3.42 \(\int \frac{(d+c d x)^4 (a+b \tanh ^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=229 \[ -\frac{c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac{88 b c^5 d^4}{105 x^2}-\frac{5 b c^4 d^4}{9 x^3}-\frac{47 b c^3 d^4}{140 x^4}-\frac{2 b c^2 d^4}{15 x^5}-\frac{5 b c^6 d^4}{3 x}+\frac{176}{105} b c^7 d^4 \log (x)-\frac{117}{70} b c^7 d^4 \log (1-c x)-\frac{1}{210} b c^7 d^4 \log (c x+1)-\frac{b c d^4}{42 x^6} \]

[Out]

-(b*c*d^4)/(42*x^6) - (2*b*c^2*d^4)/(15*x^5) - (47*b*c^3*d^4)/(140*x^4) - (5*b*c^4*d^4)/(9*x^3) - (88*b*c^5*d^
4)/(105*x^2) - (5*b*c^6*d^4)/(3*x) - (d^4*(a + b*ArcTanh[c*x]))/(7*x^7) - (2*c*d^4*(a + b*ArcTanh[c*x]))/(3*x^
6) - (6*c^2*d^4*(a + b*ArcTanh[c*x]))/(5*x^5) - (c^3*d^4*(a + b*ArcTanh[c*x]))/x^4 - (c^4*d^4*(a + b*ArcTanh[c
*x]))/(3*x^3) + (176*b*c^7*d^4*Log[x])/105 - (117*b*c^7*d^4*Log[1 - c*x])/70 - (b*c^7*d^4*Log[1 + c*x])/210

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Rubi [A]  time = 0.196895, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {43, 5936, 12, 1802} \[ -\frac{c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac{88 b c^5 d^4}{105 x^2}-\frac{5 b c^4 d^4}{9 x^3}-\frac{47 b c^3 d^4}{140 x^4}-\frac{2 b c^2 d^4}{15 x^5}-\frac{5 b c^6 d^4}{3 x}+\frac{176}{105} b c^7 d^4 \log (x)-\frac{117}{70} b c^7 d^4 \log (1-c x)-\frac{1}{210} b c^7 d^4 \log (c x+1)-\frac{b c d^4}{42 x^6} \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^8,x]

[Out]

-(b*c*d^4)/(42*x^6) - (2*b*c^2*d^4)/(15*x^5) - (47*b*c^3*d^4)/(140*x^4) - (5*b*c^4*d^4)/(9*x^3) - (88*b*c^5*d^
4)/(105*x^2) - (5*b*c^6*d^4)/(3*x) - (d^4*(a + b*ArcTanh[c*x]))/(7*x^7) - (2*c*d^4*(a + b*ArcTanh[c*x]))/(3*x^
6) - (6*c^2*d^4*(a + b*ArcTanh[c*x]))/(5*x^5) - (c^3*d^4*(a + b*ArcTanh[c*x]))/x^4 - (c^4*d^4*(a + b*ArcTanh[c
*x]))/(3*x^3) + (176*b*c^7*d^4*Log[x])/105 - (117*b*c^7*d^4*Log[1 - c*x])/70 - (b*c^7*d^4*Log[1 + c*x])/210

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5936

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1802

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

Rubi steps

\begin{align*} \int \frac{(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac{c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{d^4 \left (-15-70 c x-126 c^2 x^2-105 c^3 x^3-35 c^4 x^4\right )}{105 x^7 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac{c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{1}{105} \left (b c d^4\right ) \int \frac{-15-70 c x-126 c^2 x^2-105 c^3 x^3-35 c^4 x^4}{x^7 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac{c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{1}{105} \left (b c d^4\right ) \int \left (-\frac{15}{x^7}-\frac{70 c}{x^6}-\frac{141 c^2}{x^5}-\frac{175 c^3}{x^4}-\frac{176 c^4}{x^3}-\frac{175 c^5}{x^2}-\frac{176 c^6}{x}+\frac{351 c^7}{2 (-1+c x)}+\frac{c^7}{2 (1+c x)}\right ) \, dx\\ &=-\frac{b c d^4}{42 x^6}-\frac{2 b c^2 d^4}{15 x^5}-\frac{47 b c^3 d^4}{140 x^4}-\frac{5 b c^4 d^4}{9 x^3}-\frac{88 b c^5 d^4}{105 x^2}-\frac{5 b c^6 d^4}{3 x}-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac{2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac{6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac{c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac{176}{105} b c^7 d^4 \log (x)-\frac{117}{70} b c^7 d^4 \log (1-c x)-\frac{1}{210} b c^7 d^4 \log (1+c x)\\ \end{align*}

