3.498 \(\int (c+d x^2)^4 \tanh ^{-1}(a x) \, dx\)

Optimal. Leaf size=245 \[ \frac{d^2 x^4 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right )}{1260 a^5}+\frac{d x^2 \left (378 a^4 c^2 d+420 a^6 c^3+180 a^2 c d^2+35 d^3\right )}{630 a^7}+\frac{\left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{630 a^9}+\frac{d^3 x^6 \left (36 a^2 c+7 d\right )}{378 a^3}+\frac{6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tanh ^{-1}(a x)+c^4 x \tanh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac{d^4 x^8}{72 a}+\frac{1}{9} d^4 x^9 \tanh ^{-1}(a x) \]

[Out]

(d*(420*a^6*c^3 + 378*a^4*c^2*d + 180*a^2*c*d^2 + 35*d^3)*x^2)/(630*a^7) + (d^2*(378*a^4*c^2 + 180*a^2*c*d + 3
5*d^2)*x^4)/(1260*a^5) + (d^3*(36*a^2*c + 7*d)*x^6)/(378*a^3) + (d^4*x^8)/(72*a) + c^4*x*ArcTanh[a*x] + (4*c^3
*d*x^3*ArcTanh[a*x])/3 + (6*c^2*d^2*x^5*ArcTanh[a*x])/5 + (4*c*d^3*x^7*ArcTanh[a*x])/7 + (d^4*x^9*ArcTanh[a*x]
)/9 + ((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Log[1 - a^2*x^2])/(630*a^9)

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Rubi [A]  time = 0.179877, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {194, 5976, 1810, 260} \[ \frac{d^2 x^4 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right )}{1260 a^5}+\frac{d x^2 \left (378 a^4 c^2 d+420 a^6 c^3+180 a^2 c d^2+35 d^3\right )}{630 a^7}+\frac{\left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{630 a^9}+\frac{d^3 x^6 \left (36 a^2 c+7 d\right )}{378 a^3}+\frac{6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tanh ^{-1}(a x)+c^4 x \tanh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac{d^4 x^8}{72 a}+\frac{1}{9} d^4 x^9 \tanh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4*ArcTanh[a*x],x]

[Out]

(d*(420*a^6*c^3 + 378*a^4*c^2*d + 180*a^2*c*d^2 + 35*d^3)*x^2)/(630*a^7) + (d^2*(378*a^4*c^2 + 180*a^2*c*d + 3
5*d^2)*x^4)/(1260*a^5) + (d^3*(36*a^2*c + 7*d)*x^6)/(378*a^3) + (d^4*x^8)/(72*a) + c^4*x*ArcTanh[a*x] + (4*c^3
*d*x^3*ArcTanh[a*x])/3 + (6*c^2*d^2*x^5*ArcTanh[a*x])/5 + (4*c*d^3*x^7*ArcTanh[a*x])/7 + (d^4*x^9*ArcTanh[a*x]
)/9 + ((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Log[1 - a^2*x^2])/(630*a^9)

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5976

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^
2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x
] && (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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]

Rubi steps

\begin{align*} \int \left (c+d x^2\right )^4 \tanh ^{-1}(a x) \, dx &=c^4 x \tanh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \tanh ^{-1}(a x)-a \int \frac{c^4 x+\frac{4}{3} c^3 d x^3+\frac{6}{5} c^2 d^2 x^5+\frac{4}{7} c d^3 x^7+\frac{d^4 x^9}{9}}{1-a^2 x^2} \, dx\\ &=c^4 x \tanh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \tanh ^{-1}(a x)-a \int \left (-\frac{d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x}{315 a^8}-\frac{d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^3}{315 a^6}-\frac{d^3 \left (36 a^2 c+7 d\right ) x^5}{63 a^4}-\frac{d^4 x^7}{9 a^2}+\frac{\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) x}{315 a^8 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac{d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x^2}{630 a^7}+\frac{d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac{d^3 \left (36 a^2 c+7 d\right ) x^6}{378 a^3}+\frac{d^4 x^8}{72 a}+c^4 x \tanh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \tanh ^{-1}(a x)-\frac{\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \int \frac{x}{1-a^2 x^2} \, dx}{315 a^7}\\ &=\frac{d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x^2}{630 a^7}+\frac{d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac{d^3 \left (36 a^2 c+7 d\right ) x^6}{378 a^3}+\frac{d^4 x^8}{72 a}+c^4 x \tanh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \tanh ^{-1}(a x)+\frac{\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{630 a^9}\\ \end{align*}

