3.41 \(\int \frac{(d+c d x)^4 (a+b \tanh ^{-1}(c x))}{x^7} \, dx\)

Optimal. Leaf size=151 \[ \frac{c d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}-\frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{16 b c^4 d^4}{15 x^2}-\frac{5 b c^3 d^4}{9 x^3}-\frac{b c^2 d^4}{5 x^4}-\frac{13 b c^5 d^4}{6 x}+\frac{32}{15} b c^6 d^4 \log (x)-\frac{32}{15} b c^6 d^4 \log (1-c x)-\frac{b c d^4}{30 x^5} \]

[Out]

-(b*c*d^4)/(30*x^5) - (b*c^2*d^4)/(5*x^4) - (5*b*c^3*d^4)/(9*x^3) - (16*b*c^4*d^4)/(15*x^2) - (13*b*c^5*d^4)/(
6*x) - (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(6*x^6) + (c*d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(30*x^5) + (3
2*b*c^6*d^4*Log[x])/15 - (32*b*c^6*d^4*Log[1 - c*x])/15

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Rubi [A]  time = 0.125417, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {45, 37, 5936, 12, 148} \[ \frac{c d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}-\frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{16 b c^4 d^4}{15 x^2}-\frac{5 b c^3 d^4}{9 x^3}-\frac{b c^2 d^4}{5 x^4}-\frac{13 b c^5 d^4}{6 x}+\frac{32}{15} b c^6 d^4 \log (x)-\frac{32}{15} b c^6 d^4 \log (1-c x)-\frac{b c d^4}{30 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^7,x]

[Out]

-(b*c*d^4)/(30*x^5) - (b*c^2*d^4)/(5*x^4) - (5*b*c^3*d^4)/(9*x^3) - (16*b*c^4*d^4)/(15*x^2) - (13*b*c^5*d^4)/(
6*x) - (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(6*x^6) + (c*d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(30*x^5) + (3
2*b*c^6*d^4*Log[x])/15 - (32*b*c^6*d^4*Log[1 - c*x])/15

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 148

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g
, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))

Rubi steps

\begin{align*} \int \frac{(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^7} \, dx &=-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}+\frac{c d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}-(b c) \int \frac{(-5+c x) (d+c d x)^4}{30 x^6 (1-c x)} \, dx\\ &=-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}+\frac{c d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}-\frac{1}{30} (b c) \int \frac{(-5+c x) (d+c d x)^4}{x^6 (1-c x)} \, dx\\ &=-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}+\frac{c d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}-\frac{1}{30} (b c) \int \left (-\frac{5 d^4}{x^6}-\frac{24 c d^4}{x^5}-\frac{50 c^2 d^4}{x^4}-\frac{64 c^3 d^4}{x^3}-\frac{65 c^4 d^4}{x^2}-\frac{64 c^5 d^4}{x}+\frac{64 c^6 d^4}{-1+c x}\right ) \, dx\\ &=-\frac{b c d^4}{30 x^5}-\frac{b c^2 d^4}{5 x^4}-\frac{5 b c^3 d^4}{9 x^3}-\frac{16 b c^4 d^4}{15 x^2}-\frac{13 b c^5 d^4}{6 x}-\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}+\frac{c d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}+\frac{32}{15} b c^6 d^4 \log (x)-\frac{32}{15} b c^6 d^4 \log (1-c x)\\ \end{align*}

Mathematica [A]  time = 0.154882, size = 166, normalized size = 1.1 \[ -\frac{d^4 \left (90 a c^4 x^4+240 a c^3 x^3+270 a c^2 x^2+144 a c x+30 a+390 b c^5 x^5+192 b c^4 x^4+100 b c^3 x^3+36 b c^2 x^2-384 b c^6 x^6 \log (x)+387 b c^6 x^6 \log (1-c x)-3 b c^6 x^6 \log (c x+1)+6 b \left (15 c^4 x^4+40 c^3 x^3+45 c^2 x^2+24 c x+5\right ) \tanh ^{-1}(c x)+6 b c x\right )}{180 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^7,x]

[Out]

-(d^4*(30*a + 144*a*c*x + 6*b*c*x + 270*a*c^2*x^2 + 36*b*c^2*x^2 + 240*a*c^3*x^3 + 100*b*c^3*x^3 + 90*a*c^4*x^
4 + 192*b*c^4*x^4 + 390*b*c^5*x^5 + 6*b*(5 + 24*c*x + 45*c^2*x^2 + 40*c^3*x^3 + 15*c^4*x^4)*ArcTanh[c*x] - 384
*b*c^6*x^6*Log[x] + 387*b*c^6*x^6*Log[1 - c*x] - 3*b*c^6*x^6*Log[1 + c*x]))/(180*x^6)

