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

Optimal. Leaf size=192 \[ \frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{42} b c^2 d^3 x^6+\frac{13 b d^3 x^2}{35 c^2}+\frac{3 b d^3 x}{4 c^3}+\frac{209 b d^3 \log (1-c x)}{280 c^4}-\frac{b d^3 \log (c x+1)}{280 c^4}+\frac{1}{10} b c d^3 x^5+\frac{b d^3 x^3}{4 c}+\frac{13}{70} b d^3 x^4 \]

[Out]

(3*b*d^3*x)/(4*c^3) + (13*b*d^3*x^2)/(35*c^2) + (b*d^3*x^3)/(4*c) + (13*b*d^3*x^4)/70 + (b*c*d^3*x^5)/10 + (b*
c^2*d^3*x^6)/42 + (d^3*x^4*(a + b*ArcTanh[c*x]))/4 + (3*c*d^3*x^5*(a + b*ArcTanh[c*x]))/5 + (c^2*d^3*x^6*(a +
b*ArcTanh[c*x]))/2 + (c^3*d^3*x^7*(a + b*ArcTanh[c*x]))/7 + (209*b*d^3*Log[1 - c*x])/(280*c^4) - (b*d^3*Log[1
+ c*x])/(280*c^4)

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Rubi [A]  time = 0.18143, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {43, 5936, 12, 1802, 633, 31} \[ \frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{42} b c^2 d^3 x^6+\frac{13 b d^3 x^2}{35 c^2}+\frac{3 b d^3 x}{4 c^3}+\frac{209 b d^3 \log (1-c x)}{280 c^4}-\frac{b d^3 \log (c x+1)}{280 c^4}+\frac{1}{10} b c d^3 x^5+\frac{b d^3 x^3}{4 c}+\frac{13}{70} b d^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*d^3*x)/(4*c^3) + (13*b*d^3*x^2)/(35*c^2) + (b*d^3*x^3)/(4*c) + (13*b*d^3*x^4)/70 + (b*c*d^3*x^5)/10 + (b*
c^2*d^3*x^6)/42 + (d^3*x^4*(a + b*ArcTanh[c*x]))/4 + (3*c*d^3*x^5*(a + b*ArcTanh[c*x]))/5 + (c^2*d^3*x^6*(a +
b*ArcTanh[c*x]))/2 + (c^3*d^3*x^7*(a + b*ArcTanh[c*x]))/7 + (209*b*d^3*Log[1 - c*x])/(280*c^4) - (b*d^3*Log[1
+ c*x])/(280*c^4)

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]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int x^3 (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac{d^3 x^4 \left (35+84 c x+70 c^2 x^2+20 c^3 x^3\right )}{140 \left (1-c^2 x^2\right )} \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{140} \left (b c d^3\right ) \int \frac{x^4 \left (35+84 c x+70 c^2 x^2+20 c^3 x^3\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{140} \left (b c d^3\right ) \int \left (-\frac{105}{c^4}-\frac{104 x}{c^3}-\frac{105 x^2}{c^2}-\frac{104 x^3}{c}-70 x^4-20 c x^5+\frac{105+104 c x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{3 b d^3 x}{4 c^3}+\frac{13 b d^3 x^2}{35 c^2}+\frac{b d^3 x^3}{4 c}+\frac{13}{70} b d^3 x^4+\frac{1}{10} b c d^3 x^5+\frac{1}{42} b c^2 d^3 x^6+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )-\frac{\left (b d^3\right ) \int \frac{105+104 c x}{1-c^2 x^2} \, dx}{140 c^3}\\ &=\frac{3 b d^3 x}{4 c^3}+\frac{13 b d^3 x^2}{35 c^2}+\frac{b d^3 x^3}{4 c}+\frac{13}{70} b d^3 x^4+\frac{1}{10} b c d^3 x^5+\frac{1}{42} b c^2 d^3 x^6+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{\left (b d^3\right ) \int \frac{1}{-c-c^2 x} \, dx}{280 c^2}-\frac{\left (209 b d^3\right ) \int \frac{1}{c-c^2 x} \, dx}{280 c^2}\\ &=\frac{3 b d^3 x}{4 c^3}+\frac{13 b d^3 x^2}{35 c^2}+\frac{b d^3 x^3}{4 c}+\frac{13}{70} b d^3 x^4+\frac{1}{10} b c d^3 x^5+\frac{1}{42} b c^2 d^3 x^6+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{209 b d^3 \log (1-c x)}{280 c^4}-\frac{b d^3 \log (1+c x)}{280 c^4}\\ \end{align*}