Mathematica [A]  time = 0.111614, size = 213, normalized size = 0.87 \[ \frac{a^2 d x^2 \left (3 a^6 \left (756 c^2 d x^2+1680 c^3+240 c d^2 x^4+35 d^3 x^6\right )+4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+30 a^2 d^2 \left (72 c+7 d x^2\right )+420 d^3\right )+12 \left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )+24 a^9 x \tanh ^{-1}(a x) \left (378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4+180 c d^3 x^6+35 d^4 x^8\right )}{7560 a^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^4*ArcTanh[a*x],x]

[Out]

(a^2*d*x^2*(420*d^3 + 30*a^2*d^2*(72*c + 7*d*x^2) + 4*a^4*d*(1134*c^2 + 270*c*d*x^2 + 35*d^2*x^4) + 3*a^6*(168
0*c^3 + 756*c^2*d*x^2 + 240*c*d^2*x^4 + 35*d^3*x^6)) + 24*a^9*x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 1
80*c*d^3*x^6 + 35*d^4*x^8)*ArcTanh[a*x] + 12*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 +
35*d^4)*Log[1 - a^2*x^2])/(7560*a^9)

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Maple [A]  time = 0.035, size = 334, normalized size = 1.4 \begin{align*}{\frac{{d}^{4}{x}^{8}}{72\,a}}+{\frac{{x}^{6}{d}^{4}}{54\,{a}^{3}}}+{\frac{{x}^{2}{d}^{4}}{18\,{a}^{7}}}+{\frac{{x}^{4}{d}^{4}}{36\,{a}^{5}}}+{\frac{\ln \left ( ax-1 \right ){c}^{4}}{2\,a}}+{\frac{\ln \left ( ax-1 \right ){d}^{4}}{18\,{a}^{9}}}+{\frac{\ln \left ( ax+1 \right ){c}^{4}}{2\,a}}+{\frac{\ln \left ( ax+1 \right ){d}^{4}}{18\,{a}^{9}}}+{\frac{{d}^{4}{x}^{9}{\it Artanh} \left ( ax \right ) }{9}}+{c}^{4}x{\it Artanh} \left ( ax \right ) +{\frac{4\,{c}^{3}d{x}^{3}{\it Artanh} \left ( ax \right ) }{3}}+{\frac{6\,{c}^{2}{d}^{2}{x}^{5}{\it Artanh} \left ( ax \right ) }{5}}+{\frac{4\,c{d}^{3}{x}^{7}{\it Artanh} \left ( ax \right ) }{7}}+{\frac{{x}^{4}c{d}^{3}}{7\,{a}^{3}}}+{\frac{2\,c{x}^{2}{d}^{3}}{7\,{a}^{5}}}+{\frac{2\,\ln \left ( ax-1 \right ){c}^{3}d}{3\,{a}^{3}}}+{\frac{3\,\ln \left ( ax-1 \right ){c}^{2}{d}^{2}}{5\,{a}^{5}}}+{\frac{2\,\ln \left ( ax-1 \right ) c{d}^{3}}{7\,{a}^{7}}}+{\frac{2\,\ln \left ( ax+1 \right ){c}^{3}d}{3\,{a}^{3}}}+{\frac{3\,\ln \left ( ax+1 \right ){c}^{2}{d}^{2}}{5\,{a}^{5}}}+{\frac{2\,\ln \left ( ax+1 \right ) c{d}^{3}}{7\,{a}^{7}}}+{\frac{3\,{c}^{2}{d}^{2}{x}^{2}}{5\,{a}^{3}}}+{\frac{2\,c{d}^{3}{x}^{6}}{21\,a}}+{\frac{3\,{c}^{2}{d}^{2}{x}^{4}}{10\,a}}+{\frac{2\,{c}^{3}d{x}^{2}}{3\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^4*arctanh(a*x),x)

[Out]

