\(\int \frac {a+b \arctan (c x)}{x (d+e x)} \, dx\) [138]

Optimal result

Integrand size = 19, antiderivative size = 181 \[ \int \frac {a+b \arctan (c x)}{x (d+e x)} \, dx=\frac {a \log (x)}{d}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {4996, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{x (d+e x)} \, dx=-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d}+\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d}+\frac {a \log (x)}{d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \arctan (c x)}{x (d+e x)} \, dx=\frac {2 a \log (x)-2 a \log (d+e x)-i b \log (1-i c x) \log \left (\frac {c (d+e x)}{c d-i e}\right )+i b \log (1+i c x) \log \left (\frac {c (d+e x)}{c d+i e}\right )+i b \operatorname {PolyLog}(2,-i c x)-i b \operatorname {PolyLog}(2,i c x)-i b \operatorname {PolyLog}\left (2,\frac {e (1-i c x)}{i c d+e}\right )+i b \operatorname {PolyLog}\left (2,-\frac {e (-i+c x)}{c d+i e}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.30

method	result	size
risch	\(-\frac {i b \operatorname {dilog}\left (-i c x +1\right )}{2 d}-\frac {i b \operatorname {dilog}\left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 d}-\frac {i b \ln \left (-i c x +1\right ) \ln \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 d}+\frac {a \ln \left (-i c x \right )}{d}-\frac {a \ln \left (i c d -\left (-i c x +1\right ) e +e \right )}{d}+\frac {i b \operatorname {dilog}\left (i c x +1\right )}{2 d}+\frac {i b \operatorname {dilog}\left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 d}+\frac {i b \ln \left (i c x +1\right ) \ln \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 d}\)	\(235\)
parts	\(\frac {a \ln \left (x \right )}{d}-\frac {a \ln \left (e x +d \right )}{d}+b \left (\frac {\arctan \left (c x \right ) \ln \left (c x \right )}{d}-\frac {\arctan \left (c x \right ) \ln \left (e c x +c d \right )}{d}-c \left (\frac {-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}}{d c}-\frac {e \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{d c}\right )\right )\)	\(243\)
derivativedivides	\(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (e c x +c d \right )}{d}+b c \left (\frac {\arctan \left (c x \right ) \ln \left (c x \right )}{d c}-\frac {\arctan \left (c x \right ) \ln \left (e c x +c d \right )}{d c}-\frac {-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}}{d c}+\frac {e \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{d c}\right )\)	\(251\)
default	\(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (e c x +c d \right )}{d}+b c \left (\frac {\arctan \left (c x \right ) \ln \left (c x \right )}{d c}-\frac {\arctan \left (c x \right ) \ln \left (e c x +c d \right )}{d c}-\frac {-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}}{d c}+\frac {e \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{d c}\right )\)	\(251\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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