Mathematica [A]  time = 0.133598, size = 151, normalized size = 0.79 \[ \frac{d^3 \left (120 a c^7 x^7+420 a c^6 x^6+504 a c^5 x^5+210 a c^4 x^4+20 b c^6 x^6+84 b c^5 x^5+156 b c^4 x^4+210 b c^3 x^3+312 b c^2 x^2+6 b c^4 x^4 \left (20 c^3 x^3+70 c^2 x^2+84 c x+35\right ) \tanh ^{-1}(c x)+630 b c x+627 b \log (1-c x)-3 b \log (c x+1)\right )}{840 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(630*b*c*x + 312*b*c^2*x^2 + 210*b*c^3*x^3 + 210*a*c^4*x^4 + 156*b*c^4*x^4 + 504*a*c^5*x^5 + 84*b*c^5*x^5
+ 420*a*c^6*x^6 + 20*b*c^6*x^6 + 120*a*c^7*x^7 + 6*b*c^4*x^4*(35 + 84*c*x + 70*c^2*x^2 + 20*c^3*x^3)*ArcTanh[
c*x] + 627*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(840*c^4)

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Maple [A]  time = 0.028, size = 199, normalized size = 1. \begin{align*}{\frac{{c}^{3}{d}^{3}a{x}^{7}}{7}}+{\frac{{c}^{2}{d}^{3}a{x}^{6}}{2}}+{\frac{3\,c{d}^{3}a{x}^{5}}{5}}+{\frac{{d}^{3}a{x}^{4}}{4}}+{\frac{{c}^{3}{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{7}}{7}}+{\frac{{c}^{2}{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{6}}{2}}+{\frac{3\,c{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+{\frac{{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{4}}{4}}+{\frac{b{c}^{2}{d}^{3}{x}^{6}}{42}}+{\frac{bc{d}^{3}{x}^{5}}{10}}+{\frac{13\,b{d}^{3}{x}^{4}}{70}}+{\frac{b{d}^{3}{x}^{3}}{4\,c}}+{\frac{13\,{d}^{3}b{x}^{2}}{35\,{c}^{2}}}+{\frac{3\,b{d}^{3}x}{4\,{c}^{3}}}+{\frac{209\,{d}^{3}b\ln \left ( cx-1 \right ) }{280\,{c}^{4}}}-{\frac{{d}^{3}b\ln \left ( cx+1 \right ) }{280\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^3*d^3*a*x^7+1/2*c^2*d^3*a*x^6+3/5*c*d^3*a*x^5+1/4*d^3*a*x^4+1/7*c^3*d^3*b*arctanh(c*x)*x^7+1/2*c^2*d^3*b
*arctanh(c*x)*x^6+3/5*c*d^3*b*arctanh(c*x)*x^5+1/4*d^3*b*arctanh(c*x)*x^4+1/42*b*c^2*d^3*x^6+1/10*b*c*d^3*x^5+
13/70*b*d^3*x^4+1/4*b*d^3*x^3/c+13/35*b*d^3*x^2/c^2+3/4*b*d^3*x/c^3+209/280/c^4*d^3*b*ln(c*x-1)-1/280*b*d^3*ln
(c*x+1)/c^4

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Maxima [A]  time = 0.973141, size = 385, normalized size = 2.01 \begin{align*} \frac{1}{7} \, a c^{3} d^{3} x^{7} + \frac{1}{2} \, a c^{2} d^{3} x^{6} + \frac{3}{5} \, a c d^{3} x^{5} + \frac{1}{84} \,{\left (12 \, x^{7} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac{6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{3} + \frac{1}{4} \, a d^{3} x^{4} + \frac{1}{60} \,{\left (30 \, x^{6} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{2} d^{3} + \frac{3}{20} \,{\left (4 \, x^{5} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d^{3} + \frac{1}{24} \,{\left (6 \, x^{4} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac{3 \, \log \left (c x + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*c^3*d^3*x^7 + 1/2*a*c^2*d^3*x^6 + 3/5*a*c*d^3*x^5 + 1/84*(12*x^7*arctanh(c*x) + c*((2*c^4*x^6 + 3*c^2*x^
4 + 6*x^2)/c^6 + 6*log(c^2*x^2 - 1)/c^8))*b*c^3*d^3 + 1/4*a*d^3*x^4 + 1/60*(30*x^6*arctanh(c*x) + c*(2*(3*c^4*
x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*b*c^2*d^3 + 3/20*(4*x^5*arctanh(c*x)
+ c*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*b*c*d^3 + 1/24*(6*x^4*arctanh(c*x) + c*(2*(c^2*x^3 + 3*
x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*b*d^3