Mathematica [A]  time = 0.174554, size = 175, normalized size = 0.76 \[ -\frac{d^4 \left (420 a c^4 x^4+1260 a c^3 x^3+1512 a c^2 x^2+840 a c x+180 a+2100 b c^6 x^6+1056 b c^5 x^5+700 b c^4 x^4+423 b c^3 x^3+168 b c^2 x^2-2112 b c^7 x^7 \log (x)+2106 b c^7 x^7 \log (1-c x)+6 b c^7 x^7 \log (c x+1)+12 b \left (35 c^4 x^4+105 c^3 x^3+126 c^2 x^2+70 c x+15\right ) \tanh ^{-1}(c x)+30 b c x\right )}{1260 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^8,x]

[Out]

-(d^4*(180*a + 840*a*c*x + 30*b*c*x + 1512*a*c^2*x^2 + 168*b*c^2*x^2 + 1260*a*c^3*x^3 + 423*b*c^3*x^3 + 420*a*
c^4*x^4 + 700*b*c^4*x^4 + 1056*b*c^5*x^5 + 2100*b*c^6*x^6 + 12*b*(15 + 70*c*x + 126*c^2*x^2 + 105*c^3*x^3 + 35
*c^4*x^4)*ArcTanh[c*x] - 2112*b*c^7*x^7*Log[x] + 2106*b*c^7*x^7*Log[1 - c*x] + 6*b*c^7*x^7*Log[1 + c*x]))/(126
0*x^7)

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Maple [A]  time = 0.042, size = 245, normalized size = 1.1 \begin{align*} -{\frac{{c}^{3}{d}^{4}a}{{x}^{4}}}-{\frac{{d}^{4}a}{7\,{x}^{7}}}-{\frac{6\,{c}^{2}{d}^{4}a}{5\,{x}^{5}}}-{\frac{2\,c{d}^{4}a}{3\,{x}^{6}}}-{\frac{{c}^{4}{d}^{4}a}{3\,{x}^{3}}}-{\frac{{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ) }{{x}^{4}}}-{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ) }{7\,{x}^{7}}}-{\frac{6\,{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ) }{5\,{x}^{5}}}-{\frac{2\,c{d}^{4}b{\it Artanh} \left ( cx \right ) }{3\,{x}^{6}}}-{\frac{{c}^{4}{d}^{4}b{\it Artanh} \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{117\,{c}^{7}{d}^{4}b\ln \left ( cx-1 \right ) }{70}}-{\frac{c{d}^{4}b}{42\,{x}^{6}}}-{\frac{2\,{c}^{2}{d}^{4}b}{15\,{x}^{5}}}-{\frac{47\,{c}^{3}{d}^{4}b}{140\,{x}^{4}}}-{\frac{5\,{c}^{4}{d}^{4}b}{9\,{x}^{3}}}-{\frac{88\,b{c}^{5}{d}^{4}}{105\,{x}^{2}}}-{\frac{5\,b{c}^{6}{d}^{4}}{3\,x}}+{\frac{176\,{c}^{7}{d}^{4}b\ln \left ( cx \right ) }{105}}-{\frac{b{c}^{7}{d}^{4}\ln \left ( cx+1 \right ) }{210}} \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^8,x)

[Out]

-c^3*d^4*a/x^4-1/7*d^4*a/x^7-6/5*c^2*d^4*a/x^5-2/3*c*d^4*a/x^6-1/3*c^4*d^4*a/x^3-c^3*d^4*b*arctanh(c*x)/x^4-1/
7*d^4*b*arctanh(c*x)/x^7-6/5*c^2*d^4*b*arctanh(c*x)/x^5-2/3*c*d^4*b*arctanh(c*x)/x^6-1/3*c^4*d^4*b*arctanh(c*x
)/x^3-117/70*c^7*d^4*b*ln(c*x-1)-1/42*b*c*d^4/x^6-2/15*b*c^2*d^4/x^5-47/140*b*c^3*d^4/x^4-5/9*b*c^4*d^4/x^3-88
/105*b*c^5*d^4/x^2-5/3*b*c^6*d^4/x+176/105*c^7*d^4*b*ln(c*x)-1/210*b*c^7*d^4*ln(c*x+1)

