3.21 \(\int x^2 (d+c d x)^3 (a+b \tanh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=178 \[ \frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{30} b c^2 d^3 x^5+\frac{11 b d^3 x}{12 c^2}+\frac{37 b d^3 \log (1-c x)}{40 c^3}+\frac{b d^3 \log (c x+1)}{120 c^3}+\frac{3}{20} b c d^3 x^4+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.175664, antiderivative size = 178, 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} \[ \frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{30} b c^2 d^3 x^5+\frac{11 b d^3 x}{12 c^2}+\frac{37 b d^3 \log (1-c x)}{40 c^3}+\frac{b d^3 \log (c x+1)}{120 c^3}+\frac{3}{20} b c d^3 x^4+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]

[Out]

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

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

Mathematica [A]  time = 0.125615, size = 142, normalized size = 0.8 \[ \frac{d^3 \left (60 a c^6 x^6+216 a c^5 x^5+270 a c^4 x^4+120 a c^3 x^3+12 b c^5 x^5+54 b c^4 x^4+110 b c^3 x^3+168 b c^2 x^2+6 b c^3 x^3 \left (10 c^3 x^3+36 c^2 x^2+45 c x+20\right ) \tanh ^{-1}(c x)+330 b c x+333 b \log (1-c x)+3 b \log (c x+1)\right )}{360 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d + c*d*x)^3*(a + b*ArcTanh[c*x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.031, size = 187, normalized size = 1.1 \begin{align*}{\frac{{c}^{3}{d}^{3}a{x}^{6}}{6}}+{\frac{3\,{c}^{2}{d}^{3}a{x}^{5}}{5}}+{\frac{3\,c{d}^{3}a{x}^{4}}{4}}+{\frac{{d}^{3}a{x}^{3}}{3}}+{\frac{{c}^{3}{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{6}}{6}}+{\frac{3\,{c}^{2}{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+{\frac{3\,c{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{4}}{4}}+{\frac{{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+{\frac{b{c}^{2}{d}^{3}{x}^{5}}{30}}+{\frac{3\,bc{d}^{3}{x}^{4}}{20}}+{\frac{11\,b{d}^{3}{x}^{3}}{36}}+{\frac{7\,{d}^{3}b{x}^{2}}{15\,c}}+{\frac{11\,b{d}^{3}x}{12\,{c}^{2}}}+{\frac{37\,{d}^{3}b\ln \left ( cx-1 \right ) }{40\,{c}^{3}}}+{\frac{{d}^{3}b\ln \left ( cx+1 \right ) }{120\,{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*d*x+d)^3*(a+b*arctanh(c*x)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.965809, size = 358, normalized size = 2.01 \begin{align*} \frac{1}{6} \, a c^{3} d^{3} x^{6} + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{3}{4} \, a c d^{3} x^{4} + \frac{1}{180} \,{\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^{3} 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^{2} d^{3} + \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{8} \,{\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 c d^{3} + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*d*x+d)^3*(a+b*arctanh(c*x)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.22022, size = 413, normalized size = 2.32 \begin{align*} \frac{60 \, a c^{6} d^{3} x^{6} + 12 \,{\left (18 \, a + b\right )} c^{5} d^{3} x^{5} + 54 \,{\left (5 \, a + b\right )} c^{4} d^{3} x^{4} + 10 \,{\left (12 \, a + 11 \, b\right )} c^{3} d^{3} x^{3} + 168 \, b c^{2} d^{3} x^{2} + 330 \, b c d^{3} x + 3 \, b d^{3} \log \left (c x + 1\right ) + 333 \, b d^{3} \log \left (c x - 1\right ) + 3 \,{\left (10 \, b c^{6} d^{3} x^{6} + 36 \, b c^{5} d^{3} x^{5} + 45 \, b c^{4} d^{3} x^{4} + 20 \, b c^{3} d^{3} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{360 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*d*x+d)^3*(a+b*arctanh(c*x)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 4.32664, size = 235, normalized size = 1.32 \begin{align*} \begin{cases} \frac{a c^{3} d^{3} x^{6}}{6} + \frac{3 a c^{2} d^{3} x^{5}}{5} + \frac{3 a c d^{3} x^{4}}{4} + \frac{a d^{3} x^{3}}{3} + \frac{b c^{3} d^{3} x^{6} \operatorname{atanh}{\left (c x \right )}}{6} + \frac{3 b c^{2} d^{3} x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{b c^{2} d^{3} x^{5}}{30} + \frac{3 b c d^{3} x^{4} \operatorname{atanh}{\left (c x \right )}}{4} + \frac{3 b c d^{3} x^{4}}{20} + \frac{b d^{3} x^{3} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{11 b d^{3} x^{3}}{36} + \frac{7 b d^{3} x^{2}}{15 c} + \frac{11 b d^{3} x}{12 c^{2}} + \frac{14 b d^{3} \log{\left (x - \frac{1}{c} \right )}}{15 c^{3}} + \frac{b d^{3} \operatorname{atanh}{\left (c x \right )}}{60 c^{3}} & \text{for}\: c \neq 0 \\\frac{a d^{3} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*d*x+d)**3*(a+b*atanh(c*x)),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.25597, size = 251, normalized size = 1.41 \begin{align*} \frac{1}{6} \, a c^{3} d^{3} x^{6} + \frac{1}{30} \,{\left (18 \, a c^{2} d^{3} + b c^{2} d^{3}\right )} x^{5} + \frac{7 \, b d^{3} x^{2}}{15 \, c} + \frac{3}{20} \,{\left (5 \, a c d^{3} + b c d^{3}\right )} x^{4} + \frac{1}{36} \,{\left (12 \, a d^{3} + 11 \, b d^{3}\right )} x^{3} + \frac{11 \, b d^{3} x}{12 \, c^{2}} + \frac{b d^{3} \log \left (c x + 1\right )}{120 \, c^{3}} + \frac{37 \, b d^{3} \log \left (c x - 1\right )}{40 \, c^{3}} + \frac{1}{120} \,{\left (10 \, b c^{3} d^{3} x^{6} + 36 \, b c^{2} d^{3} x^{5} + 45 \, b c d^{3} x^{4} + 20 \, b d^{3} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*d*x+d)^3*(a+b*arctanh(c*x)),x, algorithm="giac")

[Out]

1/6*a*c^3*d^3*x^6 + 1/30*(18*a*c^2*d^3 + b*c^2*d^3)*x^5 + 7/15*b*d^3*x^2/c + 3/20*(5*a*c*d^3 + b*c*d^3)*x^4 +
1/36*(12*a*d^3 + 11*b*d^3)*x^3 + 11/12*b*d^3*x/c^2 + 1/120*b*d^3*log(c*x + 1)/c^3 + 37/40*b*d^3*log(c*x - 1)/c
^3 + 1/120*(10*b*c^3*d^3*x^6 + 36*b*c^2*d^3*x^5 + 45*b*c*d^3*x^4 + 20*b*d^3*x^3)*log(-(c*x + 1)/(c*x - 1))