3.30 \(\int \frac{(d+c d x)^3 (a+b \tanh ^{-1}(c x))}{x^7} \, dx\)

Optimal. Leaf size=196 \[ -\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{7 b c^4 d^3}{15 x^2}-\frac{11 b c^3 d^3}{36 x^3}-\frac{3 b c^2 d^3}{20 x^4}-\frac{11 b c^5 d^3}{12 x}+\frac{14}{15} b c^6 d^3 \log (x)-\frac{37}{40} b c^6 d^3 \log (1-c x)-\frac{1}{120} b c^6 d^3 \log (c x+1)-\frac{b c d^3}{30 x^5} \]

[Out]

-(b*c*d^3)/(30*x^5) - (3*b*c^2*d^3)/(20*x^4) - (11*b*c^3*d^3)/(36*x^3) - (7*b*c^4*d^3)/(15*x^2) - (11*b*c^5*d^
3)/(12*x) - (d^3*(a + b*ArcTanh[c*x]))/(6*x^6) - (3*c*d^3*(a + b*ArcTanh[c*x]))/(5*x^5) - (3*c^2*d^3*(a + b*Ar
cTanh[c*x]))/(4*x^4) - (c^3*d^3*(a + b*ArcTanh[c*x]))/(3*x^3) + (14*b*c^6*d^3*Log[x])/15 - (37*b*c^6*d^3*Log[1
- c*x])/40 - (b*c^6*d^3*Log[1 + c*x])/120

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Rubi [A]  time = 0.177313, antiderivative size = 196, 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} \[ -\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{7 b c^4 d^3}{15 x^2}-\frac{11 b c^3 d^3}{36 x^3}-\frac{3 b c^2 d^3}{20 x^4}-\frac{11 b c^5 d^3}{12 x}+\frac{14}{15} b c^6 d^3 \log (x)-\frac{37}{40} b c^6 d^3 \log (1-c x)-\frac{1}{120} b c^6 d^3 \log (c x+1)-\frac{b c d^3}{30 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7,x]

[Out]

-(b*c*d^3)/(30*x^5) - (3*b*c^2*d^3)/(20*x^4) - (11*b*c^3*d^3)/(36*x^3) - (7*b*c^4*d^3)/(15*x^2) - (11*b*c^5*d^
3)/(12*x) - (d^3*(a + b*ArcTanh[c*x]))/(6*x^6) - (3*c*d^3*(a + b*ArcTanh[c*x]))/(5*x^5) - (3*c^2*d^3*(a + b*Ar
cTanh[c*x]))/(4*x^4) - (c^3*d^3*(a + b*ArcTanh[c*x]))/(3*x^3) + (14*b*c^6*d^3*Log[x])/15 - (37*b*c^6*d^3*Log[1
- c*x])/40 - (b*c^6*d^3*Log[1 + c*x])/120

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)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^7} \, dx &=-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{d^3 \left (-10-36 c x-45 c^2 x^2-20 c^3 x^3\right )}{60 x^6 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{1}{60} \left (b c d^3\right ) \int \frac{-10-36 c x-45 c^2 x^2-20 c^3 x^3}{x^6 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac{1}{60} \left (b c d^3\right ) \int \left (-\frac{10}{x^6}-\frac{36 c}{x^5}-\frac{55 c^2}{x^4}-\frac{56 c^3}{x^3}-\frac{55 c^4}{x^2}-\frac{56 c^5}{x}+\frac{111 c^6}{2 (-1+c x)}+\frac{c^6}{2 (1+c x)}\right ) \, dx\\ &=-\frac{b c d^3}{30 x^5}-\frac{3 b c^2 d^3}{20 x^4}-\frac{11 b c^3 d^3}{36 x^3}-\frac{7 b c^4 d^3}{15 x^2}-\frac{11 b c^5 d^3}{12 x}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac{14}{15} b c^6 d^3 \log (x)-\frac{37}{40} b c^6 d^3 \log (1-c x)-\frac{1}{120} b c^6 d^3 \log (1+c x)\\ \end{align*}

Mathematica [A]  time = 0.125461, size = 149, normalized size = 0.76 \[ -\frac{d^3 \left (120 a c^3 x^3+270 a c^2 x^2+216 a c x+60 a+330 b c^5 x^5+168 b c^4 x^4+110 b c^3 x^3+54 b c^2 x^2-336 b c^6 x^6 \log (x)+333 b c^6 x^6 \log (1-c x)+3 b c^6 x^6 \log (c x+1)+6 b \left (20 c^3 x^3+45 c^2 x^2+36 c x+10\right ) \tanh ^{-1}(c x)+12 b c x\right )}{360 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7,x]

[Out]