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Maxima [A]  time = 1.01472, size = 477, normalized size = 2.08 \begin{align*} -\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 c^{4} d^{4} + \frac{1}{6} \,{\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac{2 \,{\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac{6 \, \operatorname{artanh}\left (c x\right )}{x^{4}}\right )} b c^{3} d^{4} - \frac{3}{10} \,{\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac{2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac{4 \, \operatorname{artanh}\left (c x\right )}{x^{5}}\right )} b c^{2} d^{4} + \frac{1}{45} \,{\left ({\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac{2 \,{\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c - \frac{30 \, \operatorname{artanh}\left (c x\right )}{x^{6}}\right )} b c d^{4} - \frac{1}{84} \,{\left ({\left (6 \, c^{6} \log \left (c^{2} x^{2} - 1\right ) - 6 \, c^{6} \log \left (x^{2}\right ) + \frac{6 \, c^{4} x^{4} + 3 \, c^{2} x^{2} + 2}{x^{6}}\right )} c + \frac{12 \, \operatorname{artanh}\left (c x\right )}{x^{7}}\right )} b d^{4} - \frac{a c^{4} d^{4}}{3 \, x^{3}} - \frac{a c^{3} d^{4}}{x^{4}} - \frac{6 \, a c^{2} d^{4}}{5 \, x^{5}} - \frac{2 \, a c d^{4}}{3 \, x^{6}} - \frac{a d^{4}}{7 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

-1/6*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3)*b*c^4*d^4 + 1/6*((3*c^3*log(c*x +
1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x^2 + 1)/x^3)*c - 6*arctanh(c*x)/x^4)*b*c^3*d^4 - 3/10*((2*c^4*log(c^2*x^2
- 1) - 2*c^4*log(x^2) + (2*c^2*x^2 + 1)/x^4)*c + 4*arctanh(c*x)/x^5)*b*c^2*d^4 + 1/45*((15*c^5*log(c*x + 1) -
15*c^5*log(c*x - 1) - 2*(15*c^4*x^4 + 5*c^2*x^2 + 3)/x^5)*c - 30*arctanh(c*x)/x^6)*b*c*d^4 - 1/84*((6*c^6*log(
c^2*x^2 - 1) - 6*c^6*log(x^2) + (6*c^4*x^4 + 3*c^2*x^2 + 2)/x^6)*c + 12*arctanh(c*x)/x^7)*b*d^4 - 1/3*a*c^4*d^
4/x^3 - a*c^3*d^4/x^4 - 6/5*a*c^2*d^4/x^5 - 2/3*a*c*d^4/x^6 - 1/7*a*d^4/x^7

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Fricas [A]  time = 2.31061, size = 524, normalized size = 2.29 \begin{align*} -\frac{6 \, b c^{7} d^{4} x^{7} \log \left (c x + 1\right ) + 2106 \, b c^{7} d^{4} x^{7} \log \left (c x - 1\right ) - 2112 \, b c^{7} d^{4} x^{7} \log \left (x\right ) + 2100 \, b c^{6} d^{4} x^{6} + 1056 \, b c^{5} d^{4} x^{5} + 140 \,{\left (3 \, a + 5 \, b\right )} c^{4} d^{4} x^{4} + 9 \,{\left (140 \, a + 47 \, b\right )} c^{3} d^{4} x^{3} + 168 \,{\left (9 \, a + b\right )} c^{2} d^{4} x^{2} + 30 \,{\left (28 \, a + b\right )} c d^{4} x + 180 \, a d^{4} + 6 \,{\left (35 \, b c^{4} d^{4} x^{4} + 105 \, b c^{3} d^{4} x^{3} + 126 \, b c^{2} d^{4} x^{2} + 70 \, b c d^{4} x + 15 \, b d^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{1260 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/1260*(6*b*c^7*d^4*x^7*log(c*x + 1) + 2106*b*c^7*d^4*x^7*log(c*x - 1) - 2112*b*c^7*d^4*x^7*log(x) + 2100*b*c
^6*d^4*x^6 + 1056*b*c^5*d^4*x^5 + 140*(3*a + 5*b)*c^4*d^4*x^4 + 9*(140*a + 47*b)*c^3*d^4*x^3 + 168*(9*a + b)*c
^2*d^4*x^2 + 30*(28*a + b)*c*d^4*x + 180*a*d^4 + 6*(35*b*c^4*d^4*x^4 + 105*b*c^3*d^4*x^3 + 126*b*c^2*d^4*x^2 +
70*b*c*d^4*x + 15*b*d^4)*log(-(c*x + 1)/(c*x - 1)))/x^7