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Maple [A]  time = 0.04, size = 233, normalized size = 1.5 \begin{align*} -{\frac{3\,{c}^{2}{d}^{4}a}{2\,{x}^{4}}}-{\frac{4\,c{d}^{4}a}{5\,{x}^{5}}}-{\frac{{c}^{4}{d}^{4}a}{2\,{x}^{2}}}-{\frac{{d}^{4}a}{6\,{x}^{6}}}-{\frac{4\,{c}^{3}{d}^{4}a}{3\,{x}^{3}}}-{\frac{3\,{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ) }{2\,{x}^{4}}}-{\frac{4\,c{d}^{4}b{\it Artanh} \left ( cx \right ) }{5\,{x}^{5}}}-{\frac{{c}^{4}{d}^{4}b{\it Artanh} \left ( cx \right ) }{2\,{x}^{2}}}-{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ) }{6\,{x}^{6}}}-{\frac{4\,{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{43\,{c}^{6}{d}^{4}b\ln \left ( cx-1 \right ) }{20}}-{\frac{c{d}^{4}b}{30\,{x}^{5}}}-{\frac{{c}^{2}{d}^{4}b}{5\,{x}^{4}}}-{\frac{5\,{c}^{3}{d}^{4}b}{9\,{x}^{3}}}-{\frac{16\,{c}^{4}{d}^{4}b}{15\,{x}^{2}}}-{\frac{13\,b{c}^{5}{d}^{4}}{6\,x}}+{\frac{32\,{c}^{6}{d}^{4}b\ln \left ( cx \right ) }{15}}+{\frac{{c}^{6}{d}^{4}b\ln \left ( cx+1 \right ) }{60}} \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^7,x)

[Out]

-3/2*c^2*d^4*a/x^4-4/5*c*d^4*a/x^5-1/2*c^4*d^4*a/x^2-1/6*d^4*a/x^6-4/3*c^3*d^4*a/x^3-3/2*c^2*d^4*b*arctanh(c*x
)/x^4-4/5*c*d^4*b*arctanh(c*x)/x^5-1/2*c^4*d^4*b*arctanh(c*x)/x^2-1/6*d^4*b*arctanh(c*x)/x^6-4/3*c^3*d^4*b*arc
tanh(c*x)/x^3-43/20*c^6*d^4*b*ln(c*x-1)-1/30*b*c*d^4/x^5-1/5*b*c^2*d^4/x^4-5/9*b*c^3*d^4/x^3-16/15*b*c^4*d^4/x
^2-13/6*b*c^5*d^4/x+32/15*c^6*d^4*b*ln(c*x)+1/60*c^6*d^4*b*ln(c*x+1)

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Maxima [B]  time = 0.9885, size = 444, normalized size = 2.94 \begin{align*} \frac{1}{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^{4} d^{4} - \frac{2}{3} \,{\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^{4} + \frac{1}{4} \,{\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^{4} - \frac{1}{5} \,{\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^{4} - \frac{a c^{4} d^{4}}{2 \, x^{2}} + \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^{4} - \frac{4 \, a c^{3} d^{4}}{3 \, x^{3}} - \frac{3 \, a c^{2} d^{4}}{2 \, x^{4}} - \frac{4 \, a c d^{4}}{5 \, x^{5}} - \frac{a d^{4}}{6 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

1/4*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*b*c^4*d^4 - 2/3*((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^4 + 1/4*((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^4 - 1/5*((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^4 - 1/2*a*c^4*d^4/x^2 + 1/180*((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*d^4 - 4/3*a*c^3*d^4/x^3 - 3/2*a*c^2*d
^4/x^4 - 4/5*a*c*d^4/x^5 - 1/6*a*d^4/x^6

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Fricas [A]  time = 2.20086, size = 483, normalized size = 3.2 \begin{align*} \frac{3 \, b c^{6} d^{4} x^{6} \log \left (c x + 1\right ) - 387 \, b c^{6} d^{4} x^{6} \log \left (c x - 1\right ) + 384 \, b c^{6} d^{4} x^{6} \log \left (x\right ) - 390 \, b c^{5} d^{4} x^{5} - 6 \,{\left (15 \, a + 32 \, b\right )} c^{4} d^{4} x^{4} - 20 \,{\left (12 \, a + 5 \, b\right )} c^{3} d^{4} x^{3} - 18 \,{\left (15 \, a + 2 \, b\right )} c^{2} d^{4} x^{2} - 6 \,{\left (24 \, a + b\right )} c d^{4} x - 30 \, a d^{4} - 3 \,{\left (15 \, b c^{4} d^{4} x^{4} + 40 \, b c^{3} d^{4} x^{3} + 45 \, b c^{2} d^{4} x^{2} + 24 \, b c d^{4} x + 5 \, b d^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{180 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

1/180*(3*b*c^6*d^4*x^6*log(c*x + 1) - 387*b*c^6*d^4*x^6*log(c*x - 1) + 384*b*c^6*d^4*x^6*log(x) - 390*b*c^5*d^
4*x^5 - 6*(15*a + 32*b)*c^4*d^4*x^4 - 20*(12*a + 5*b)*c^3*d^4*x^3 - 18*(15*a + 2*b)*c^2*d^4*x^2 - 6*(24*a + b)
*c*d^4*x - 30*a*d^4 - 3*(15*b*c^4*d^4*x^4 + 40*b*c^3*d^4*x^3 + 45*b*c^2*d^4*x^2 + 24*b*c*d^4*x + 5*b*d^4)*log(
-(c*x + 1)/(c*x - 1)))/x^6

