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

Optimal. Leaf size=61 \[ -b c d^2 \text{PolyLog}(2,-c x)+b c d^2 \text{PolyLog}(2,c x)+\frac{d^2 \left (c^2 x^2-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{x}+c d^2 (2 a+b) \log (x) \]

[Out]

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

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Rubi [A]  time = 0.125832, antiderivative size = 80, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {5940, 5910, 260, 5916, 266, 36, 29, 31, 5912} \[ -b c d^2 \text{PolyLog}(2,-c x)+b c d^2 \text{PolyLog}(2,c x)-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 x+2 a c d^2 \log (x)+b c^2 d^2 x \tanh ^{-1}(c x)+b c d^2 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5910

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

Rule 260

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

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]

Rubi steps

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

Mathematica [A]  time = 0.106028, size = 73, normalized size = 1.2 \[ \frac{d^2 \left (-b c x \text{PolyLog}(2,-c x)+b c x \text{PolyLog}(2,c x)+a c^2 x^2+2 a c x \log (x)-a+b c^2 x^2 \tanh ^{-1}(c x)+b c x \log (c x)-b \tanh ^{-1}(c x)\right )}{x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 123, normalized size = 2. \begin{align*}{d}^{2}a{c}^{2}x-{\frac{{d}^{2}a}{x}}+2\,c{d}^{2}a\ln \left ( cx \right ) +b{c}^{2}{d}^{2}x{\it Artanh} \left ( cx \right ) -{\frac{{d}^{2}b{\it Artanh} \left ( cx \right ) }{x}}+2\,c{d}^{2}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) +c{d}^{2}b\ln \left ( cx \right ) -c{d}^{2}b{\it dilog} \left ( cx \right ) -c{d}^{2}b{\it dilog} \left ( cx+1 \right ) -c{d}^{2}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*(Integral(a*c**2, x) + Integral(a/x**2, x) + Integral(2*a*c/x, x) + Integral(b*c**2*atanh(c*x), x) + Inte
gral(b*atanh(c*x)/x**2, x) + Integral(2*b*c*atanh(c*x)/x, x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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