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

Optimal. Leaf size=171 \[ \frac{b c \text{PolyLog}(2,-c x)}{d^2}-\frac{b c \text{PolyLog}(2,c x)}{d^2}+\frac{b c \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{d^2}-\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}-\frac{a+b \tanh ^{-1}(c x)}{d^2 x}-\frac{2 c \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{2 a c \log (x)}{d^2}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d^2}-\frac{b c}{2 d^2 (c x+1)}+\frac{b c \log (x)}{d^2}+\frac{b c \tanh ^{-1}(c x)}{2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.221096, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 14, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5940, 5916, 266, 36, 29, 31, 5912, 5926, 627, 44, 207, 5918, 2402, 2315} \[ \frac{b c \text{PolyLog}(2,-c x)}{d^2}-\frac{b c \text{PolyLog}(2,c x)}{d^2}+\frac{b c \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{d^2}-\frac{c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}-\frac{a+b \tanh ^{-1}(c x)}{d^2 x}-\frac{2 c \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{2 a c \log (x)}{d^2}-\frac{b c \log \left (1-c^2 x^2\right )}{2 d^2}-\frac{b c}{2 d^2 (c x+1)}+\frac{b c \log (x)}{d^2}+\frac{b c \tanh ^{-1}(c x)}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c)/(2*d^2*(1 + c*x)) + (b*c*ArcTanh[c*x])/(2*d^2) - (a + b*ArcTanh[c*x])/(d^2*x) - (c*(a + b*ArcTanh[c*x])
)/(d^2*(1 + c*x)) - (2*a*c*Log[x])/d^2 + (b*c*Log[x])/d^2 - (2*c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 -
(b*c*Log[1 - c^2*x^2])/(2*d^2) + (b*c*PolyLog[2, -(c*x)])/d^2 - (b*c*PolyLog[2, c*x])/d^2 + (b*c*PolyLog[2, 1
- 2/(1 + c*x)])/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 5926

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 5918

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

Rubi steps

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

Mathematica [A]  time = 0.813649, size = 140, normalized size = 0.82 \[ \frac{b c \left (4 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+4 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+\sinh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )+\tanh ^{-1}(c x) \left (-\frac{4}{c x}-8 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+2 \sinh \left (2 \tanh ^{-1}(c x)\right )-2 \cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )-\frac{4 a c}{c x+1}-8 a c \log (x)+8 a c \log (c x+1)-\frac{4 a}{x}}{4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*a)/x - (4*a*c)/(1 + c*x) - 8*a*c*Log[x] + 8*a*c*Log[1 + c*x] + b*c*(-Cosh[2*ArcTanh[c*x]] + 4*Log[(c*x)/S
qrt[1 - c^2*x^2]] + 4*PolyLog[2, E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + ArcTanh[c*x]*(-4/(c*x) - 2*Cosh
[2*ArcTanh[c*x]] - 8*Log[1 - E^(-2*ArcTanh[c*x])] + 2*Sinh[2*ArcTanh[c*x]])))/(4*d^2)

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Maple [A]  time = 0.058, size = 269, normalized size = 1.6 \begin{align*} -{\frac{a}{{d}^{2}x}}-2\,{\frac{ac\ln \left ( cx \right ) }{{d}^{2}}}-{\frac{ac}{{d}^{2} \left ( cx+1 \right ) }}+2\,{\frac{ac\ln \left ( cx+1 \right ) }{{d}^{2}}}-{\frac{b{\it Artanh} \left ( cx \right ) }{{d}^{2}x}}-2\,{\frac{bc{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) }{{d}^{2}}}-{\frac{bc{\it Artanh} \left ( cx \right ) }{{d}^{2} \left ( cx+1 \right ) }}+2\,{\frac{bc{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{{d}^{2}}}-{\frac{3\,bc\ln \left ( cx-1 \right ) }{4\,{d}^{2}}}+{\frac{bc\ln \left ( cx \right ) }{{d}^{2}}}-{\frac{bc}{2\,{d}^{2} \left ( cx+1 \right ) }}-{\frac{bc\ln \left ( cx+1 \right ) }{4\,{d}^{2}}}+{\frac{bc{\it dilog} \left ( cx \right ) }{{d}^{2}}}+{\frac{bc{\it dilog} \left ( cx+1 \right ) }{{d}^{2}}}+{\frac{bc\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{{d}^{2}}}-{\frac{bc}{{d}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{bc\ln \left ( cx+1 \right ) }{{d}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{\frac{bc}{{d}^{2}}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{bc \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{2\,{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/d^2/x-2*c*a/d^2*ln(c*x)-c*a/d^2/(c*x+1)+2*c*a/d^2*ln(c*x+1)-b/d^2*arctanh(c*x)/x-2*c*b/d^2*arctanh(c*x)*ln(
c*x)-c*b/d^2*arctanh(c*x)/(c*x+1)+2*c*b/d^2*arctanh(c*x)*ln(c*x+1)-3/4*c*b/d^2*ln(c*x-1)+c*b/d^2*ln(c*x)-1/2*b
*c/d^2/(c*x+1)-1/4*c*b/d^2*ln(c*x+1)+c*b/d^2*dilog(c*x)+c*b/d^2*dilog(c*x+1)+c*b/d^2*ln(c*x)*ln(c*x+1)-c*b/d^2
*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)+c*b/d^2*ln(-1/2*c*x+1/2)*ln(c*x+1)-c*b/d^2*dilog(1/2+1/2*c*x)-1/2*c*b/d^2*ln
(c*x+1)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{2 \, c x + 1}{c d^{2} x^{2} + d^{2} x} - \frac{2 \, c \log \left (c x + 1\right )}{d^{2}} + \frac{2 \, c \log \left (x\right )}{d^{2}}\right )} + \frac{1}{2} \, b \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c^{2} d^{2} x^{4} + 2 \, c d^{2} x^{3} + d^{2} 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/(c*d*x+d)^2,x, algorithm="maxima")

[Out]

-a*((2*c*x + 1)/(c*d^2*x^2 + d^2*x) - 2*c*log(c*x + 1)/d^2 + 2*c*log(x)/d^2) + 1/2*b*integrate((log(c*x + 1) -
log(-c*x + 1))/(c^2*d^2*x^4 + 2*c*d^2*x^3 + d^2*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (c x\right ) + a}{c^{2} d^{2} x^{4} + 2 \, c d^{2} x^{3} + d^{2} 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/(c*d*x+d)^2,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x) + a)/(c^2*d^2*x^4 + 2*c*d^2*x^3 + d^2*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a/(c**2*x**4 + 2*c*x**3 + x**2), x) + Integral(b*atanh(c*x)/(c**2*x**4 + 2*c*x**3 + x**2), x))/d**2

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

[Out]

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