Integrand size = 14, antiderivative size = 185 \[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\frac {b^3 x}{4 c^3}-\frac {b^3 \text {arctanh}(c x)}{4 c^4}+\frac {b^2 x^2 (a+b \text {arctanh}(c x))}{4 c^2}+\frac {b (a+b \text {arctanh}(c x))^2}{c^4}+\frac {3 b x (a+b \text {arctanh}(c x))^2}{4 c^3}+\frac {b x^3 (a+b \text {arctanh}(c x))^2}{4 c}-\frac {(a+b \text {arctanh}(c x))^3}{4 c^4}+\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {2 b^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^4}-\frac {b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^4} \] Output:
1/4*b^3*x/c^3-1/4*b^3*arctanh(c*x)/c^4+1/4*b^2*x^2*(a+b*arctanh(c*x))/c^2+ b*(a+b*arctanh(c*x))^2/c^4+3/4*b*x*(a+b*arctanh(c*x))^2/c^3+1/4*b*x^3*(a+b *arctanh(c*x))^2/c-1/4*(a+b*arctanh(c*x))^3/c^4+1/4*x^4*(a+b*arctanh(c*x)) ^3-2*b^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4-b^3*polylog(2,1-2/(-c*x+1)) /c^4
Time = 0.33 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.32 \[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\frac {-2 a b^2+6 a^2 b c x+2 b^3 c x+2 a b^2 c^2 x^2+2 a^2 b c^3 x^3+2 a^3 c^4 x^4+2 b^2 \left (b \left (-4+3 c x+c^3 x^3\right )+3 a \left (-1+c^4 x^4\right )\right ) \text {arctanh}(c x)^2+2 b^3 \left (-1+c^4 x^4\right ) \text {arctanh}(c x)^3+2 b \text {arctanh}(c x) \left (3 a^2 c^4 x^4+b^2 \left (-1+c^2 x^2\right )+2 a b c x \left (3+c^2 x^2\right )-8 b^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 a^2 b \log (1-c x)-3 a^2 b \log (1+c x)+8 a b^2 \log \left (1-c^2 x^2\right )+8 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{8 c^4} \] Input:
Integrate[x^3*(a + b*ArcTanh[c*x])^3,x]
Output:
(-2*a*b^2 + 6*a^2*b*c*x + 2*b^3*c*x + 2*a*b^2*c^2*x^2 + 2*a^2*b*c^3*x^3 + 2*a^3*c^4*x^4 + 2*b^2*(b*(-4 + 3*c*x + c^3*x^3) + 3*a*(-1 + c^4*x^4))*ArcT anh[c*x]^2 + 2*b^3*(-1 + c^4*x^4)*ArcTanh[c*x]^3 + 2*b*ArcTanh[c*x]*(3*a^2 *c^4*x^4 + b^2*(-1 + c^2*x^2) + 2*a*b*c*x*(3 + c^2*x^2) - 8*b^2*Log[1 + E^ (-2*ArcTanh[c*x])]) + 3*a^2*b*Log[1 - c*x] - 3*a^2*b*Log[1 + c*x] + 8*a*b^ 2*Log[1 - c^2*x^2] + 8*b^3*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(8*c^4)
Time = 3.06 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.57, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6452, 6542, 6452, 6542, 6436, 6452, 262, 219, 6510, 6546, 6470, 2849, 2752}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 (a+b \text {arctanh}(c x))^3 \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int x^2 (a+b \text {arctanh}(c x))^2dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int (a+b \text {arctanh}(c x))^2dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \int \frac {x^2}{1-c^2 x^2}dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x^2}dx}{c^2}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-\frac {2}{1-c x}}d\frac {1}{1-c x}}{c}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-\frac {2}{1-c x}}d\frac {1}{1-c x}}{c}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{4} x^4 (a+b \text {arctanh}(c x))^3-\frac {3}{4} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}\right )\) |
Input:
Int[x^3*(a + b*ArcTanh[c*x])^3,x]
Output:
(x^4*(a + b*ArcTanh[c*x])^3)/4 - (3*b*c*(-(((x^3*(a + b*ArcTanh[c*x])^2)/3 - (2*b*c*(-(((x^2*(a + b*ArcTanh[c*x]))/2 - (b*c*(-(x/c^2) + ArcTanh[c*x] /c^3))/2)/c^2) + (-1/2*(a + b*ArcTanh[c*x])^2/(b*c^2) + (((a + b*ArcTanh[c *x])*Log[2/(1 - c*x)])/c + (b*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c))/c)/c^2)) /3)/c^2) + ((a + b*ArcTanh[c*x])^3/(3*b*c^3) - (x*(a + b*ArcTanh[c*x])^2 - 2*b*c*(-1/2*(a + b*ArcTanh[c*x])^2/(b*c^2) + (((a + b*ArcTanh[c*x])*Log[2 /(1 - c*x)])/c + (b*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c))/c))/c^2)/c^2))/4
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(666\) vs. \(2(171)=342\).
