3.40 \(\int \frac{(d+c d x)^4 (a+b \tanh ^{-1}(c x))}{x^6} \, dx\)

Optimal. Leaf size=109 \[ -\frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{11 b c^3 d^4}{10 x^2}-\frac{b c^2 d^4}{3 x^3}-\frac{3 b c^4 d^4}{x}+\frac{16}{5} b c^5 d^4 \log (x)-\frac{16}{5} b c^5 d^4 \log (1-c x)-\frac{b c d^4}{20 x^4} \]

[Out]

-(b*c*d^4)/(20*x^4) - (b*c^2*d^4)/(3*x^3) - (11*b*c^3*d^4)/(10*x^2) - (3*b*c^4*d^4)/x - (d^4*(1 + c*x)^5*(a +
b*ArcTanh[c*x]))/(5*x^5) + (16*b*c^5*d^4*Log[x])/5 - (16*b*c^5*d^4*Log[1 - c*x])/5

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Rubi [A]  time = 0.104959, antiderivative size = 109, 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 = {37, 5936, 12, 88} \[ -\frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{11 b c^3 d^4}{10 x^2}-\frac{b c^2 d^4}{3 x^3}-\frac{3 b c^4 d^4}{x}+\frac{16}{5} b c^5 d^4 \log (x)-\frac{16}{5} b c^5 d^4 \log (1-c x)-\frac{b c d^4}{20 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^6,x]

[Out]

-(b*c*d^4)/(20*x^4) - (b*c^2*d^4)/(3*x^3) - (11*b*c^3*d^4)/(10*x^2) - (3*b*c^4*d^4)/x - (d^4*(1 + c*x)^5*(a +
b*ArcTanh[c*x]))/(5*x^5) + (16*b*c^5*d^4*Log[x])/5 - (16*b*c^5*d^4*Log[1 - c*x])/5

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-(b c) \int \frac{(d+c d x)^4}{5 x^5 (-1+c x)} \, dx\\ &=-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{1}{5} (b c) \int \frac{(d+c d x)^4}{x^5 (-1+c x)} \, dx\\ &=-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{1}{5} (b c) \int \left (-\frac{d^4}{x^5}-\frac{5 c d^4}{x^4}-\frac{11 c^2 d^4}{x^3}-\frac{15 c^3 d^4}{x^2}-\frac{16 c^4 d^4}{x}+\frac{16 c^5 d^4}{-1+c x}\right ) \, dx\\ &=-\frac{b c d^4}{20 x^4}-\frac{b c^2 d^4}{3 x^3}-\frac{11 b c^3 d^4}{10 x^2}-\frac{3 b c^4 d^4}{x}-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}+\frac{16}{5} b c^5 d^4 \log (x)-\frac{16}{5} b c^5 d^4 \log (1-c x)\\ \end{align*}

Mathematica [A]  time = 0.147057, size = 157, normalized size = 1.44 \[ -\frac{d^4 \left (60 a c^4 x^4+120 a c^3 x^3+120 a c^2 x^2+60 a c x+12 a+180 b c^4 x^4+66 b c^3 x^3+20 b c^2 x^2-192 b c^5 x^5 \log (x)+186 b c^5 x^5 \log (1-c x)+6 b c^5 x^5 \log (c x+1)+12 b \left (5 c^4 x^4+10 c^3 x^3+10 c^2 x^2+5 c x+1\right ) \tanh ^{-1}(c x)+3 b c x\right )}{60 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^6,x]

[Out]

-(d^4*(12*a + 60*a*c*x + 3*b*c*x + 120*a*c^2*x^2 + 20*b*c^2*x^2 + 120*a*c^3*x^3 + 66*b*c^3*x^3 + 60*a*c^4*x^4
+ 180*b*c^4*x^4 + 12*b*(1 + 5*c*x + 10*c^2*x^2 + 10*c^3*x^3 + 5*c^4*x^4)*ArcTanh[c*x] - 192*b*c^5*x^5*Log[x] +
186*b*c^5*x^5*Log[1 - c*x] + 6*b*c^5*x^5*Log[1 + c*x]))/(60*x^5)