4/3*c^3*d*x^3*arctanh(a*x)+6/5*c^2*d^2*x^5*arctanh(a*x)+4/7*c*d^3*x^7*arctanh(a*x)+1/72*d^4*x^8/a+1/9*d^4*x^9*
arctanh(a*x)+1/54/a^3*x^6*d^4+1/18/a^7*x^2*d^4+1/36/a^5*x^4*d^4+1/2/a*ln(a*x-1)*c^4+1/18/a^9*ln(a*x-1)*d^4+1/2
/a*ln(a*x+1)*c^4+1/18/a^9*ln(a*x+1)*d^4+1/7/a^3*x^4*c*d^3+2/7/a^5*x^2*c*d^3+2/3/a^3*ln(a*x-1)*c^3*d+3/5/a^5*ln
(a*x-1)*c^2*d^2+2/7/a^7*ln(a*x-1)*c*d^3+2/3/a^3*ln(a*x+1)*c^3*d+3/5/a^5*ln(a*x+1)*c^2*d^2+2/7/a^7*ln(a*x+1)*c*
d^3+3/5/a^3*c^2*d^2*x^2+2/21/a*c*d^3*x^6+3/10/a*c^2*d^2*x^4+2/3/a*c^3*d*x^2+c^4*x*arctanh(a*x)

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Maxima [A]  time = 0.966531, size = 373, normalized size = 1.52 \begin{align*} \frac{1}{7560} \, a{\left (\frac{105 \, a^{6} d^{4} x^{8} + 20 \,{\left (36 \, a^{6} c d^{3} + 7 \, a^{4} d^{4}\right )} x^{6} + 6 \,{\left (378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \,{\left (420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac{12 \,{\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a x + 1\right )}{a^{10}} + \frac{12 \,{\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a x - 1\right )}{a^{10}}\right )} + \frac{1}{315} \,{\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname{artanh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arctanh(a*x),x, algorithm="maxima")

[Out]

1/7560*a*((105*a^6*d^4*x^8 + 20*(36*a^6*c*d^3 + 7*a^4*d^4)*x^6 + 6*(378*a^6*c^2*d^2 + 180*a^4*c*d^3 + 35*a^2*d
^4)*x^4 + 12*(420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*x^2)/a^8 + 12*(315*a^8*c^4 + 420*a^6*c
^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(a*x + 1)/a^10 + 12*(315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4
*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(a*x - 1)/a^10) + 1/315*(35*d^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 +
420*c^3*d*x^3 + 315*c^4*x)*arctanh(a*x)

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Fricas [A]  time = 1.94526, size = 559, normalized size = 2.28 \begin{align*} \frac{105 \, a^{8} d^{4} x^{8} + 20 \,{\left (36 \, a^{8} c d^{3} + 7 \, a^{6} d^{4}\right )} x^{6} + 6 \,{\left (378 \, a^{8} c^{2} d^{2} + 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \,{\left (420 \, a^{8} c^{3} d + 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{2} + 12 \,{\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} - 1\right ) + 12 \,{\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{7560 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arctanh(a*x),x, algorithm="fricas")

[Out]

1/7560*(105*a^8*d^4*x^8 + 20*(36*a^8*c*d^3 + 7*a^6*d^4)*x^6 + 6*(378*a^8*c^2*d^2 + 180*a^6*c*d^3 + 35*a^4*d^4)
*x^4 + 12*(420*a^8*c^3*d + 378*a^6*c^2*d^2 + 180*a^4*c*d^3 + 35*a^2*d^4)*x^2 + 12*(315*a^8*c^4 + 420*a^6*c^3*d
+ 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(a^2*x^2 - 1) + 12*(35*a^9*d^4*x^9 + 180*a^9*c*d^3*x^7 + 378*a
^9*c^2*d^2*x^5 + 420*a^9*c^3*d*x^3 + 315*a^9*c^4*x)*log(-(a*x + 1)/(a*x - 1)))/a^9