Time = 8.06 (sec) , antiderivative size = 667, normalized size of antiderivative = 3.61
| method | result | size |
| risch | \(\frac {b^{3} x}{4 c^{3}}+\frac {3 \ln \left (-c x +1\right )^{2} a \,b^{2} x^{4}}{16}-\frac {3 \ln \left (-c x +1\right ) a^{2} b \,x^{4}}{8}-\frac {b^{3}}{4 c^{4}}-\frac {3 a \,b^{2} \ln \left (-c x +1\right ) x}{4 c^{3}}+\frac {a \,b^{2} x^{2}}{4 c^{2}}+\frac {a^{2} b \,x^{3}}{4 c}+\frac {3 a^{2} b x}{4 c^{3}}+\frac {3 a^{2} b \ln \left (-c x +1\right )}{8 c^{4}}-\frac {3 a \,b^{2} \ln \left (-c x +1\right )^{2}}{16 c^{4}}+\frac {a \,b^{2} \ln \left (-c x +1\right )}{c^{4}}+\frac {b^{3} \ln \left (-c x +1\right )^{2} x^{3}}{16 c}+\frac {3 b^{3} \ln \left (-c x +1\right )^{2} x}{16 c^{3}}-\frac {b^{3} \ln \left (-c x +1\right ) x^{2}}{8 c^{2}}+\frac {a^{3} x^{4}}{4}-\frac {a \,b^{2}}{4 c^{4}}-\frac {a^{2} b}{c^{4}}-\frac {a^{3}}{4 c^{4}}-\frac {b^{3} \ln \left (-c x -1\right )}{8 c^{4}}+\frac {b^{3} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{c^{4}}+\frac {b^{3} \ln \left (-c x +1\right )^{3}}{32 c^{4}}-\frac {b^{3} \ln \left (-c x +1\right )^{2}}{4 c^{4}}+\frac {b^{3} \ln \left (-c x +1\right )}{8 c^{4}}+\left (\frac {3 b^{3} \left (c^{4} x^{4}-1\right ) \ln \left (-c x +1\right )^{2}}{32 c^{4}}-\frac {b^{2} x \left (3 c^{3} x^{3} a +b \,c^{2} x^{2}+3 b \right ) \ln \left (-c x +1\right )}{8 c^{3}}-\frac {b \left (-3 a^{2} c^{4} x^{4}-2 a b \,c^{3} x^{3}-b^{2} c^{2} x^{2}-6 a b c x -3 b \ln \left (-c x +1\right ) a -4 b^{2} \ln \left (-c x +1\right )\right )}{8 c^{4}}\right ) \ln \left (c x +1\right )-\frac {a \,b^{2} \ln \left (-c x +1\right ) x^{3}}{4 c}+\frac {b^{3} \left (c^{4} x^{4}-1\right ) \ln \left (c x +1\right )^{3}}{32 c^{4}}+\frac {b^{2} \left (-3 b \,x^{4} \ln \left (-c x +1\right ) c^{4}+6 a \,c^{4} x^{4}+2 b \,c^{3} x^{3}+6 b c x +3 b \ln \left (-c x +1\right )-6 a +8 b \right ) \ln \left (c x +1\right )^{2}}{32 c^{4}}-\frac {\ln \left (-c x +1\right )^{3} b^{3} x^{4}}{32}+\frac {b^{2} \ln \left (-c x -1\right ) a}{c^{4}}-\frac {3 b \ln \left (-c x -1\right ) a^{2}}{8 c^{4}}+\frac {b^{3} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{c^{4}}-\frac {b^{3} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{c^{4}}\) | \(667\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1042\) |
| default | \(\text {Expression too large to display}\) | \(1042\) |
| parts | \(\text {Expression too large to display}\) | \(1044\) |
Input:
