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

Optimal. Leaf size=114 \[ -\frac{1}{2} b d^2 \text{PolyLog}(2,-c x)+\frac{1}{2} b d^2 \text{PolyLog}(2,c x)+\frac{1}{2} c^2 d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+2 a c d^2 x+a d^2 \log (x)+b d^2 \log \left (1-c^2 x^2\right )+\frac{1}{2} b c d^2 x-\frac{1}{2} b d^2 \tanh ^{-1}(c x)+2 b c d^2 x \tanh ^{-1}(c x) \]

[Out]

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

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Rubi [A]  time = 0.113866, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {5940, 5910, 260, 5912, 5916, 321, 206} \[ -\frac{1}{2} b d^2 \text{PolyLog}(2,-c x)+\frac{1}{2} b d^2 \text{PolyLog}(2,c x)+\frac{1}{2} c^2 d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )+2 a c d^2 x+a d^2 \log (x)+b d^2 \log \left (1-c^2 x^2\right )+\frac{1}{2} b c d^2 x-\frac{1}{2} b d^2 \tanh ^{-1}(c x)+2 b c d^2 x \tanh ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*c*d^2*x + (b*c*d^2*x)/2 - (b*d^2*ArcTanh[c*x])/2 + 2*b*c*d^2*x*ArcTanh[c*x] + (c^2*d^2*x^2*(a + b*ArcTanh[
c*x]))/2 + a*d^2*Log[x] + b*d^2*Log[1 - c^2*x^2] - (b*d^2*PolyLog[2, -(c*x)])/2 + (b*d^2*PolyLog[2, c*x])/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 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 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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.10277, size = 103, normalized size = 0.9 \[ \frac{1}{4} d^2 \left (-2 b \text{PolyLog}(2,-c x)+2 b \text{PolyLog}(2,c x)+2 a c^2 x^2+8 a c x+4 a \log (x)+4 b \log \left (1-c^2 x^2\right )+2 b c^2 x^2 \tanh ^{-1}(c x)+2 b c x+b \log (1-c x)-b \log (c x+1)+8 b c x \tanh ^{-1}(c x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 142, normalized size = 1.3 \begin{align*}{\frac{{d}^{2}a{c}^{2}{x}^{2}}{2}}+2\,ac{d}^{2}x+{d}^{2}a\ln \left ( cx \right ) +{\frac{{d}^{2}b{\it Artanh} \left ( cx \right ){c}^{2}{x}^{2}}{2}}+2\,bc{d}^{2}x{\it Artanh} \left ( cx \right ) +{d}^{2}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -{\frac{{d}^{2}b{\it dilog} \left ( cx \right ) }{2}}-{\frac{{d}^{2}b{\it dilog} \left ( cx+1 \right ) }{2}}-{\frac{{d}^{2}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2}}+{\frac{bc{d}^{2}x}{2}}+{\frac{5\,{d}^{2}b\ln \left ( cx-1 \right ) }{4}}+{\frac{3\,{d}^{2}b\ln \left ( cx+1 \right ) }{4}} \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,x)

[Out]

1/2*d^2*a*c^2*x^2+2*a*c*d^2*x+d^2*a*ln(c*x)+1/2*d^2*b*arctanh(c*x)*c^2*x^2+2*b*c*d^2*x*arctanh(c*x)+d^2*b*arct
anh(c*x)*ln(c*x)-1/2*d^2*b*dilog(c*x)-1/2*d^2*b*dilog(c*x+1)-1/2*d^2*b*ln(c*x)*ln(c*x+1)+1/2*b*c*d^2*x+5/4*d^2
*b*ln(c*x-1)+3/4*d^2*b*ln(c*x+1)

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Maxima [A]  time = 1.45441, size = 234, normalized size = 2.05 \begin{align*} \frac{1}{4} \, b c^{2} d^{2} x^{2} \log \left (c x + 1\right ) - \frac{1}{4} \, b c^{2} d^{2} x^{2} \log \left (-c x + 1\right ) + \frac{1}{2} \, a c^{2} d^{2} x^{2} + 2 \, a c d^{2} x + \frac{1}{2} \, b c d^{2} x +{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{2} - \frac{1}{2} \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} b d^{2} + \frac{1}{2} \,{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} b d^{2} - \frac{1}{4} \, b d^{2} \log \left (c x + 1\right ) + \frac{1}{4} \, b d^{2} \log \left (c x - 1\right ) + a d^{2} \log \left (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,x, algorithm="maxima")

[Out]

1/4*b*c^2*d^2*x^2*log(c*x + 1) - 1/4*b*c^2*d^2*x^2*log(-c*x + 1) + 1/2*a*c^2*d^2*x^2 + 2*a*c*d^2*x + 1/2*b*c*d
^2*x + (2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*b*d^2 - 1/2*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b*d^2 +
1/2*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b*d^2 - 1/4*b*d^2*log(c*x + 1) + 1/4*b*d^2*log(c*x - 1) + a*d^2
*log(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}, 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,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, x)

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

[Out]

d**2*(Integral(2*a*c, x) + Integral(a/x, x) + Integral(a*c**2*x, x) + Integral(2*b*c*atanh(c*x), x) + Integral
(b*atanh(c*x)/x, x) + Integral(b*c**2*x*atanh(c*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}\,{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,x, algorithm="giac")

[Out]

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