-(d^3*(60*a + 216*a*c*x + 12*b*c*x + 270*a*c^2*x^2 + 54*b*c^2*x^2 + 120*a*c^3*x^3 + 110*b*c^3*x^3 + 168*b*c^4*
x^4 + 330*b*c^5*x^5 + 6*b*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3)*ArcTanh[c*x] - 336*b*c^6*x^6*Log[x] + 333*b*
c^6*x^6*Log[1 - c*x] + 3*b*c^6*x^6*Log[1 + c*x]))/(360*x^6)

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Maple [A]  time = 0.039, size = 205, normalized size = 1.1 \begin{align*} -{\frac{3\,{c}^{2}{d}^{3}a}{4\,{x}^{4}}}-{\frac{3\,c{d}^{3}a}{5\,{x}^{5}}}-{\frac{{d}^{3}a}{6\,{x}^{6}}}-{\frac{{c}^{3}{d}^{3}a}{3\,{x}^{3}}}-{\frac{3\,{d}^{3}b{c}^{2}{\it Artanh} \left ( cx \right ) }{4\,{x}^{4}}}-{\frac{3\,c{d}^{3}b{\it Artanh} \left ( cx \right ) }{5\,{x}^{5}}}-{\frac{{d}^{3}b{\it Artanh} \left ( cx \right ) }{6\,{x}^{6}}}-{\frac{{c}^{3}{d}^{3}b{\it Artanh} \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{37\,{c}^{6}{d}^{3}b\ln \left ( cx-1 \right ) }{40}}-{\frac{c{d}^{3}b}{30\,{x}^{5}}}-{\frac{3\,{d}^{3}b{c}^{2}}{20\,{x}^{4}}}-{\frac{11\,{c}^{3}{d}^{3}b}{36\,{x}^{3}}}-{\frac{7\,b{c}^{4}{d}^{3}}{15\,{x}^{2}}}-{\frac{11\,b{c}^{5}{d}^{3}}{12\,x}}+{\frac{14\,{c}^{6}{d}^{3}b\ln \left ( cx \right ) }{15}}-{\frac{b{c}^{6}{d}^{3}\ln \left ( cx+1 \right ) }{120}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x)

[Out]

-3/4*c^2*d^3*a/x^4-3/5*c*d^3*a/x^5-1/6*d^3*a/x^6-1/3*c^3*d^3*a/x^3-3/4*c^2*d^3*b*arctanh(c*x)/x^4-3/5*c*d^3*b*
arctanh(c*x)/x^5-1/6*d^3*b*arctanh(c*x)/x^6-1/3*c^3*d^3*b*arctanh(c*x)/x^3-37/40*c^6*d^3*b*ln(c*x-1)-1/30*b*c*
d^3/x^5-3/20*b*c^2*d^3/x^4-11/36*b*c^3*d^3/x^3-7/15*b*c^4*d^3/x^2-11/12*b*c^5*d^3/x+14/15*c^6*d^3*b*ln(c*x)-1/
120*b*c^6*d^3*ln(c*x+1)

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Maxima [A]  time = 0.973585, size = 369, normalized size = 1.88 \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^{3} d^{3} + \frac{1}{8} \,{\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^{2} d^{3} - \frac{3}{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 c d^{3} + \frac{1}{180} \,{\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 d^{3} - \frac{a c^{3} d^{3}}{3 \, x^{3}} - \frac{3 \, a c^{2} d^{3}}{4 \, x^{4}} - \frac{3 \, a c d^{3}}{5 \, x^{5}} - \frac{a d^{3}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,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^3*d^3 + 1/8*((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^2*d^3 - 3/20*((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*c*d^3 + 1/180*((15*c^5*log(c*x + 1) - 1
5*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*d^3 - 1/3*a*c^3*d^3/x^3 -
3/4*a*c^2*d^3/x^4 - 3/5*a*c*d^3/x^5 - 1/6*a*d^3/x^6

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Fricas [A]  time = 2.09426, size = 446, normalized size = 2.28 \begin{align*} -\frac{3 \, b c^{6} d^{3} x^{6} \log \left (c x + 1\right ) + 333 \, b c^{6} d^{3} x^{6} \log \left (c x - 1\right ) - 336 \, b c^{6} d^{3} x^{6} \log \left (x\right ) + 330 \, b c^{5} d^{3} x^{5} + 168 \, b c^{4} d^{3} x^{4} + 10 \,{\left (12 \, a + 11 \, b\right )} c^{3} d^{3} x^{3} + 54 \,{\left (5 \, a + b\right )} c^{2} d^{3} x^{2} + 12 \,{\left (18 \, a + b\right )} c d^{3} x + 60 \, a d^{3} + 3 \,{\left (20 \, b c^{3} d^{3} x^{3} + 45 \, b c^{2} d^{3} x^{2} + 36 \, b c d^{3} x + 10 \, b d^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{360 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="fricas")

[Out]