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Sympy [A]  time = 6.81763, size = 291, normalized size = 1.93 \begin{align*} \begin{cases} - \frac{a c^{4} d^{4}}{2 x^{2}} - \frac{4 a c^{3} d^{4}}{3 x^{3}} - \frac{3 a c^{2} d^{4}}{2 x^{4}} - \frac{4 a c d^{4}}{5 x^{5}} - \frac{a d^{4}}{6 x^{6}} + \frac{32 b c^{6} d^{4} \log{\left (x \right )}}{15} - \frac{32 b c^{6} d^{4} \log{\left (x - \frac{1}{c} \right )}}{15} + \frac{b c^{6} d^{4} \operatorname{atanh}{\left (c x \right )}}{30} - \frac{13 b c^{5} d^{4}}{6 x} - \frac{b c^{4} d^{4} \operatorname{atanh}{\left (c x \right )}}{2 x^{2}} - \frac{16 b c^{4} d^{4}}{15 x^{2}} - \frac{4 b c^{3} d^{4} \operatorname{atanh}{\left (c x \right )}}{3 x^{3}} - \frac{5 b c^{3} d^{4}}{9 x^{3}} - \frac{3 b c^{2} d^{4} \operatorname{atanh}{\left (c x \right )}}{2 x^{4}} - \frac{b c^{2} d^{4}}{5 x^{4}} - \frac{4 b c d^{4} \operatorname{atanh}{\left (c x \right )}}{5 x^{5}} - \frac{b c d^{4}}{30 x^{5}} - \frac{b d^{4} \operatorname{atanh}{\left (c x \right )}}{6 x^{6}} & \text{for}\: c \neq 0 \\- \frac{a d^{4}}{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)**4*(a+b*atanh(c*x))/x**7,x)

[Out]

Piecewise((-a*c**4*d**4/(2*x**2) - 4*a*c**3*d**4/(3*x**3) - 3*a*c**2*d**4/(2*x**4) - 4*a*c*d**4/(5*x**5) - a*d
**4/(6*x**6) + 32*b*c**6*d**4*log(x)/15 - 32*b*c**6*d**4*log(x - 1/c)/15 + b*c**6*d**4*atanh(c*x)/30 - 13*b*c*
*5*d**4/(6*x) - b*c**4*d**4*atanh(c*x)/(2*x**2) - 16*b*c**4*d**4/(15*x**2) - 4*b*c**3*d**4*atanh(c*x)/(3*x**3)
- 5*b*c**3*d**4/(9*x**3) - 3*b*c**2*d**4*atanh(c*x)/(2*x**4) - b*c**2*d**4/(5*x**4) - 4*b*c*d**4*atanh(c*x)/(
5*x**5) - b*c*d**4/(30*x**5) - b*d**4*atanh(c*x)/(6*x**6), Ne(c, 0)), (-a*d**4/(6*x**6), True))

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Giac [A]  time = 1.63756, size = 304, normalized size = 2.01 \begin{align*} \frac{1}{60} \, b c^{6} d^{4} \log \left (c x + 1\right ) - \frac{43}{20} \, b c^{6} d^{4} \log \left (c x - 1\right ) + \frac{32}{15} \, b c^{6} d^{4} \log \left (x\right ) - \frac{{\left (15 \, b c^{4} d^{4} x^{4} + 40 \, b c^{3} d^{4} x^{3} + 45 \, b c^{2} d^{4} x^{2} + 24 \, b c d^{4} x + 5 \, b d^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{60 \, x^{6}} - \frac{195 \, b c^{5} d^{4} x^{5} + 45 \, a c^{4} d^{4} x^{4} + 96 \, b c^{4} d^{4} x^{4} + 120 \, a c^{3} d^{4} x^{3} + 50 \, b c^{3} d^{4} x^{3} + 135 \, a c^{2} d^{4} x^{2} + 18 \, b c^{2} d^{4} x^{2} + 72 \, a c d^{4} x + 3 \, b c d^{4} x + 15 \, a d^{4}}{90 \, x^{6}} \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^7,x, algorithm="giac")

[Out]

1/60*b*c^6*d^4*log(c*x + 1) - 43/20*b*c^6*d^4*log(c*x - 1) + 32/15*b*c^6*d^4*log(x) - 1/60*(15*b*c^4*d^4*x^4 +
40*b*c^3*d^4*x^3 + 45*b*c^2*d^4*x^2 + 24*b*c*d^4*x + 5*b*d^4)*log(-(c*x + 1)/(c*x - 1))/x^6 - 1/90*(195*b*c^5
*d^4*x^5 + 45*a*c^4*d^4*x^4 + 96*b*c^4*d^4*x^4 + 120*a*c^3*d^4*x^3 + 50*b*c^3*d^4*x^3 + 135*a*c^2*d^4*x^2 + 18
*b*c^2*d^4*x^2 + 72*a*c*d^4*x + 3*b*c*d^4*x + 15*a*d^4)/x^6