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Sympy [A]  time = 17.4093, size = 301, normalized size = 1.31 \begin{align*} \begin{cases} - \frac{a c^{4} d^{4}}{3 x^{3}} - \frac{a c^{3} d^{4}}{x^{4}} - \frac{6 a c^{2} d^{4}}{5 x^{5}} - \frac{2 a c d^{4}}{3 x^{6}} - \frac{a d^{4}}{7 x^{7}} + \frac{176 b c^{7} d^{4} \log{\left (x \right )}}{105} - \frac{176 b c^{7} d^{4} \log{\left (x - \frac{1}{c} \right )}}{105} - \frac{b c^{7} d^{4} \operatorname{atanh}{\left (c x \right )}}{105} - \frac{5 b c^{6} d^{4}}{3 x} - \frac{88 b c^{5} d^{4}}{105 x^{2}} - \frac{b c^{4} d^{4} \operatorname{atanh}{\left (c x \right )}}{3 x^{3}} - \frac{5 b c^{4} d^{4}}{9 x^{3}} - \frac{b c^{3} d^{4} \operatorname{atanh}{\left (c x \right )}}{x^{4}} - \frac{47 b c^{3} d^{4}}{140 x^{4}} - \frac{6 b c^{2} d^{4} \operatorname{atanh}{\left (c x \right )}}{5 x^{5}} - \frac{2 b c^{2} d^{4}}{15 x^{5}} - \frac{2 b c d^{4} \operatorname{atanh}{\left (c x \right )}}{3 x^{6}} - \frac{b c d^{4}}{42 x^{6}} - \frac{b d^{4} \operatorname{atanh}{\left (c x \right )}}{7 x^{7}} & \text{for}\: c \neq 0 \\- \frac{a d^{4}}{7 x^{7}} & \text{otherwise} \end{cases} \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**8,x)

[Out]

Piecewise((-a*c**4*d**4/(3*x**3) - a*c**3*d**4/x**4 - 6*a*c**2*d**4/(5*x**5) - 2*a*c*d**4/(3*x**6) - a*d**4/(7
*x**7) + 176*b*c**7*d**4*log(x)/105 - 176*b*c**7*d**4*log(x - 1/c)/105 - b*c**7*d**4*atanh(c*x)/105 - 5*b*c**6
*d**4/(3*x) - 88*b*c**5*d**4/(105*x**2) - b*c**4*d**4*atanh(c*x)/(3*x**3) - 5*b*c**4*d**4/(9*x**3) - b*c**3*d*
*4*atanh(c*x)/x**4 - 47*b*c**3*d**4/(140*x**4) - 6*b*c**2*d**4*atanh(c*x)/(5*x**5) - 2*b*c**2*d**4/(15*x**5) -
2*b*c*d**4*atanh(c*x)/(3*x**6) - b*c*d**4/(42*x**6) - b*d**4*atanh(c*x)/(7*x**7), Ne(c, 0)), (-a*d**4/(7*x**7
), True))

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Giac [A]  time = 2.60587, size = 320, normalized size = 1.4 \begin{align*} -\frac{1}{210} \, b c^{7} d^{4} \log \left (c x + 1\right ) - \frac{117}{70} \, b c^{7} d^{4} \log \left (c x - 1\right ) + \frac{176}{105} \, b c^{7} d^{4} \log \left (x\right ) - \frac{{\left (35 \, b c^{4} d^{4} x^{4} + 105 \, b c^{3} d^{4} x^{3} + 126 \, b c^{2} d^{4} x^{2} + 70 \, b c d^{4} x + 15 \, b d^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{210 \, x^{7}} - \frac{2100 \, b c^{6} d^{4} x^{6} + 1056 \, b c^{5} d^{4} x^{5} + 420 \, a c^{4} d^{4} x^{4} + 700 \, b c^{4} d^{4} x^{4} + 1260 \, a c^{3} d^{4} x^{3} + 423 \, b c^{3} d^{4} x^{3} + 1512 \, a c^{2} d^{4} x^{2} + 168 \, b c^{2} d^{4} x^{2} + 840 \, a c d^{4} x + 30 \, b c d^{4} x + 180 \, a d^{4}}{1260 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

-1/210*b*c^7*d^4*log(c*x + 1) - 117/70*b*c^7*d^4*log(c*x - 1) + 176/105*b*c^7*d^4*log(x) - 1/210*(35*b*c^4*d^4
*x^4 + 105*b*c^3*d^4*x^3 + 126*b*c^2*d^4*x^2 + 70*b*c*d^4*x + 15*b*d^4)*log(-(c*x + 1)/(c*x - 1))/x^7 - 1/1260
*(2100*b*c^6*d^4*x^6 + 1056*b*c^5*d^4*x^5 + 420*a*c^4*d^4*x^4 + 700*b*c^4*d^4*x^4 + 1260*a*c^3*d^4*x^3 + 423*b
*c^3*d^4*x^3 + 1512*a*c^2*d^4*x^2 + 168*b*c^2*d^4*x^2 + 840*a*c*d^4*x + 30*b*c*d^4*x + 180*a*d^4)/x^7