\(\int \frac {a+b \arctan (c x)}{d+e x^2} \, dx\) [1156]

Optimal result

Integrand size = 18, antiderivative size = 517 \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}} \]

[Out]

-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,(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,(
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)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5030, 211, 5028, 2456, 2441, 2440, 2438} \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} \sqrt {e}} \]

[In]

Int[(a + b*ArcTan[c*x])/(d + e*x^2),x]

[Out]

(a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) - ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*
Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d
] + I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) - ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*S
qrt[e])])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])
])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e])
- ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) - ((I/4)*b*PolyLog[2
, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I + c*x)
)/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[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]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 5028

Int[ArcTan[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[I/2, Int[Log[1 - I*c*x]/(d + e*x^2), x], x] -
Dist[I/2, Int[Log[1 + I*c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

Rule 5030

Int[(ArcTan[(c_.)*(x_)]*(b_.) + (a_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[a, Int[1/(d + e*x^2), x], x] +
Dist[b, Int[ArcTan[c*x]/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]

Rubi steps \begin{align*} \text {integral}& = a \int \frac {1}{d+e x^2} \, dx+b \int \frac {\arctan (c x)}{d+e x^2} \, dx \\ & = \frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+\frac {1}{2} (i b) \int \frac {\log (1-i c x)}{d+e x^2} \, dx-\frac {1}{2} (i b) \int \frac {\log (1+i c x)}{d+e x^2} \, dx \\ & = \frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+\frac {1}{2} (i b) \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 {-d}+\sqrt {e} x\right )}\right ) \, dx-\frac {1}{2} (i b) \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 {-d}+\sqrt {e} x\right )}\right ) \, dx \\ & = \frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}-\frac {(i b) \int \frac {\log (1-i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 \sqrt {-d}}-\frac {(i b) \int \frac {\log (1-i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 \sqrt {-d}}+\frac {(i b) \int \frac {\log (1+i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 \sqrt {-d}}+\frac {(i b) \int \frac {\log (1+i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 \sqrt {-d}} \\ & = \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 {(b c) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{-i c \sqrt {-d}+\sqrt {e}}\right )}{1-i c x} \, dx}{4 \sqrt {-d} \sqrt {e}}-\frac {(b c) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{i c \sqrt {-d}+\sqrt {e}}\right )}{1+i c x} \, dx}{4 \sqrt {-d} \sqrt {e}}+\frac {(b c) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{-i c \sqrt {-d}-\sqrt {e}}\right )}{1-i c x} \, dx}{4 \sqrt {-d} \sqrt {e}}+\frac {(b c) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{i c \sqrt {-d}-\sqrt {e}}\right )}{1+i c x} \, dx}{4 \sqrt {-d} \sqrt {e}} \\ & = \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) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{-i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 \sqrt {-d} \sqrt {e}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{-i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 \sqrt {-d} \sqrt {e}}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 \sqrt {-d} \sqrt {e}} \\ & = \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\frac {4 a \sqrt {-d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-i b \sqrt {d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+i b \sqrt {d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-i b \sqrt {d} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+i b \sqrt {d} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )+i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )-i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )-i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )+i b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d^2} \sqrt {e}} \]

[In]

Integrate[(a + b*ArcTan[c*x])/(d + e*x^2),x]

[Out]

(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*
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]*L
og[(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]*PolyLo
g[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*Sqrt[e])])/(4*Sqrt[-d^2]*Sqrt[e])

Maple [A] (verified)

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}\)	\(886\)

[In]

int((a+b*arctan(c*x))/(e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

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)*arct
anh(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))

Fricas [F]

\[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{e x^{2} + d} \,d x } \]

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d),x, algorithm="fricas")

[Out]

integral((b*arctan(c*x) + a)/(e*x^2 + d), x)

Sympy [F]

\[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{d + e x^{2}}\, dx \]

[In]

integrate((a+b*atan(c*x))/(e*x**2+d),x)

[Out]

Integral((a + b*atan(c*x))/(d + e*x**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{e x^{2} + d} \,d x } \]

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{e\,x^2+d} \,d x \]

[In]

int((a + b*atan(c*x))/(d + e*x^2),x)

[Out]

int((a + b*atan(c*x))/(d + e*x^2), x)