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

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

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

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

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 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 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 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]))

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*}

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Mathematica [A]  time = 0.14, 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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fricas [A]  time = 0.44, size = 190, normalized size = 0.99 \[ \frac {120 \, a c^{7} d^{3} x^{7} + 20 \, {\left (21 \, a + b\right )} c^{6} d^{3} x^{6} + 84 \, {\left (6 \, a + b\right )} c^{5} d^{3} x^{5} + 6 \, {\left (35 \, a + 26 \, b\right )} c^{4} d^{3} x^{4} + 210 \, b c^{3} d^{3} x^{3} + 312 \, b c^{2} d^{3} x^{2} + 630 \, b c d^{3} x - 3 \, b d^{3} \log \left (c x + 1\right ) + 627 \, b d^{3} \log \left (c x - 1\right ) + 3 \, {\left (20 \, b c^{7} d^{3} x^{7} + 70 \, b c^{6} d^{3} x^{6} + 84 \, b c^{5} d^{3} x^{5} + 35 \, b c^{4} d^{3} x^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{840 \, c^{4}} \]

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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giac [B]  time = 0.27, size = 722, normalized size = 3.76 \[ \frac {1}{105} \, c {\left (\frac {6 \, {\left (\frac {140 \, {\left (c x + 1\right )}^{6} b d^{3}}{{\left (c x - 1\right )}^{6}} - \frac {210 \, {\left (c x + 1\right )}^{5} b d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {490 \, {\left (c x + 1\right )}^{4} b d^{3}}{{\left (c x - 1\right )}^{4}} - \frac {455 \, {\left (c x + 1\right )}^{3} b d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {273 \, {\left (c x + 1\right )}^{2} b d^{3}}{{\left (c x - 1\right )}^{2}} - \frac {91 \, {\left (c x + 1\right )} b d^{3}}{c x - 1} + 13 \, b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{7} c^{5}}{{\left (c x - 1\right )}^{7}} - \frac {7 \, {\left (c x + 1\right )}^{6} c^{5}}{{\left (c x - 1\right )}^{6}} + \frac {21 \, {\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} - \frac {35 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} + \frac {35 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} - \frac {21 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )} c^{5}}{c x - 1} - c^{5}} + \frac {\frac {1680 \, {\left (c x + 1\right )}^{6} a d^{3}}{{\left (c x - 1\right )}^{6}} - \frac {2520 \, {\left (c x + 1\right )}^{5} a d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {5880 \, {\left (c x + 1\right )}^{4} a d^{3}}{{\left (c x - 1\right )}^{4}} - \frac {5460 \, {\left (c x + 1\right )}^{3} a d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {3276 \, {\left (c x + 1\right )}^{2} a d^{3}}{{\left (c x - 1\right )}^{2}} - \frac {1092 \, {\left (c x + 1\right )} a d^{3}}{c x - 1} + 156 \, a d^{3} + \frac {762 \, {\left (c x + 1\right )}^{6} b d^{3}}{{\left (c x - 1\right )}^{6}} - \frac {3063 \, {\left (c x + 1\right )}^{5} b d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {5959 \, {\left (c x + 1\right )}^{4} b d^{3}}{{\left (c x - 1\right )}^{4}} - \frac {6694 \, {\left (c x + 1\right )}^{3} b d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {4344 \, {\left (c x + 1\right )}^{2} b d^{3}}{{\left (c x - 1\right )}^{2}} - \frac {1539 \, {\left (c x + 1\right )} b d^{3}}{c x - 1} + 231 \, b d^{3}}{\frac {{\left (c x + 1\right )}^{7} c^{5}}{{\left (c x - 1\right )}^{7}} - \frac {7 \, {\left (c x + 1\right )}^{6} c^{5}}{{\left (c x - 1\right )}^{6}} + \frac {21 \, {\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} - \frac {35 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} + \frac {35 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} - \frac {21 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )} c^{5}}{c x - 1} - c^{5}} - \frac {78 \, b d^{3} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{5}} + \frac {78 \, b d^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{5}}\right )} \]

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]

1/105*c*(6*(140*(c*x + 1)^6*b*d^3/(c*x - 1)^6 - 210*(c*x + 1)^5*b*d^3/(c*x - 1)^5 + 490*(c*x + 1)^4*b*d^3/(c*x
- 1)^4 - 455*(c*x + 1)^3*b*d^3/(c*x - 1)^3 + 273*(c*x + 1)^2*b*d^3/(c*x - 1)^2 - 91*(c*x + 1)*b*d^3/(c*x - 1)
+ 13*b*d^3)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^7*c^5/(c*x - 1)^7 - 7*(c*x + 1)^6*c^5/(c*x - 1)^6 + 21*(c*x
+ 1)^5*c^5/(c*x - 1)^5 - 35*(c*x + 1)^4*c^5/(c*x - 1)^4 + 35*(c*x + 1)^3*c^5/(c*x - 1)^3 - 21*(c*x + 1)^2*c^5/
(c*x - 1)^2 + 7*(c*x + 1)*c^5/(c*x - 1) - c^5) + (1680*(c*x + 1)^6*a*d^3/(c*x - 1)^6 - 2520*(c*x + 1)^5*a*d^3/
(c*x - 1)^5 + 5880*(c*x + 1)^4*a*d^3/(c*x - 1)^4 - 5460*(c*x + 1)^3*a*d^3/(c*x - 1)^3 + 3276*(c*x + 1)^2*a*d^3
/(c*x - 1)^2 - 1092*(c*x + 1)*a*d^3/(c*x - 1) + 156*a*d^3 + 762*(c*x + 1)^6*b*d^3/(c*x - 1)^6 - 3063*(c*x + 1)
^5*b*d^3/(c*x - 1)^5 + 5959*(c*x + 1)^4*b*d^3/(c*x - 1)^4 - 6694*(c*x + 1)^3*b*d^3/(c*x - 1)^3 + 4344*(c*x + 1
)^2*b*d^3/(c*x - 1)^2 - 1539*(c*x + 1)*b*d^3/(c*x - 1) + 231*b*d^3)/((c*x + 1)^7*c^5/(c*x - 1)^7 - 7*(c*x + 1)
^6*c^5/(c*x - 1)^6 + 21*(c*x + 1)^5*c^5/(c*x - 1)^5 - 35*(c*x + 1)^4*c^5/(c*x - 1)^4 + 35*(c*x + 1)^3*c^5/(c*x
- 1)^3 - 21*(c*x + 1)^2*c^5/(c*x - 1)^2 + 7*(c*x + 1)*c^5/(c*x - 1) - c^5) - 78*b*d^3*log(-(c*x + 1)/(c*x - 1
) + 1)/c^5 + 78*b*d^3*log(-(c*x + 1)/(c*x - 1))/c^5)

