Integrand size = 18, antiderivative size = 517 \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}} \]
-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))) /(-d)^(1/2)/e^(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)))/(-d)^(1/2)/e^(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)))/(-d)^(1/2)/e^(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)))/(-d)^(1/2)/e^(1/ 2)+1/4*I*b*polylog(2,(I-c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))/(-d)^(1/2)/ e^(1/2)+1/4*I*b*polylog(2,(c*x+I)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))/(-d)^( 1/2)/e^(1/2)-1/4*I*b*polylog(2,(1-I*c*x)*e^(1/2)/(I*c*(-d)^(1/2)+e^(1/2))) /(-d)^(1/2)/e^(1/2)-1/4*I*b*polylog(2,(1+I*c*x)*e^(1/2)/(I*c*(-d)^(1/2)+e^ (1/2)))/(-d)^(1/2)/e^(1/2)+a*arctan(x*e^(1/2)/d^(1/2))/d^(1/2)/e^(1/2)
Time = 0.17 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\frac {4 a \sqrt {-d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-i b \sqrt {d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+i b \sqrt {d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-i b \sqrt {d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+i b \sqrt {d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )+i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )-i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )-i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )+i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d^2} \sqrt {e}} \]
(4*a*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - I*b*Sqrt[d]*Log[1 + I*c*x]*Log [(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + I*b*Sqrt[d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*Sq rt[d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e ])] + I*b*Sqrt[d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d ] + I*Sqrt[e])] + I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] - I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] + I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sq rt[e])])/(4*Sqrt[-d^2]*Sqrt[e])
Time = 0.69 (sec) , antiderivative size = 510, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5445, 218, 5443, 2856, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 5445 |
| \(\displaystyle a \int \frac {1}{e x^2+d}dx+b \int \frac {\arctan (c x)}{e x^2+d}dx\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle b \int \frac {\arctan (c x)}{e x^2+d}dx+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}\) |
| \(\Big \downarrow \) 5443 |
| \(\displaystyle \frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+b \left (\frac {1}{2} i \int \frac {\log (1-i c x)}{e x^2+d}dx-\frac {1}{2} i \int \frac {\log (i c x+1)}{e x^2+d}dx\right )\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+b \left (\frac {1}{2} i \int \left (\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {e} x+\sqrt {-d}\right )}\right )dx-\frac {1}{2} i \int \left (\frac {\sqrt {-d} \log (i c x+1)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (i c x+1)}{2 d \left (\sqrt {e} x+\sqrt {-d}\right )}\right )dx\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+b \left (\frac {1}{2} i \left (-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}\right )-\frac {1}{2} i \left (-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}\right )\right )\) |
(a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) + b*((-1/2*I)*((Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(2*Sqrt[- d]*Sqrt[e]) - (Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqr t[-d] + I*Sqrt[e])]/(2*Sqrt[-d]*Sqrt[e]) + PolyLog[2, (Sqrt[e]*(1 + I*c*x) )/(I*c*Sqrt[-d] + Sqrt[e])]/(2*Sqrt[-d]*Sqrt[e])) + (I/2)*((Log[1 - I*c*x] *Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(2*Sqrt[-d]*Sqr t[e]) - (Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqr t[e])])/(2*Sqrt[-d]*Sqrt[e]) - PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[ -d] + Sqrt[e])]/(2*Sqrt[-d]*Sqrt[e]) + PolyLog[2, (Sqrt[e]*(I + c*x))/(c*S qrt[-d] + I*Sqrt[e])]/(2*Sqrt[-d]*Sqrt[e])))
3.12.56.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Int[ArcTan[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[I/2 Int[ Log[1 - I*c*x]/(d + e*x^2), x], x] - Simp[I/2 Int[Log[1 + I*c*x]/(d + e*x ^2), x], x] /; FreeQ[{c, d, e}, x]
Int[(ArcTan[(c_.)*(x_)]*(b_.) + (a_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[a Int[1/(d + e*x^2), x], x] + Simp[b Int[ArcTan[c*x]/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.52 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(\frac {b \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 \sqrt {e d}}-\frac {b \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 \sqrt {e d}}+\frac {b \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 \sqrt {e d}}-\frac {b \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 \sqrt {e d}}+\frac {i a \,\operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right )}{\sqrt {e d}}+\frac {b \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 \sqrt {e d}}-\frac {b \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 \sqrt {e d}}+\frac {b \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 \sqrt {e d}}-\frac {b \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 \sqrt {e d}}\) | \(400\) |