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Maple [B]  time = 0.04, size = 221, normalized size = 2. \begin{align*} -{\frac{c{d}^{4}a}{{x}^{4}}}-{\frac{{c}^{4}{d}^{4}a}{x}}-{\frac{{d}^{4}a}{5\,{x}^{5}}}-2\,{\frac{{c}^{3}{d}^{4}a}{{x}^{2}}}-2\,{\frac{{c}^{2}{d}^{4}a}{{x}^{3}}}-{\frac{c{d}^{4}b{\it Artanh} \left ( cx \right ) }{{x}^{4}}}-{\frac{{c}^{4}{d}^{4}b{\it Artanh} \left ( cx \right ) }{x}}-{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ) }{5\,{x}^{5}}}-2\,{\frac{{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ) }{{x}^{2}}}-2\,{\frac{{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ) }{{x}^{3}}}-{\frac{31\,{c}^{5}{d}^{4}b\ln \left ( cx-1 \right ) }{10}}-{\frac{c{d}^{4}b}{20\,{x}^{4}}}-{\frac{{c}^{2}{d}^{4}b}{3\,{x}^{3}}}-{\frac{11\,{c}^{3}{d}^{4}b}{10\,{x}^{2}}}-3\,{\frac{{c}^{4}{d}^{4}b}{x}}+{\frac{16\,{c}^{5}{d}^{4}b\ln \left ( cx \right ) }{5}}-{\frac{{c}^{5}{d}^{4}b\ln \left ( cx+1 \right ) }{10}} \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^6,x)

[Out]

-c*d^4*a/x^4-c^4*d^4*a/x-1/5*d^4*a/x^5-2*c^3*d^4*a/x^2-2*c^2*d^4*a/x^3-c*d^4*b*arctanh(c*x)/x^4-c^4*d^4*b*arct
anh(c*x)/x-1/5*d^4*b*arctanh(c*x)/x^5-2*c^3*d^4*b*arctanh(c*x)/x^2-2*c^2*d^4*b*arctanh(c*x)/x^3-31/10*c^5*d^4*
b*ln(c*x-1)-1/20*b*c*d^4/x^4-1/3*b*c^2*d^4/x^3-11/10*b*c^3*d^4/x^2-3*b*c^4*d^4/x+16/5*c^5*d^4*b*ln(c*x)-1/10*c
^5*d^4*b*ln(c*x+1)

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Maxima [B]  time = 0.980416, size = 404, normalized size = 3.71 \begin{align*} -\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 c^{4} 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^{3} d^{4} -{\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^{2} d^{4} - \frac{a c^{4} d^{4}}{x} + \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 d^{4} - \frac{1}{20} \,{\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 d^{4} - \frac{2 \, a c^{3} d^{4}}{x^{2}} - \frac{2 \, a c^{2} d^{4}}{x^{3}} - \frac{a c d^{4}}{x^{4}} - \frac{a d^{4}}{5 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

-1/2*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*c^4*d^4 + ((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)
*c - 2*arctanh(c*x)/x^2)*b*c^3*d^4 - ((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^2*d^4 - a*c^4*d^4/x + 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*d^4 - 1/20*((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*d^4 - 2*a*c^3*d^4/x^2 - 2*a*c^2*d^4/x^3 - a*c*d^4/x^4 - 1/5*a*d^4/x^5

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Fricas [A]  time = 2.15719, size = 443, normalized size = 4.06 \begin{align*} -\frac{6 \, b c^{5} d^{4} x^{5} \log \left (c x + 1\right ) + 186 \, b c^{5} d^{4} x^{5} \log \left (c x - 1\right ) - 192 \, b c^{5} d^{4} x^{5} \log \left (x\right ) + 60 \,{\left (a + 3 \, b\right )} c^{4} d^{4} x^{4} + 6 \,{\left (20 \, a + 11 \, b\right )} c^{3} d^{4} x^{3} + 20 \,{\left (6 \, a + b\right )} c^{2} d^{4} x^{2} + 3 \,{\left (20 \, a + b\right )} c d^{4} x + 12 \, a d^{4} + 6 \,{\left (5 \, b c^{4} d^{4} x^{4} + 10 \, b c^{3} d^{4} x^{3} + 10 \, b c^{2} d^{4} x^{2} + 5 \, b c d^{4} x + b d^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{60 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/60*(6*b*c^5*d^4*x^5*log(c*x + 1) + 186*b*c^5*d^4*x^5*log(c*x - 1) - 192*b*c^5*d^4*x^5*log(x) + 60*(a + 3*b)
*c^4*d^4*x^4 + 6*(20*a + 11*b)*c^3*d^4*x^3 + 20*(6*a + b)*c^2*d^4*x^2 + 3*(20*a + b)*c*d^4*x + 12*a*d^4 + 6*(5
*b*c^4*d^4*x^4 + 10*b*c^3*d^4*x^3 + 10*b*c^2*d^4*x^2 + 5*b*c*d^4*x + b*d^4)*log(-(c*x + 1)/(c*x - 1)))/x^5