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maple [A]  time = 0.03, size = 199, normalized size = 1.04 \[ \frac {c^{3} d^{3} a \,x^{7}}{7}+\frac {c^{2} d^{3} a \,x^{6}}{2}+\frac {3 c \,d^{3} a \,x^{5}}{5}+\frac {d^{3} a \,x^{4}}{4}+\frac {c^{3} d^{3} b \arctanh \left (c x \right ) x^{7}}{7}+\frac {c^{2} d^{3} b \arctanh \left (c x \right ) x^{6}}{2}+\frac {3 c \,d^{3} b \arctanh \left (c x \right ) x^{5}}{5}+\frac {d^{3} b \arctanh \left (c x \right ) x^{4}}{4}+\frac {b \,c^{2} d^{3} x^{6}}{42}+\frac {b c \,d^{3} x^{5}}{10}+\frac {13 b \,d^{3} x^{4}}{70}+\frac {b \,d^{3} x^{3}}{4 c}+\frac {13 b \,d^{3} x^{2}}{35 c^{2}}+\frac {3 b \,d^{3} x}{4 c^{3}}+\frac {209 d^{3} b \ln \left (c x -1\right )}{280 c^{4}}-\frac {b \,d^{3} \ln \left (c x +1\right )}{280 c^{4}} \]

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 = 0.32, size = 285, normalized size = 1.48 \[ \frac {1}{7} \, a c^{3} d^{3} x^{7} + \frac {1}{2} \, a c^{2} d^{3} x^{6} + \frac {3}{5} \, a c d^{3} x^{5} + \frac {1}{84} \, {\left (12 \, x^{7} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{3} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{60} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{2} d^{3} + \frac {3}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d^{3} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d^{3} \]

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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mupad [B]  time = 1.05, size = 177, normalized size = 0.92 \[ \frac {\frac {13\,b\,c^2\,d^3\,x^2}{35}-\frac {d^3\,\left (315\,b\,\mathrm {atanh}\left (c\,x\right )-156\,b\,\ln \left (c^2\,x^2-1\right )\right )}{420}+\frac {b\,c^3\,d^3\,x^3}{4}+\frac {3\,b\,c\,d^3\,x}{4}}{c^4}+\frac {d^3\,\left (105\,a\,x^4+78\,b\,x^4+105\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{420}+\frac {c^3\,d^3\,\left (60\,a\,x^7+60\,b\,x^7\,\mathrm {atanh}\left (c\,x\right )\right )}{420}+\frac {c\,d^3\,\left (252\,a\,x^5+42\,b\,x^5+252\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{420}+\frac {c^2\,d^3\,\left (210\,a\,x^6+10\,b\,x^6+210\,b\,x^6\,\mathrm {atanh}\left (c\,x\right )\right )}{420} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((13*b*c^2*d^3*x^2)/35 - (d^3*(315*b*atanh(c*x) - 156*b*log(c^2*x^2 - 1)))/420 + (b*c^3*d^3*x^3)/4 + (3*b*c*d^
3*x)/4)/c^4 + (d^3*(105*a*x^4 + 78*b*x^4 + 105*b*x^4*atanh(c*x)))/420 + (c^3*d^3*(60*a*x^7 + 60*b*x^7*atanh(c*
x)))/420 + (c*d^3*(252*a*x^5 + 42*b*x^5 + 252*b*x^5*atanh(c*x)))/420 + (c^2*d^3*(210*a*x^6 + 10*b*x^6 + 210*b*
x^6*atanh(c*x)))/420

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sympy [A]  time = 2.84, size = 243, normalized size = 1.27 \[ \begin {cases} \frac {a c^{3} d^{3} x^{7}}{7} + \frac {a c^{2} d^{3} x^{6}}{2} + \frac {3 a c d^{3} x^{5}}{5} + \frac {a d^{3} x^{4}}{4} + \frac {b c^{3} d^{3} x^{7} \operatorname {atanh}{\left (c x \right )}}{7} + \frac {b c^{2} d^{3} x^{6} \operatorname {atanh}{\left (c x \right )}}{2} + \frac {b c^{2} d^{3} x^{6}}{42} + \frac {3 b c d^{3} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b c d^{3} x^{5}}{10} + \frac {b d^{3} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {13 b d^{3} x^{4}}{70} + \frac {b d^{3} x^{3}}{4 c} + \frac {13 b d^{3} x^{2}}{35 c^{2}} + \frac {3 b d^{3} x}{4 c^{3}} + \frac {26 b d^{3} \log {\left (x - \frac {1}{c} \right )}}{35 c^{4}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{140 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]

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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