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Sympy [A]  time = 15.0584, size = 372, normalized size = 1.52 \begin{align*} \begin{cases} c^{4} x \operatorname{atanh}{\left (a x \right )} + \frac{4 c^{3} d x^{3} \operatorname{atanh}{\left (a x \right )}}{3} + \frac{6 c^{2} d^{2} x^{5} \operatorname{atanh}{\left (a x \right )}}{5} + \frac{4 c d^{3} x^{7} \operatorname{atanh}{\left (a x \right )}}{7} + \frac{d^{4} x^{9} \operatorname{atanh}{\left (a x \right )}}{9} + \frac{c^{4} \log{\left (x - \frac{1}{a} \right )}}{a} + \frac{c^{4} \operatorname{atanh}{\left (a x \right )}}{a} + \frac{2 c^{3} d x^{2}}{3 a} + \frac{3 c^{2} d^{2} x^{4}}{10 a} + \frac{2 c d^{3} x^{6}}{21 a} + \frac{d^{4} x^{8}}{72 a} + \frac{4 c^{3} d \log{\left (x - \frac{1}{a} \right )}}{3 a^{3}} + \frac{4 c^{3} d \operatorname{atanh}{\left (a x \right )}}{3 a^{3}} + \frac{3 c^{2} d^{2} x^{2}}{5 a^{3}} + \frac{c d^{3} x^{4}}{7 a^{3}} + \frac{d^{4} x^{6}}{54 a^{3}} + \frac{6 c^{2} d^{2} \log{\left (x - \frac{1}{a} \right )}}{5 a^{5}} + \frac{6 c^{2} d^{2} \operatorname{atanh}{\left (a x \right )}}{5 a^{5}} + \frac{2 c d^{3} x^{2}}{7 a^{5}} + \frac{d^{4} x^{4}}{36 a^{5}} + \frac{4 c d^{3} \log{\left (x - \frac{1}{a} \right )}}{7 a^{7}} + \frac{4 c d^{3} \operatorname{atanh}{\left (a x \right )}}{7 a^{7}} + \frac{d^{4} x^{2}}{18 a^{7}} + \frac{d^{4} \log{\left (x - \frac{1}{a} \right )}}{9 a^{9}} + \frac{d^{4} \operatorname{atanh}{\left (a x \right )}}{9 a^{9}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**4*atanh(a*x),x)

[Out]

Piecewise((c**4*x*atanh(a*x) + 4*c**3*d*x**3*atanh(a*x)/3 + 6*c**2*d**2*x**5*atanh(a*x)/5 + 4*c*d**3*x**7*atan
h(a*x)/7 + d**4*x**9*atanh(a*x)/9 + c**4*log(x - 1/a)/a + c**4*atanh(a*x)/a + 2*c**3*d*x**2/(3*a) + 3*c**2*d**
2*x**4/(10*a) + 2*c*d**3*x**6/(21*a) + d**4*x**8/(72*a) + 4*c**3*d*log(x - 1/a)/(3*a**3) + 4*c**3*d*atanh(a*x)
/(3*a**3) + 3*c**2*d**2*x**2/(5*a**3) + c*d**3*x**4/(7*a**3) + d**4*x**6/(54*a**3) + 6*c**2*d**2*log(x - 1/a)/
(5*a**5) + 6*c**2*d**2*atanh(a*x)/(5*a**5) + 2*c*d**3*x**2/(7*a**5) + d**4*x**4/(36*a**5) + 4*c*d**3*log(x - 1
/a)/(7*a**7) + 4*c*d**3*atanh(a*x)/(7*a**7) + d**4*x**2/(18*a**7) + d**4*log(x - 1/a)/(9*a**9) + d**4*atanh(a*
x)/(9*a**9), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.1733, size = 331, normalized size = 1.35 \begin{align*} \frac{1}{630} \,{\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + \frac{105 \, a^{7} d^{4} x^{8} + 720 \, a^{7} c d^{3} x^{6} + 2268 \, a^{7} c^{2} d^{2} x^{4} + 140 \, a^{5} d^{4} x^{6} + 5040 \, a^{7} c^{3} d x^{2} + 1080 \, a^{5} c d^{3} x^{4} + 4536 \, a^{5} c^{2} d^{2} x^{2} + 210 \, a^{3} d^{4} x^{4} + 2160 \, a^{3} c d^{3} x^{2} + 420 \, a d^{4} x^{2}}{7560 \, a^{8}} + \frac{{\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{630 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arctanh(a*x),x, algorithm="giac")

[Out]

1/630*(35*d^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 + 420*c^3*d*x^3 + 315*c^4*x)*log(-(a*x + 1)/(a*x - 1)) + 1
/7560*(105*a^7*d^4*x^8 + 720*a^7*c*d^3*x^6 + 2268*a^7*c^2*d^2*x^4 + 140*a^5*d^4*x^6 + 5040*a^7*c^3*d*x^2 + 108
0*a^5*c*d^3*x^4 + 4536*a^5*c^2*d^2*x^2 + 210*a^3*d^4*x^4 + 2160*a^3*c*d^3*x^2 + 420*a*d^4*x^2)/a^8 + 1/630*(31
5*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*log(abs(a^2*x^2 - 1))/a^9