3.38 \(\int \frac{(d+c d x)^4 (a+b \tanh ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=189 \[ -2 b c^3 d^4 \text{PolyLog}(2,-c x)+2 b c^3 d^4 \text{PolyLog}(2,c x)-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+a c^4 d^4 x+4 a c^3 d^4 \log (x)-\frac{8}{3} b c^3 d^4 \log \left (1-c^2 x^2\right )-\frac{2 b c^2 d^4}{x}+\frac{19}{3} b c^3 d^4 \log (x)+2 b c^3 d^4 \tanh ^{-1}(c x)+b c^4 d^4 x \tanh ^{-1}(c x)-\frac{b c d^4}{6 x^2} \]

[Out]

-(b*c*d^4)/(6*x^2) - (2*b*c^2*d^4)/x + a*c^4*d^4*x + 2*b*c^3*d^4*ArcTanh[c*x] + b*c^4*d^4*x*ArcTanh[c*x] - (d^
4*(a + b*ArcTanh[c*x]))/(3*x^3) - (2*c*d^4*(a + b*ArcTanh[c*x]))/x^2 - (6*c^2*d^4*(a + b*ArcTanh[c*x]))/x + 4*
a*c^3*d^4*Log[x] + (19*b*c^3*d^4*Log[x])/3 - (8*b*c^3*d^4*Log[1 - c^2*x^2])/3 - 2*b*c^3*d^4*PolyLog[2, -(c*x)]
+ 2*b*c^3*d^4*PolyLog[2, c*x]

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Rubi [A]  time = 0.216642, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5940, 5910, 260, 5916, 266, 44, 325, 206, 36, 29, 31, 5912} \[ -2 b c^3 d^4 \text{PolyLog}(2,-c x)+2 b c^3 d^4 \text{PolyLog}(2,c x)-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+a c^4 d^4 x+4 a c^3 d^4 \log (x)-\frac{8}{3} b c^3 d^4 \log \left (1-c^2 x^2\right )-\frac{2 b c^2 d^4}{x}+\frac{19}{3} b c^3 d^4 \log (x)+2 b c^3 d^4 \tanh ^{-1}(c x)+b c^4 d^4 x \tanh ^{-1}(c x)-\frac{b c d^4}{6 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^4,x]

[Out]

-(b*c*d^4)/(6*x^2) - (2*b*c^2*d^4)/x + a*c^4*d^4*x + 2*b*c^3*d^4*ArcTanh[c*x] + b*c^4*d^4*x*ArcTanh[c*x] - (d^
4*(a + b*ArcTanh[c*x]))/(3*x^3) - (2*c*d^4*(a + b*ArcTanh[c*x]))/x^2 - (6*c^2*d^4*(a + b*ArcTanh[c*x]))/x + 4*
a*c^3*d^4*Log[x] + (19*b*c^3*d^4*Log[x])/3 - (8*b*c^3*d^4*Log[1 - c^2*x^2])/3 - 2*b*c^3*d^4*PolyLog[2, -(c*x)]
+ 2*b*c^3*d^4*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 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 325

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

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)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4} \, dx &=\int \left (c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}+\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^3}+\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac{4 c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^4 \int \frac{a+b \tanh ^{-1}(c x)}{x^4} \, dx+\left (4 c d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (6 c^2 d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (4 c^3 d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx+\left (c^4 d^4\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=a c^4 d^4 x-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)-2 b c^3 d^4 \text{Li}_2(-c x)+2 b c^3 d^4 \text{Li}_2(c x)+\frac{1}{3} \left (b c d^4\right ) \int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (2 b c^2 d^4\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^4\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx+\left (b c^4 d^4\right ) \int \tanh ^{-1}(c x) \, dx\\ &=-\frac{2 b c^2 d^4}{x}+a c^4 d^4 x+b c^4 d^4 x \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)-2 b c^3 d^4 \text{Li}_2(-c x)+2 b c^3 d^4 \text{Li}_2(c x)+\frac{1}{6} \left (b c d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\left (3 b c^3 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\left (2 b c^4 d^4\right ) \int \frac{1}{1-c^2 x^2} \, dx-\left (b c^5 d^4\right ) \int \frac{x}{1-c^2 x^2} \, dx\\ &=-\frac{2 b c^2 d^4}{x}+a c^4 d^4 x+2 b c^3 d^4 \tanh ^{-1}(c x)+b c^4 d^4 x \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)+\frac{1}{2} b c^3 d^4 \log \left (1-c^2 x^2\right )-2 b c^3 d^4 \text{Li}_2(-c x)+2 b c^3 d^4 \text{Li}_2(c x)+\frac{1}{6} \left (b c d^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c^2}{x}-\frac{c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\left (3 b c^3 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\left (3 b c^5 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^4}{6 x^2}-\frac{2 b c^2 d^4}{x}+a c^4 d^4 x+2 b c^3 d^4 \tanh ^{-1}(c x)+b c^4 d^4 x \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 a c^3 d^4 \log (x)+\frac{19}{3} b c^3 d^4 \log (x)-\frac{8}{3} b c^3 d^4 \log \left (1-c^2 x^2\right )-2 b c^3 d^4 \text{Li}_2(-c x)+2 b c^3 d^4 \text{Li}_2(c x)\\ \end{align*}

