3.152 \(\int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+e x)} \, dx\)

Optimal. Leaf size=200 \[ \frac{b e \text{PolyLog}(2,-c x)}{2 d^2}-\frac{b e \text{PolyLog}(2,c x)}{2 d^2}+\frac{b e \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 d^2}-\frac{b e \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 d^2}-\frac{e \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d}+\frac{b c \log (x)}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.20372, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {5940, 5916, 266, 36, 29, 31, 5912, 5920, 2402, 2315, 2447} \[ \frac{b e \text{PolyLog}(2,-c x)}{2 d^2}-\frac{b e \text{PolyLog}(2,c x)}{2 d^2}+\frac{b e \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 d^2}-\frac{b e \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 d^2}-\frac{e \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d}+\frac{b c \log (x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5940

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*ArcTanh[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])

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 266

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 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 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 5912

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

Rule 5920

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

Rule 2402

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 2315

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

Rule 2447

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]]

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{x^2 (d+e x)} \, dx &=\int \left (\frac{a+b \tanh ^{-1}(c x)}{d x^2}-\frac{e \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac{e^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}-\frac{e \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{d^2}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}+\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac{b e \text{Li}_2(-c x)}{2 d^2}-\frac{b e \text{Li}_2(c x)}{2 d^2}+\frac{(b c) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac{(b c e) \int \frac{\log \left (\frac{2}{1+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+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}+\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac{b e \text{Li}_2(-c x)}{2 d^2}-\frac{b e \text{Li}_2(c x)}{2 d^2}-\frac{b e \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{d^2}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}+\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac{b e \text{Li}_2(-c x)}{2 d^2}-\frac{b e \text{Li}_2(c x)}{2 d^2}+\frac{b e \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 d^2}-\frac{b e \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{d x}+\frac{b c \log (x)}{d}-\frac{a e \log (x)}{d^2}-\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{d^2}+\frac{e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d}+\frac{b e \text{Li}_2(-c x)}{2 d^2}-\frac{b e \text{Li}_2(c x)}{2 d^2}+\frac{b e \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 d^2}-\frac{b e \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2}\\ \end{align*}

Mathematica [C]  time = 3.22214, size = 360, normalized size = 1.8 \[ -\frac{-b d e \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+b d e \text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+\frac{2 a d^2}{x}+2 a d e \log (x)-2 a d e \log (d+e x)+\frac{b e^2 \sqrt{1-\frac{c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left (\frac{c d}{e}\right )}}{c}-2 b c d^2 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+\frac{1}{2} i \pi b d e \log \left (1-c^2 x^2\right )+\frac{2 b d^2 \tanh ^{-1}(c x)}{x}+b d e \tanh ^{-1}(c x)^2-2 b d e \tanh ^{-1}(c x) \tanh ^{-1}\left (\frac{c d}{e}\right )-i \pi b d e \tanh ^{-1}(c x)+2 b d e \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+i \pi b d e \log \left (e^{2 \tanh ^{-1}(c x)}+1\right )-2 b d e \tanh ^{-1}\left (\frac{c d}{e}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-2 b d e \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+2 b d e \tanh ^{-1}\left (\frac{c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac{c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )-\frac{b e^2 \tanh ^{-1}(c x)^2}{c}}{2 d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((2*a*d^2)/x - I*b*d*e*Pi*ArcTanh[c*x] + (2*b*d^2*ArcTanh[c*x])/x - 2*b*d*e*ArcTanh[(c*d)/e]*ArcTanh[c*x] + b
*d*e*ArcTanh[c*x]^2 - (b*e^2*ArcTanh[c*x]^2)/c + (b*Sqrt[1 - (c^2*d^2)/e^2]*e^2*ArcTanh[c*x]^2)/(c*E^ArcTanh[(
c*d)/e]) + 2*b*d*e*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + I*b*d*e*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*b*d*
e*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 2*b*d*e*ArcTanh[c*x]*Log[1 - E^(-2*(Arc
Tanh[(c*d)/e] + ArcTanh[c*x]))] + 2*a*d*e*Log[x] - 2*a*d*e*Log[d + e*x] - 2*b*c*d^2*Log[(c*x)/Sqrt[1 - c^2*x^2
]] + (I/2)*b*d*e*Pi*Log[1 - c^2*x^2] + 2*b*d*e*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] -
b*d*e*PolyLog[2, E^(-2*ArcTanh[c*x])] + b*d*e*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))])/(2*d^3)

________________________________________________________________________________________

Maple [A]  time = 0.134, size = 279, normalized size = 1.4 \begin{align*}{\frac{ae\ln \left ( cxe+cd \right ) }{{d}^{2}}}-{\frac{a}{dx}}-{\frac{ae\ln \left ( cx \right ) }{{d}^{2}}}+{\frac{b{\it Artanh} \left ( cx \right ) e\ln \left ( cxe+cd \right ) }{{d}^{2}}}-{\frac{b{\it Artanh} \left ( cx \right ) }{dx}}-{\frac{b{\it Artanh} \left ( cx \right ) e\ln \left ( cx \right ) }{{d}^{2}}}-{\frac{bc\ln \left ( cx-1 \right ) }{2\,d}}+{\frac{bc\ln \left ( cx \right ) }{d}}-{\frac{bc\ln \left ( cx+1 \right ) }{2\,d}}+{\frac{be{\it dilog} \left ( cx \right ) }{2\,{d}^{2}}}+{\frac{be{\it dilog} \left ( cx+1 \right ) }{2\,{d}^{2}}}+{\frac{be\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2\,{d}^{2}}}-{\frac{b\ln \left ( cxe+cd \right ) e}{2\,{d}^{2}}\ln \left ({\frac{cxe+e}{-cd+e}} \right ) }-{\frac{be}{2\,{d}^{2}}{\it dilog} \left ({\frac{cxe+e}{-cd+e}} \right ) }+{\frac{b\ln \left ( cxe+cd \right ) e}{2\,{d}^{2}}\ln \left ({\frac{cxe-e}{-cd-e}} \right ) }+{\frac{be}{2\,{d}^{2}}{\it dilog} \left ({\frac{cxe-e}{-cd-e}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/d^2*e*ln(c*e*x+c*d)-a/d/x-a/d^2*e*ln(c*x)+b*arctanh(c*x)/d^2*e*ln(c*e*x+c*d)-b*arctanh(c*x)/d/x-b*arctanh(c*
x)/d^2*e*ln(c*x)-1/2*c*b/d*ln(c*x-1)+c*b/d*ln(c*x)-1/2*c*b/d*ln(c*x+1)+1/2*b/d^2*e*dilog(c*x)+1/2*b/d^2*e*dilo
g(c*x+1)+1/2*b/d^2*e*ln(c*x)*ln(c*x+1)-1/2*b/d^2*ln((c*e*x+e)/(-c*d+e))*ln(c*e*x+c*d)*e-1/2*b/d^2*dilog((c*e*x
+e)/(-c*d+e))*e+1/2*b/d^2*ln((c*e*x-e)/(-c*d-e))*ln(c*e*x+c*d)*e+1/2*b/d^2*dilog((c*e*x-e)/(-c*d-e))*e

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a{\left (\frac{e \log \left (e x + d\right )}{d^{2}} - \frac{e \log \left (x\right )}{d^{2}} - \frac{1}{d x}\right )} + \frac{1}{2} \, b \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{e x^{3} + d x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (c x\right ) + a}{e x^{3} + d x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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