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Fricas [A]  time = 2.30813, size = 440, normalized size = 2.29 \begin{align*} \frac{120 \, a c^{7} d^{3} x^{7} + 20 \,{\left (21 \, a + b\right )} c^{6} d^{3} x^{6} + 84 \,{\left (6 \, a + b\right )} c^{5} d^{3} x^{5} + 6 \,{\left (35 \, a + 26 \, b\right )} c^{4} d^{3} x^{4} + 210 \, b c^{3} d^{3} x^{3} + 312 \, b c^{2} d^{3} x^{2} + 630 \, b c d^{3} x - 3 \, b d^{3} \log \left (c x + 1\right ) + 627 \, b d^{3} \log \left (c x - 1\right ) + 3 \,{\left (20 \, b c^{7} d^{3} x^{7} + 70 \, b c^{6} d^{3} x^{6} + 84 \, b c^{5} d^{3} x^{5} + 35 \, b c^{4} d^{3} x^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{840 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(120*a*c^7*d^3*x^7 + 20*(21*a + b)*c^6*d^3*x^6 + 84*(6*a + b)*c^5*d^3*x^5 + 6*(35*a + 26*b)*c^4*d^3*x^4
+ 210*b*c^3*d^3*x^3 + 312*b*c^2*d^3*x^2 + 630*b*c*d^3*x - 3*b*d^3*log(c*x + 1) + 627*b*d^3*log(c*x - 1) + 3*(2
0*b*c^7*d^3*x^7 + 70*b*c^6*d^3*x^6 + 84*b*c^5*d^3*x^5 + 35*b*c^4*d^3*x^4)*log(-(c*x + 1)/(c*x - 1)))/c^4

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Sympy [A]  time = 6.68272, size = 243, normalized size = 1.27 \begin{align*} \begin{cases} \frac{a c^{3} d^{3} x^{7}}{7} + \frac{a c^{2} d^{3} x^{6}}{2} + \frac{3 a c d^{3} x^{5}}{5} + \frac{a d^{3} x^{4}}{4} + \frac{b c^{3} d^{3} x^{7} \operatorname{atanh}{\left (c x \right )}}{7} + \frac{b c^{2} d^{3} x^{6} \operatorname{atanh}{\left (c x \right )}}{2} + \frac{b c^{2} d^{3} x^{6}}{42} + \frac{3 b c d^{3} x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{b c d^{3} x^{5}}{10} + \frac{b d^{3} x^{4} \operatorname{atanh}{\left (c x \right )}}{4} + \frac{13 b d^{3} x^{4}}{70} + \frac{b d^{3} x^{3}}{4 c} + \frac{13 b d^{3} x^{2}}{35 c^{2}} + \frac{3 b d^{3} x}{4 c^{3}} + \frac{26 b d^{3} \log{\left (x - \frac{1}{c} \right )}}{35 c^{4}} - \frac{b d^{3} \operatorname{atanh}{\left (c x \right )}}{140 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d^{3} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**3*d**3*x**7/7 + a*c**2*d**3*x**6/2 + 3*a*c*d**3*x**5/5 + a*d**3*x**4/4 + b*c**3*d**3*x**7*atan
h(c*x)/7 + b*c**2*d**3*x**6*atanh(c*x)/2 + b*c**2*d**3*x**6/42 + 3*b*c*d**3*x**5*atanh(c*x)/5 + b*c*d**3*x**5/
10 + b*d**3*x**4*atanh(c*x)/4 + 13*b*d**3*x**4/70 + b*d**3*x**3/(4*c) + 13*b*d**3*x**2/(35*c**2) + 3*b*d**3*x/
(4*c**3) + 26*b*d**3*log(x - 1/c)/(35*c**4) - b*d**3*atanh(c*x)/(140*c**4), Ne(c, 0)), (a*d**3*x**4/4, True))

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Giac [A]  time = 1.33519, size = 267, normalized size = 1.39 \begin{align*} \frac{1}{7} \, a c^{3} d^{3} x^{7} + \frac{1}{42} \,{\left (21 \, a c^{2} d^{3} + b c^{2} d^{3}\right )} x^{6} + \frac{b d^{3} x^{3}}{4 \, c} + \frac{1}{10} \,{\left (6 \, a c d^{3} + b c d^{3}\right )} x^{5} + \frac{1}{140} \,{\left (35 \, a d^{3} + 26 \, b d^{3}\right )} x^{4} + \frac{13 \, b d^{3} x^{2}}{35 \, c^{2}} + \frac{3 \, b d^{3} x}{4 \, c^{3}} + \frac{1}{280} \,{\left (20 \, b c^{3} d^{3} x^{7} + 70 \, b c^{2} d^{3} x^{6} + 84 \, b c d^{3} x^{5} + 35 \, b d^{3} x^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) - \frac{b d^{3} \log \left (c x + 1\right )}{280 \, c^{4}} + \frac{209 \, b d^{3} \log \left (c x - 1\right )}{280 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*c^3*d^3*x^7 + 1/42*(21*a*c^2*d^3 + b*c^2*d^3)*x^6 + 1/4*b*d^3*x^3/c + 1/10*(6*a*c*d^3 + b*c*d^3)*x^5 + 1
/140*(35*a*d^3 + 26*b*d^3)*x^4 + 13/35*b*d^3*x^2/c^2 + 3/4*b*d^3*x/c^3 + 1/280*(20*b*c^3*d^3*x^7 + 70*b*c^2*d^
3*x^6 + 84*b*c*d^3*x^5 + 35*b*d^3*x^4)*log(-(c*x + 1)/(c*x - 1)) - 1/280*b*d^3*log(c*x + 1)/c^4 + 209/280*b*d^
3*log(c*x - 1)/c^4