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Sympy [A]  time = 5.92803, size = 253, normalized size = 2.32 \begin{align*} \begin{cases} - \frac{a c^{4} d^{4}}{x} - \frac{2 a c^{3} d^{4}}{x^{2}} - \frac{2 a c^{2} d^{4}}{x^{3}} - \frac{a c d^{4}}{x^{4}} - \frac{a d^{4}}{5 x^{5}} + \frac{16 b c^{5} d^{4} \log{\left (x \right )}}{5} - \frac{16 b c^{5} d^{4} \log{\left (x - \frac{1}{c} \right )}}{5} - \frac{b c^{5} d^{4} \operatorname{atanh}{\left (c x \right )}}{5} - \frac{b c^{4} d^{4} \operatorname{atanh}{\left (c x \right )}}{x} - \frac{3 b c^{4} d^{4}}{x} - \frac{2 b c^{3} d^{4} \operatorname{atanh}{\left (c x \right )}}{x^{2}} - \frac{11 b c^{3} d^{4}}{10 x^{2}} - \frac{2 b c^{2} d^{4} \operatorname{atanh}{\left (c x \right )}}{x^{3}} - \frac{b c^{2} d^{4}}{3 x^{3}} - \frac{b c d^{4} \operatorname{atanh}{\left (c x \right )}}{x^{4}} - \frac{b c d^{4}}{20 x^{4}} - \frac{b d^{4} \operatorname{atanh}{\left (c x \right )}}{5 x^{5}} & \text{for}\: c \neq 0 \\- \frac{a d^{4}}{5 x^{5}} & \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**6,x)

[Out]

Piecewise((-a*c**4*d**4/x - 2*a*c**3*d**4/x**2 - 2*a*c**2*d**4/x**3 - a*c*d**4/x**4 - a*d**4/(5*x**5) + 16*b*c
**5*d**4*log(x)/5 - 16*b*c**5*d**4*log(x - 1/c)/5 - b*c**5*d**4*atanh(c*x)/5 - b*c**4*d**4*atanh(c*x)/x - 3*b*
c**4*d**4/x - 2*b*c**3*d**4*atanh(c*x)/x**2 - 11*b*c**3*d**4/(10*x**2) - 2*b*c**2*d**4*atanh(c*x)/x**3 - b*c**
2*d**4/(3*x**3) - b*c*d**4*atanh(c*x)/x**4 - b*c*d**4/(20*x**4) - b*d**4*atanh(c*x)/(5*x**5), Ne(c, 0)), (-a*d
**4/(5*x**5), True))

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Giac [B]  time = 1.34178, size = 286, normalized size = 2.62 \begin{align*} -\frac{1}{10} \, b c^{5} d^{4} \log \left (c x + 1\right ) - \frac{31}{10} \, b c^{5} d^{4} \log \left (c x - 1\right ) + \frac{16}{5} \, b c^{5} d^{4} \log \left (x\right ) - \frac{{\left (5 \, b c^{4} d^{4} x^{4} + 10 \, b c^{3} d^{4} x^{3} + 10 \, b c^{2} d^{4} x^{2} + 5 \, b c d^{4} x + b d^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{10 \, x^{5}} - \frac{60 \, a c^{4} d^{4} x^{4} + 180 \, b c^{4} d^{4} x^{4} + 120 \, a c^{3} d^{4} x^{3} + 66 \, b c^{3} d^{4} x^{3} + 120 \, a c^{2} d^{4} x^{2} + 20 \, b c^{2} d^{4} x^{2} + 60 \, a c d^{4} x + 3 \, b c d^{4} x + 12 \, a d^{4}}{60 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

-1/10*b*c^5*d^4*log(c*x + 1) - 31/10*b*c^5*d^4*log(c*x - 1) + 16/5*b*c^5*d^4*log(x) - 1/10*(5*b*c^4*d^4*x^4 +
10*b*c^3*d^4*x^3 + 10*b*c^2*d^4*x^2 + 5*b*c*d^4*x + b*d^4)*log(-(c*x + 1)/(c*x - 1))/x^5 - 1/60*(60*a*c^4*d^4*
x^4 + 180*b*c^4*d^4*x^4 + 120*a*c^3*d^4*x^3 + 66*b*c^3*d^4*x^3 + 120*a*c^2*d^4*x^2 + 20*b*c^2*d^4*x^2 + 60*a*c
*d^4*x + 3*b*c*d^4*x + 12*a*d^4)/x^5