Mathematica [A]  time = 0.174583, size = 197, normalized size = 1.04 \[ \frac{d^4 \left (-12 b c^3 x^3 \text{PolyLog}(2,-c x)+12 b c^3 x^3 \text{PolyLog}(2,c x)+6 a c^4 x^4-36 a c^2 x^2+24 a c^3 x^3 \log (x)-12 a c x-2 a-12 b c^2 x^2+38 b c^3 x^3 \log (c x)-6 b c^3 x^3 \log (1-c x)+6 b c^3 x^3 \log (c x+1)-16 b c^3 x^3 \log \left (1-c^2 x^2\right )+6 b c^4 x^4 \tanh ^{-1}(c x)-36 b c^2 x^2 \tanh ^{-1}(c x)-b c x-12 b c x \tanh ^{-1}(c x)-2 b \tanh ^{-1}(c x)\right )}{6 x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^4,x]

[Out]

(d^4*(-2*a - 12*a*c*x - b*c*x - 36*a*c^2*x^2 - 12*b*c^2*x^2 + 6*a*c^4*x^4 - 2*b*ArcTanh[c*x] - 12*b*c*x*ArcTan
h[c*x] - 36*b*c^2*x^2*ArcTanh[c*x] + 6*b*c^4*x^4*ArcTanh[c*x] + 24*a*c^3*x^3*Log[x] + 38*b*c^3*x^3*Log[c*x] -
6*b*c^3*x^3*Log[1 - c*x] + 6*b*c^3*x^3*Log[1 + c*x] - 16*b*c^3*x^3*Log[1 - c^2*x^2] - 12*b*c^3*x^3*PolyLog[2,
-(c*x)] + 12*b*c^3*x^3*PolyLog[2, c*x]))/(6*x^3)

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Maple [A]  time = 0.049, size = 240, normalized size = 1.3 \begin{align*} a{c}^{4}{d}^{4}x-6\,{\frac{{c}^{2}{d}^{4}a}{x}}+4\,{c}^{3}{d}^{4}a\ln \left ( cx \right ) -2\,{\frac{ca{d}^{4}}{{x}^{2}}}-{\frac{{d}^{4}a}{3\,{x}^{3}}}+b{c}^{4}{d}^{4}x{\it Artanh} \left ( cx \right ) -6\,{\frac{{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ) }{x}}+4\,{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -2\,{\frac{{d}^{4}bc{\it Artanh} \left ( cx \right ) }{{x}^{2}}}-{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ) }{3\,{x}^{3}}}-2\,{c}^{3}{d}^{4}b{\it dilog} \left ( cx \right ) -2\,{c}^{3}{d}^{4}b{\it dilog} \left ( cx+1 \right ) -2\,{c}^{3}{d}^{4}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) -{\frac{11\,{c}^{3}{d}^{4}b\ln \left ( cx-1 \right ) }{3}}-{\frac{{d}^{4}bc}{6\,{x}^{2}}}-2\,{\frac{{c}^{2}{d}^{4}b}{x}}+{\frac{19\,{c}^{3}{d}^{4}b\ln \left ( cx \right ) }{3}}-{\frac{5\,{c}^{3}{d}^{4}b\ln \left ( cx+1 \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^4*(a+b*arctanh(c*x))/x^4,x)

