\(\int \frac {a+b \arctan (c x)}{x^2 (d+e x)} \, dx\) [139]

Optimal result

Integrand size = 19, antiderivative size = 232 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+e x)} \, dx=-\frac {a+b \arctan (c x)}{d x}+\frac {b c \log (x)}{d}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2} \]

[Out]

(-a-b*arctan(c*x))/d/x+b*c*ln(x)/d-a*e*ln(x)/d^2-e*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/d^2+e*(a+b*arctan(c*x))*l
n(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/d^2-1/2*b*c*ln(c^2*x^2+1)/d-1/2*I*b*e*polylog(2,-I*c*x)/d^2+1/2*I*b*e*polyl
og(2,I*c*x)/d^2+1/2*I*b*e*polylog(2,1-2/(1-I*c*x))/d^2-1/2*I*b*e*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/
d^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {4996, 4946, 272, 36, 29, 31, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{x^2 (d+e x)} \, dx=-\frac {e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^2}-\frac {a+b \arctan (c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2}+\frac {b c \log (x)}{d} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

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 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4946

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] - Dist[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]

Rule 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((
d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcTan[c*x])*(Log[2*c*((d + e*x)/((c*
d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4996

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \arctan (c x)}{d x^2}-\frac {e (a+b \arctan (c x))}{d^2 x}+\frac {e^2 (a+b \arctan (c x))}{d^2 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{x^2} \, dx}{d}-\frac {e \int \frac {a+b \arctan (c x)}{x} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \arctan (c x)}{d+e x} \, dx}{d^2} \\ & = -\frac {a+b \arctan (c x)}{d x}-\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 (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^2}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}-\frac {(i b e) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^2}+\frac {(i b e) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^2}+\frac {(b c e) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac {(b c e) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{d^2} \\ & = -\frac {a+b \arctan (c x)}{d x}-\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 (d+e x)}{(c d+i e) (1-i c x)}\right )}{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 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d}+\frac {(i b e) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{d^2} \\ & = -\frac {a+b \arctan (c x)}{d x}-\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 (d+e x)}{(c d+i e) (1-i c x)}\right )}{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 (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {a+b \arctan (c x)}{d x}+\frac {b c \log (x)}{d}-\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 (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\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 (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+e x)} \, dx=-\frac {2 a d+2 b d \arctan (c x)-2 b c d x \log (x)+2 a e x \log (x)-2 a e x \log (d+e x)-i b e x \log (1-i c x) \log \left (\frac {c (d+e x)}{c d-i e}\right )+i b e x \log (1+i c x) \log \left (\frac {c (d+e x)}{c d+i e}\right )+b c d x \log \left (1+c^2 x^2\right )+i b e x \operatorname {PolyLog}(2,-i c x)-i b e x \operatorname {PolyLog}(2,i c x)-i b e x \operatorname {PolyLog}\left (2,\frac {e (1-i c x)}{i c d+e}\right )+i b e x \operatorname {PolyLog}\left (2,-\frac {e (-i+c x)}{c d+i e}\right )}{2 d^2 x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.32

method	result	size
parts	\(a \left (-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}+\frac {e \ln \left (e x +d \right )}{d^{2}}\right )+b c \left (-\frac {\arctan \left (c x \right )}{d c x}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{c \,d^{2}}+\frac {\arctan \left (c x \right ) e \ln \left (e c x +c d \right )}{c \,d^{2}}-c \left (\frac {e^{2} \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{d^{2} c^{2}}-\frac {-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )}{d c}-\frac {e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{2}}\right )\right )\)	\(306\)
derivativedivides	\(c \left (-\frac {a}{d c x}-\frac {a e \ln \left (c x \right )}{c \,d^{2}}+\frac {a e \ln \left (e c x +c d \right )}{c \,d^{2}}+b c \left (-\frac {\arctan \left (c x \right )}{d \,c^{2} x}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{2}}+\frac {\arctan \left (c x \right ) e \ln \left (e c x +c d \right )}{d^{2} c^{2}}-\frac {e^{2} \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{d^{2} c^{2}}+\frac {-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )}{d c}+\frac {e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{2}}\right )\right )\)	\(317\)
default	\(c \left (-\frac {a}{d c x}-\frac {a e \ln \left (c x \right )}{c \,d^{2}}+\frac {a e \ln \left (e c x +c d \right )}{c \,d^{2}}+b c \left (-\frac {\arctan \left (c x \right )}{d \,c^{2} x}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{2}}+\frac {\arctan \left (c x \right ) e \ln \left (e c x +c d \right )}{d^{2} c^{2}}-\frac {e^{2} \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{d^{2} c^{2}}+\frac {-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )}{d c}+\frac {e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{2}}\right )\right )\)	\(317\)
risch	\(\frac {i b e \operatorname {dilog}\left (-i c x +1\right )}{2 d^{2}}+\frac {i b e \operatorname {dilog}\left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 d^{2}}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 d^{2}}+\frac {c b \ln \left (-i c x \right )}{2 d}-\frac {c b \ln \left (-i c x +1\right )}{2 d}-\frac {i b \ln \left (-i c x +1\right )}{2 d x}+\frac {a e \ln \left (i c d -\left (-i c x +1\right ) e +e \right )}{d^{2}}-\frac {a e \ln \left (-i c x \right )}{d^{2}}-\frac {a}{d x}-\frac {i b e \operatorname {dilog}\left (i c x +1\right )}{2 d^{2}}-\frac {i b e \operatorname {dilog}\left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 d^{2}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 d^{2}}+\frac {b c \ln \left (i c x \right )}{2 d}-\frac {b c \ln \left (i c x +1\right )}{2 d}+\frac {i b \ln \left (i c x +1\right )}{2 d x}\)	\(344\)

[In]

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

[Out]

a*(-1/d/x-e/d^2*ln(x)+e/d^2*ln(e*x+d))+b*c*(-arctan(c*x)/d/c/x-1/c*arctan(c*x)*e/d^2*ln(c*x)+1/c*arctan(c*x)*e
/d^2*ln(c*e*x+c*d)-c*(1/d^2/c^2*e^2*(-1/2*I*ln(c*e*x+c*d)*(ln((I*e-e*c*x)/(c*d+I*e))-ln((I*e+e*c*x)/(I*e-c*d))
)/e-1/2*I*(dilog((I*e-e*c*x)/(c*d+I*e))-dilog((I*e+e*c*x)/(I*e-c*d)))/e)-1/d/c*(-1/2*ln(c^2*x^2+1)+ln(c*x))-1/
d^2/c^2*e*(-1/2*I*ln(c*x)*ln(1+I*c*x)+1/2*I*ln(c*x)*ln(1-I*c*x)-1/2*I*dilog(1+I*c*x)+1/2*I*dilog(1-I*c*x))))

Fricas [F]

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

[In]

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

[Out]

integral((b*arctan(c*x) + a)/(e*x^3 + d*x^2), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

a*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + 2*b*integrate(1/2*arctan(c*x)/(e*x^3 + d*x^2), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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