-1/360*(3*b*c^6*d^3*x^6*log(c*x + 1) + 333*b*c^6*d^3*x^6*log(c*x - 1) - 336*b*c^6*d^3*x^6*log(x) + 330*b*c^5*d
^3*x^5 + 168*b*c^4*d^3*x^4 + 10*(12*a + 11*b)*c^3*d^3*x^3 + 54*(5*a + b)*c^2*d^3*x^2 + 12*(18*a + b)*c*d^3*x +
60*a*d^3 + 3*(20*b*c^3*d^3*x^3 + 45*b*c^2*d^3*x^2 + 36*b*c*d^3*x + 10*b*d^3)*log(-(c*x + 1)/(c*x - 1)))/x^6

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Sympy [A]  time = 8.38368, size = 257, normalized size = 1.31 \begin{align*} \begin{cases} - \frac{a c^{3} d^{3}}{3 x^{3}} - \frac{3 a c^{2} d^{3}}{4 x^{4}} - \frac{3 a c d^{3}}{5 x^{5}} - \frac{a d^{3}}{6 x^{6}} + \frac{14 b c^{6} d^{3} \log{\left (x \right )}}{15} - \frac{14 b c^{6} d^{3} \log{\left (x - \frac{1}{c} \right )}}{15} - \frac{b c^{6} d^{3} \operatorname{atanh}{\left (c x \right )}}{60} - \frac{11 b c^{5} d^{3}}{12 x} - \frac{7 b c^{4} d^{3}}{15 x^{2}} - \frac{b c^{3} d^{3} \operatorname{atanh}{\left (c x \right )}}{3 x^{3}} - \frac{11 b c^{3} d^{3}}{36 x^{3}} - \frac{3 b c^{2} d^{3} \operatorname{atanh}{\left (c x \right )}}{4 x^{4}} - \frac{3 b c^{2} d^{3}}{20 x^{4}} - \frac{3 b c d^{3} \operatorname{atanh}{\left (c x \right )}}{5 x^{5}} - \frac{b c d^{3}}{30 x^{5}} - \frac{b d^{3} \operatorname{atanh}{\left (c x \right )}}{6 x^{6}} & \text{for}\: c \neq 0 \\- \frac{a d^{3}}{6 x^{6}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)**3*(a+b*atanh(c*x))/x**7,x)

[Out]

Piecewise((-a*c**3*d**3/(3*x**3) - 3*a*c**2*d**3/(4*x**4) - 3*a*c*d**3/(5*x**5) - a*d**3/(6*x**6) + 14*b*c**6*
d**3*log(x)/15 - 14*b*c**6*d**3*log(x - 1/c)/15 - b*c**6*d**3*atanh(c*x)/60 - 11*b*c**5*d**3/(12*x) - 7*b*c**4
*d**3/(15*x**2) - b*c**3*d**3*atanh(c*x)/(3*x**3) - 11*b*c**3*d**3/(36*x**3) - 3*b*c**2*d**3*atanh(c*x)/(4*x**
4) - 3*b*c**2*d**3/(20*x**4) - 3*b*c*d**3*atanh(c*x)/(5*x**5) - b*c*d**3/(30*x**5) - b*d**3*atanh(c*x)/(6*x**6
), Ne(c, 0)), (-a*d**3/(6*x**6), True))

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Giac [A]  time = 1.76978, size = 271, normalized size = 1.38 \begin{align*} -\frac{1}{120} \, b c^{6} d^{3} \log \left (c x + 1\right ) - \frac{37}{40} \, b c^{6} d^{3} \log \left (c x - 1\right ) + \frac{14}{15} \, b c^{6} d^{3} \log \left (x\right ) - \frac{{\left (20 \, b c^{3} d^{3} x^{3} + 45 \, b c^{2} d^{3} x^{2} + 36 \, b c d^{3} x + 10 \, b d^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{120 \, x^{6}} - \frac{165 \, b c^{5} d^{3} x^{5} + 84 \, b c^{4} d^{3} x^{4} + 60 \, a c^{3} d^{3} x^{3} + 55 \, b c^{3} d^{3} x^{3} + 135 \, a c^{2} d^{3} x^{2} + 27 \, b c^{2} d^{3} x^{2} + 108 \, a c d^{3} x + 6 \, b c d^{3} x + 30 \, a d^{3}}{180 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="giac")

[Out]

-1/120*b*c^6*d^3*log(c*x + 1) - 37/40*b*c^6*d^3*log(c*x - 1) + 14/15*b*c^6*d^3*log(x) - 1/120*(20*b*c^3*d^3*x^
3 + 45*b*c^2*d^3*x^2 + 36*b*c*d^3*x + 10*b*d^3)*log(-(c*x + 1)/(c*x - 1))/x^6 - 1/180*(165*b*c^5*d^3*x^5 + 84*
b*c^4*d^3*x^4 + 60*a*c^3*d^3*x^3 + 55*b*c^3*d^3*x^3 + 135*a*c^2*d^3*x^2 + 27*b*c^2*d^3*x^2 + 108*a*c*d^3*x + 6
*b*c*d^3*x + 30*a*d^3)/x^6