[Out]

a*c^4*d^4*x-6*c^2*d^4*a/x+4*c^3*d^4*a*ln(c*x)-2*c*d^4*a/x^2-1/3*d^4*a/x^3+b*c^4*d^4*x*arctanh(c*x)-6*c^2*d^4*b
*arctanh(c*x)/x+4*c^3*d^4*b*arctanh(c*x)*ln(c*x)-2*c*d^4*b*arctanh(c*x)/x^2-1/3*d^4*b*arctanh(c*x)/x^3-2*c^3*d
^4*b*dilog(c*x)-2*c^3*d^4*b*dilog(c*x+1)-2*c^3*d^4*b*ln(c*x)*ln(c*x+1)-11/3*c^3*d^4*b*ln(c*x-1)-1/6*b*c*d^4/x^
2-2*b*c^2*d^4/x+19/3*c^3*d^4*b*ln(c*x)-5/3*c^3*d^4*b*ln(c*x+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a c^{4} d^{4} 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^{3} d^{4} + 2 \, b c^{3} d^{4} \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + 4 \, a c^{3} d^{4} \log \left (x\right ) - 3 \,{\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 c^{2} d^{4} +{\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac{2}{x}\right )} c - \frac{2 \, \operatorname{artanh}\left (c x\right )}{x^{2}}\right )} b c d^{4} - \frac{1}{6} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac{1}{x^{2}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x\right )}{x^{3}}\right )} b d^{4} - \frac{6 \, a c^{2} d^{4}}{x} - \frac{2 \, a c d^{4}}{x^{2}} - \frac{a d^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^4*(a+b*arctanh(c*x))/x^4,x, algorithm="maxima")

[Out]

a*c^4*d^4*x + 1/2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*b*c^3*d^4 + 2*b*c^3*d^4*integrate((log(c*x + 1) - l
og(-c*x + 1))/x, x) + 4*a*c^3*d^4*log(x) - 3*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*c^2*d^4 +
((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*b*c*d^4 - 1/6*((c^2*log(c^2*x^2 - 1) - c^2*lo
g(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3)*b*d^4 - 6*a*c^2*d^4/x - 2*a*c*d^4/x^2 - 1/3*a*d^4/x^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a c^{4} d^{4} x^{4} + 4 \, a c^{3} d^{4} x^{3} + 6 \, a c^{2} d^{4} x^{2} + 4 \, a c d^{4} x + a d^{4} +{\left (b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 4 \, b c d^{4} x + b d^{4}\right )} \operatorname{artanh}\left (c x\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^4*(a+b*arctanh(c*x))/x^4,x, algorithm="fricas")

[Out]

integral((a*c^4*d^4*x^4 + 4*a*c^3*d^4*x^3 + 6*a*c^2*d^4*x^2 + 4*a*c*d^4*x + a*d^4 + (b*c^4*d^4*x^4 + 4*b*c^3*d
^4*x^3 + 6*b*c^2*d^4*x^2 + 4*b*c*d^4*x + b*d^4)*arctanh(c*x))/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{4} \left (\int a c^{4}\, dx + \int \frac{a}{x^{4}}\, dx + \int \frac{4 a c}{x^{3}}\, dx + \int \frac{6 a c^{2}}{x^{2}}\, dx + \int \frac{4 a c^{3}}{x}\, dx + \int b c^{4} \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{4 b c \operatorname{atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{6 b c^{2} \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{4 b c^{3} \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)**4*(a+b*atanh(c*x))/x**4,x)

[Out]

d**4*(Integral(a*c**4, x) + Integral(a/x**4, x) + Integral(4*a*c/x**3, x) + Integral(6*a*c**2/x**2, x) + Integ
ral(4*a*c**3/x, x) + Integral(b*c**4*atanh(c*x), x) + Integral(b*atanh(c*x)/x**4, x) + Integral(4*b*c*atanh(c*
x)/x**3, x) + Integral(6*b*c**2*atanh(c*x)/x**2, x) + Integral(4*b*c**3*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 )}^{4}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^4*(a+b*arctanh(c*x))/x^4,x, algorithm="giac")

[Out]

integrate((c*d*x + d)^4*(b*arctanh(c*x) + a)/x^4, x)