| derivativedivides | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{\sqrt {e d}}+\frac {i b \,c^{4} \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}\, d}{2 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {i b \,c^{2} \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}}{c^{4} d^{2}-2 c^{2} d e +e^{2}}+\frac {b \,c^{4} \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}\, d}{2 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b \,c^{2} \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}}{c^{4} d^{2}-2 c^{2} d e +e^{2}}-\frac {i b \sqrt {c^{2} d e}\, \arctan \left (c x \right ) \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d +2 \sqrt {c^{2} d e}-e \right )}\right )}{2 d e}+\frac {b \,c^{4} \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}\, d}{4 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b \,c^{2} \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}}{2 \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {b \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}\, e}{2 d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {b \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}\, e}{4 d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {i b \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}\, e}{2 d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b \sqrt {c^{2} d e}\, \arctan \left (c x \right )^{2}}{2 d e}-\frac {b \sqrt {c^{2} d e}\, \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d +2 \sqrt {c^{2} d e}-e \right )}\right )}{4 d e}}{c}\) | \(879\) |
| default | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{\sqrt {e d}}+\frac {i b \,c^{4} \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}\, d}{2 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {i b \,c^{2} \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}}{c^{4} d^{2}-2 c^{2} d e +e^{2}}+\frac {b \,c^{4} \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}\, d}{2 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b \,c^{2} \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}}{c^{4} d^{2}-2 c^{2} d e +e^{2}}-\frac {i b \sqrt {c^{2} d e}\, \arctan \left (c x \right ) \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d +2 \sqrt {c^{2} d e}-e \right )}\right )}{2 d e}+\frac {b \,c^{4} \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}\, d}{4 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b \,c^{2} \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}}{2 \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {b \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}\, e}{2 d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {b \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}\, e}{4 d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {i b \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}\, e}{2 d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b \sqrt {c^{2} d e}\, \arctan \left (c x \right )^{2}}{2 d e}-\frac {b \sqrt {c^{2} d e}\, \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d +2 \sqrt {c^{2} d e}-e \right )}\right )}{4 d e}}{c}\) | \(879\) |
| parts | \(\frac {a \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{\sqrt {e d}}-\frac {i b \sqrt {c^{2} d e}\, \arctan \left (c x \right ) \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d +2 \sqrt {c^{2} d e}-e \right )}\right )}{2 c d e}+\frac {b \,c^{3} \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}\, d}{2 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b c \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}}{c^{4} d^{2}-2 c^{2} d e +e^{2}}-\frac {i b c \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}}{c^{4} d^{2}-2 c^{2} d e +e^{2}}+\frac {i b \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}\, e}{2 c d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {b \,c^{3} \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}\, d}{4 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b c \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}}{2 \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {b \arctan \left (c x \right )^{2} \sqrt {c^{2} d e}\, e}{2 c d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {b \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \sqrt {c^{2} d e}\, e}{4 c d \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}+\frac {i b \,c^{3} \ln \left (1-\frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d -2 \sqrt {c^{2} d e}-e \right )}\right ) \arctan \left (c x \right ) \sqrt {c^{2} d e}\, d}{2 e \left (c^{4} d^{2}-2 c^{2} d e +e^{2}\right )}-\frac {b \sqrt {c^{2} d e}\, \arctan \left (c x \right )^{2}}{2 c d e}-\frac {b \sqrt {c^{2} d e}\, \operatorname {polylog}\left (2, \frac {\left (c^{2} d -e \right ) \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (-c^{2} d +2 \sqrt {c^{2} d e}-e \right )}\right )}{4 c d e}\) | \(886\) |
1/4*b*ln(1-I*c*x)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1 /2)+e))-1/4*b*ln(1-I*c*x)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1-I*c*x)*e-e)/(c* (e*d)^(1/2)-e))+1/4*b/(e*d)^(1/2)*dilog((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*( e*d)^(1/2)+e))-1/4*b/(e*d)^(1/2)*dilog((c*(e*d)^(1/2)+(1-I*c*x)*e-e)/(c*(e *d)^(1/2)-e))+I*a/(e*d)^(1/2)*arctanh(1/2*(2*(1-I*c*x)*e-2*e)/c/(e*d)^(1/2 ))+1/4*b*ln(1+I*c*x)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/(c*(e*d) ^(1/2)+e))-1/4*b*ln(1+I*c*x)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/ (c*(e*d)^(1/2)-e))+1/4*b/(e*d)^(1/2)*dilog((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/( c*(e*d)^(1/2)+e))-1/4*b/(e*d)^(1/2)*dilog((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c *(e*d)^(1/2)-e))
\[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{e x^{2} + d} \,d x } \]
\[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{e x^{2} + d} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{e\,x^2+d} \,d x \]