I*e^(1/2))/(1-I*c*x))/d^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.23 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.26 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {2 a d+2 b d \arctan (c x)+2 b c d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )+4 a e x^2 \log (x)+i b e x^2 \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )-i b e x^2 \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-i b e x^2 \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+i b e x^2 \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-2 a e x^2 \log \left (d+e x^2\right )+2 i b e x^2 \operatorname {PolyLog}(2,-i c x)-2 i b e x^2 \operatorname {PolyLog}(2,i c x)+i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )-i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )+i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )-i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 d^2 x^2} \] Input:
Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)),x]
Output:
-1/4*(2*a*d + 2*b*d*ArcTan[c*x] + 2*b*c*d*x*Hypergeometric2F1[-1/2, 1, 1/2 , -(c^2*x^2)] + 4*a*e*x^2*Log[x] + I*b*e*x^2*Log[1 + I*c*x]*Log[(c*(Sqrt[- d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] - I*b*e*x^2*Log[1 - I*c*x]*Log[ (c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*e*x^2*Log[1 - I *c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + I*b*e*x^2 *Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] - 2*a*e*x^2*Log[d + e*x^2] + (2*I)*b*e*x^2*PolyLog[2, (-I)*c*x] - (2*I)*b*e *x^2*PolyLog[2, I*c*x] + I*b*e*x^2*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[ -d] + I*Sqrt[e])] - I*b*e*x^2*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[- d] + Sqrt[e])] + I*b*e*x^2*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] - I*b*e*x^2*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqr t[e])])/(d^2*x^2)
Time = 0.94 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5453, 5361, 264, 216, 5463, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {a+b \arctan (c x)}{x^3}dx}{d}-\frac {e \int \frac {a+b \arctan (c x)}{x \left (e x^2+d\right )}dx}{d}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx-\frac {a+b \arctan (c x)}{2 x^2}}{d}-\frac {e \int \frac {a+b \arctan (c x)}{x \left (e x^2+d\right )}dx}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {1}{2} b c \left (c^2 \left (-\int \frac {1}{c^2 x^2+1}dx\right )-\frac {1}{x}\right )-\frac {a+b \arctan (c x)}{2 x^2}}{d}-\frac {e \int \frac {a+b \arctan (c x)}{x \left (e x^2+d\right )}dx}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )-\frac {a+b \arctan (c x)}{2 x^2}}{d}-\frac {e \int \frac {a+b \arctan (c x)}{x \left (e x^2+d\right )}dx}{d}\) |
| \(\Big \downarrow \) 5463 |
| \(\displaystyle \frac {\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )-\frac {a+b \arctan (c x)}{2 x^2}}{d}-\frac {e \int \left (\frac {a+b \arctan (c x)}{d x}-\frac {e x (a+b \arctan (c x))}{d \left (e x^2+d\right )}\right )dx}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )-\frac {a+b \arctan (c x)}{2 x^2}}{d}-\frac {e \left (-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d}-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d}+\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d}+\frac {a \log (x)}{d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d}\right )}{d}\) |
Input:
Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)),x]
Output:
(-1/2*(a + b*ArcTan[c*x])/x^2 + (b*c*(-x^(-1) - c*ArcTan[c*x]))/2)/d - (e* ((a*Log[x])/d + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTa n[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I* c*x))])/(2*d) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c* Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d) + ((I/2)*b*PolyLog[2, (-I)*c*x] )/d - ((I/2)*b*PolyLog[2, I*c*x])/d - ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x )])/d + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + S qrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a + b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a, 0])
Time = 0.59 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.40
| method | result | size |
| risch | \(\frac {i b e \operatorname {dilog}\left (-i c x +1\right )}{2 d^{2}}-\frac {i b e \operatorname {dilog}\left (i c x +1\right )}{2 d^{2}}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {d e}-\left (-i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 d^{2}}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {d e}+\left (-i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 d^{2}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {d e}-\left (i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 d^{2}}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {d e}-\left (-i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 d^{2}}-\frac {b c}{2 d x}+\frac {i b \ln \left (i c x +1\right )}{4 d \,x^{2}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {d e}+\left (i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 