int(x^3*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/4*b^3*x/c^3+3/16*ln(-c*x+1)^2*a*b^2*x^4-3/8*ln(-c*x+1)*a^2*b*x^4-1/4*b^3 /c^4-3/4/c^3*a*b^2*ln(-c*x+1)*x+1/4/c^2*a*b^2*x^2+1/4/c*a^2*b*x^3+3/4/c^3* a^2*b*x+3/8/c^4*a^2*b*ln(-c*x+1)-3/16/c^4*a*b^2*ln(-c*x+1)^2+1/c^4*a*b^2*l n(-c*x+1)+1/16/c*b^3*ln(-c*x+1)^2*x^3+3/16/c^3*b^3*ln(-c*x+1)^2*x-1/8/c^2* b^3*ln(-c*x+1)*x^2+1/4*a^3*x^4-1/4/c^4*a*b^2-1/c^4*a^2*b-1/4/c^4*a^3-1/8/c ^4*b^3*ln(-c*x-1)+b^3/c^4*dilog(-1/2*c*x+1/2)+1/32/c^4*b^3*ln(-c*x+1)^3-1/ 4/c^4*b^3*ln(-c*x+1)^2+1/8/c^4*b^3*ln(-c*x+1)+(3/32*b^3*(c^4*x^4-1)/c^4*ln (-c*x+1)^2-1/8*b^2*x*(3*a*c^3*x^3+b*c^2*x^2+3*b)/c^3*ln(-c*x+1)-1/8*b*(-3* a^2*c^4*x^4-2*a*b*c^3*x^3-b^2*c^2*x^2-6*a*b*c*x-3*b*ln(-c*x+1)*a-4*b^2*ln( -c*x+1))/c^4)*ln(c*x+1)-1/4/c*a*b^2*ln(-c*x+1)*x^3+1/32*b^3*(c^4*x^4-1)/c^ 4*ln(c*x+1)^3+1/32*b^2*(-3*b*x^4*ln(-c*x+1)*c^4+6*a*c^4*x^4+2*b*c^3*x^3+6* b*c*x+3*b*ln(-c*x+1)-6*a+8*b)/c^4*ln(c*x+1)^2-1/32*ln(-c*x+1)^3*b^3*x^4+b^ 2/c^4*ln(-c*x-1)*a-3/8*b/c^4*ln(-c*x-1)*a^2+b^3/c^4*ln(-1/2*c*x+1/2)*ln(1/ 2*c*x+1/2)-b^3/c^4*ln(1/2*c*x+1/2)*ln(-c*x+1)
\[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*arctanh(c*x))^3,x, algorithm="fricas")
Output:
integral(b^3*x^3*arctanh(c*x)^3 + 3*a*b^2*x^3*arctanh(c*x)^2 + 3*a^2*b*x^3 *arctanh(c*x) + a^3*x^3, x)
\[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\int x^{3} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, dx \] Input:
integrate(x**3*(a+b*atanh(c*x))**3,x)
Output:
Integral(x**3*(a + b*atanh(c*x))**3, x)
\[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*arctanh(c*x))^3,x, algorithm="maxima")
Output:
3/4*a*b^2*x^4*arctanh(c*x)^2 + 1/4*a^3*x^4 + 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))*a^2*b + 1/16*(4*c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5 )*arctanh(c*x) + (4*c^2*x^2 - 2*(3*log(c*x - 1) - 8)*log(c*x + 1) + 3*log( c*x + 1)^2 + 3*log(c*x - 1)^2 + 16*log(c*x - 1))/c^4)*a*b^2 - 1/9216*(27*c ^5*((c^2*x^4 + 2*x^2)/c^7 + 2*log(c^2*x^2 - 1)/c^9) + 74*c^4*(2*(c^2*x^3 + 3*x)/c^7 - 3*log(c*x + 1)/c^8 + 3*log(c*x - 1)/c^8) + 60*c^3*(x^2/c^5 + l og(c^2*x^2 - 1)/c^7) - 221184*c^3*integrate(1/96*x^3*log(c*x + 1)/(c^5*x^2 - c^3), x) + 1692*c^2*(2*x/c^5 - log(c*x + 1)/c^6 + log(c*x - 1)/c^6) - 1 105920*c*integrate(1/96*x*log(c*x + 1)/(c^5*x^2 - c^3), x) + (9*(32*log(-c *x + 1)^3 - 24*log(-c*x + 1)^2 + 12*log(-c*x + 1) - 3)*(c*x - 1)^4 + 128*( 9*log(-c*x + 1)^3 - 9*log(-c*x + 1)^2 + 6*log(-c*x + 1) - 2)*(c*x - 1)^3 + 432*(4*log(-c*x + 1)^3 - 6*log(-c*x + 1)^2 + 6*log(-c*x + 1) - 3)*(c*x - 1)^2 + 1152*(log(-c*x + 1)^3 - 3*log(-c*x + 1)^2 + 6*log(-c*x + 1) - 6)*(c *x - 1))/c^4 - 12*(24*(c^4*x^4 - 1)*log(c*x + 1)^3 + 48*(c^3*x^3 + 3*c*x)* log(c*x + 1)^2 - 6*(3*c^4*x^4 - 4*c^3*x^3 + 6*c^2*x^2 - 12*c*x - 12*(c^4*x ^4 - 1)*log(c*x + 1) + 7)*log(-c*x + 1)^2 + (9*c^4*x^4 + 28*c^3*x^3 - 18*c ^2*x^2 - 72*(c^4*x^4 - 1)*log(c*x + 1)^2 + 300*c*x - 96*(c^3*x^3 + 3*c*x + 4)*log(c*x + 1))*log(-c*x + 1))/c^4 + 1800*log(96*c^5*x^2 - 96*c^3)/c^4 - 442368*integrate(1/96*log(c*x + 1)/(c^5*x^2 - c^3), x))*b^3
\[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{3} \,d x } \] Input:
integrate(x^3*(a+b*arctanh(c*x))^3,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^3*x^3, x)
Timed out. \[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\int x^3\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3 \,d x \] Input:
int(x^3*(a + b*atanh(c*x))^3,x)
Output:
int(x^3*(a + b*atanh(c*x))^3, x)
\[ \int x^3 (a+b \text {arctanh}(c x))^3 \, dx=\frac {\mathit {atanh} \left (c x \right )^{3} b^{3} c^{4} x^{4}-\mathit {atanh} \left (c x \right )^{3} b^{3}+3 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c^{4} x^{4}-3 \mathit {atanh} \left (c x \right )^{2} a \,b^{2}+\mathit {atanh} \left (c x \right )^{2} b^{3} c^{3} x^{3}+3 \mathit {atanh} \left (c x \right )^{2} b^{3} c x +3 \mathit {atanh} \left (c x \right ) a^{2} b \,c^{4} x^{4}-3 \mathit {atanh} \left (c x \right ) a^{2} b +2 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{3} x^{3}+6 \mathit {atanh} \left (c x \right ) a \,b^{2} c x +8 \mathit {atanh} \left (c x \right ) a \,b^{2}+\mathit {atanh} \left (c x \right ) b^{3} c^{2} x^{2}-\mathit {atanh} \left (c x \right ) b^{3}+8 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) b^{3} c^{2}+8 \,\mathrm {log}\left (c^{2} x -c \right ) a \,b^{2}+a^{3} c^{4} x^{4}+a^{2} b \,c^{3} x^{3}+3 a^{2} b c x +a \,b^{2} c^{2} x^{2}+b^{3} c x}{4 c^{4}} \] Input:
int(x^3*(a+b*atanh(c*x))^3,x)
Output:
(atanh(c*x)**3*b**3*c**4*x**4 - atanh(c*x)**3*b**3 + 3*atanh(c*x)**2*a*b** 2*c**4*x**4 - 3*atanh(c*x)**2*a*b**2 + atanh(c*x)**2*b**3*c**3*x**3 + 3*at anh(c*x)**2*b**3*c*x + 3*atanh(c*x)*a**2*b*c**4*x**4 - 3*atanh(c*x)*a**2*b + 2*atanh(c*x)*a*b**2*c**3*x**3 + 6*atanh(c*x)*a*b**2*c*x + 8*atanh(c*x)* a*b**2 + atanh(c*x)*b**3*c**2*x**2 - atanh(c*x)*b**3 + 8*int((atanh(c*x)*x )/(c**2*x**2 - 1),x)*b**3*c**2 + 8*log(c**2*x - c)*a*b**2 + a**3*c**4*x**4 + a**2*b*c**3*x**3 + 3*a**2*b*c*x + a*b**2*c**2*x**2 + b**3*c*x)/(4*c**4)