d^{2}}-\frac {a}{2 d \,x^{2}}-\frac {a e \ln \left (-i c x \right )}{d^{2}}+\frac {a e \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{2 d^{2}}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {d e}+\left (-i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 d^{2}}+\frac {i c^{2} b \ln \left (-i c x \right )}{4 d}-\frac {i b \ln \left (-i c x +1\right )}{4 d \,x^{2}}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {d e}-\left (i c x +1\right ) e +e}{c \sqrt {d e}+e}\right )}{4 d^{2}}-\frac {i c^{2} b \ln \left (i c x \right )}{4 d}+\frac {i c^{2} b \ln \left (i c x +1\right )}{4 d}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {d e}+\left (i c x +1\right ) e -e}{c \sqrt {d e}-e}\right )}{4 d^{2}}-\frac {i c^{2} b \ln \left (-i c x +1\right )}{4 d}\) | \(571\) |
| parts | \(a \left (\frac {e \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {1}{2 d \,x^{2}}-\frac {e \ln \left (x \right )}{d^{2}}\right )+b \,c^{2} \left (\frac {\arctan \left (c x \right ) e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {\arctan \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{c^{2} d^{2}}-\frac {c^{2} \left (\frac {e \left (-\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}\right )}{d^{2} c^{4}}-\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{d \,c^{2}}-\frac {2 e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{4}}\right )}{2}\right )\) | \(757\) |
| derivativedivides | \(c^{2} \left (-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}+b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 d \,c^{4} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{4}}+\frac {\arctan \left (c x \right ) e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2} c^{4}}-\frac {e \left (-\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}\right )}{2 d^{2} c^{4}}+\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{2 d \,c^{2}}+\frac {e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{4}}\right )\right )\) | \(773\) |
| default | \(c^{2} \left (-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}+b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 d \,c^{4} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{4}}+\frac {\arctan \left (c x \right ) e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2} c^{4}}-\frac {e \left (-\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}\right )}{2 d^{2} c^{4}}+\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{2 d \,c^{2}}+\frac {e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{4}}\right )\right )\) | \(773\) |
Input:
int((a+b*arctan(c*x))/x^3/(e*x^2+d),x,method=_RETURNVERBOSE)
Output:
1/2*I*b/d^2*e*dilog(1-I*c*x)-1/2*I*b*e/d^2*dilog(1+I*c*x)+1/4*I*b*e/d^2*di log((c*(d*e)^(1/2)-(1-I*c*x)*e+e)/(c*(d*e)^(1/2)+e))+1/4*I*b*e/d^2*dilog(( c*(d*e)^(1/2)+(1-I*c*x)*e-e)/(c*(d*e)^(1/2)-e))-1/4*I*b*e/d^2*ln(1+I*c*x)* ln((c*(d*e)^(1/2)-(1+I*c*x)*e+e)/(c*(d*e)^(1/2)+e))+1/4*I*b*e/d^2*ln(1-I*c *x)*ln((c*(d*e)^(1/2)-(1-I*c*x)*e+e)/(c*(d*e)^(1/2)+e))-1/2*b*c/d/x+1/4*I* b/d*ln(1+I*c*x)/x^2-1/4*I*b*e/d^2*ln(1+I*c*x)*ln((c*(d*e)^(1/2)+(1+I*c*x)* e-e)/(c*(d*e)^(1/2)-e))-1/2*a/d/x^2-a/d^2*e*ln(-I*c*x)+1/2*a*e/d^2*ln((1-I *c*x)^2*e-c^2*d-2*(1-I*c*x)*e+e)+1/4*I*b*e/d^2*ln(1-I*c*x)*ln((c*(d*e)^(1/ 2)+(1-I*c*x)*e-e)/(c*(d*e)^(1/2)-e))+1/4*I*c^2*b/d*ln(-I*c*x)-1/4*I*b/d*ln (1-I*c*x)/x^2-1/4*I*b*e/d^2*dilog((c*(d*e)^(1/2)-(1+I*c*x)*e+e)/(c*(d*e)^( 1/2)+e))-1/4*I*c^2*b/d*ln(I*c*x)+1/4*I*c^2*b/d*ln(1+I*c*x)-1/4*I*b*e/d^2*d ilog((c*(d*e)^(1/2)+(1+I*c*x)*e-e)/(c*(d*e)^(1/2)-e))-1/4*I*c^2*b/d*ln(1-I *c*x)
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctan(c*x))/x^3/(e*x^2+d),x, algorithm="fricas")
Output:
integral((b*arctan(c*x) + a)/(e*x^5 + d*x^3), x)
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{3} \left (d + e x^{2}\right )}\, dx \] Input:
integrate((a+b*atan(c*x))/x**3/(e*x**2+d),x)
Output:
Integral((a + b*atan(c*x))/(x**3*(d + e*x**2)), x)
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctan(c*x))/x^3/(e*x^2+d),x, algorithm="maxima")
Output:
1/2*a*(e*log(e*x^2 + d)/d^2 - 2*e*log(x)/d^2 - 1/(d*x^2)) + 2*b*integrate( 1/2*arctan(c*x)/(e*x^5 + d*x^3), x)
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctan(c*x))/x^3/(e*x^2+d),x, algorithm="giac")
Output:
integrate((b*arctan(c*x) + a)/((e*x^2 + d)*x^3), x)
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,\left (e\,x^2+d\right )} \,d x \] Input:
int((a + b*atan(c*x))/(x^3*(d + e*x^2)),x)
Output:
int((a + b*atan(c*x))/(x^3*(d + e*x^2)), x)
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\frac {2 \left (\int \frac {\mathit {atan} \left (c x \right )}{e \,x^{5}+d \,x^{3}}d x \right ) b \,d^{2} x^{2}+\mathrm {log}\left (e \,x^{2}+d \right ) a e \,x^{2}-2 \,\mathrm {log}\left (x \right ) a e \,x^{2}-a d}{2 d^{2} x^{2}} \] Input:
int((a+b*atan(c*x))/x^3/(e*x^2+d),x)
Output:
(2*int(atan(c*x)/(d*x**3 + e*x**5),x)*b*d**2*x**2 + log(d + e*x**2)*a*e*x* *2 - 2*log(x)*a*e*x**2 - a*d)/(2*d**2*x**2)