Integrand size = 12, antiderivative size = 101 \[ \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx=-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b}+\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b}+\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^2}-\frac {\operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^3} \]
-1/2*x^2*polylog(2,-exp(b*x+a))/b+1/2*x^2*polylog(2,exp(b*x+a))/b+x*polylo g(3,-exp(b*x+a))/b^2-x*polylog(3,exp(b*x+a))/b^2-polylog(4,-exp(b*x+a))/b^ 3+polylog(4,exp(b*x+a))/b^3
Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.48 \[ \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx=\frac {2 b^3 x^3 \text {arctanh}\left (e^{a+b x}\right )+b^3 x^3 \log \left (1-e^{a+b x}\right )-b^3 x^3 \log \left (1+e^{a+b x}\right )-3 b^2 x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+3 b^2 x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )+6 b x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-6 b x \operatorname {PolyLog}\left (3,e^{a+b x}\right )-6 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )+6 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{6 b^3} \]
(2*b^3*x^3*ArcTanh[E^(a + b*x)] + b^3*x^3*Log[1 - E^(a + b*x)] - b^3*x^3*L og[1 + E^(a + b*x)] - 3*b^2*x^2*PolyLog[2, -E^(a + b*x)] + 3*b^2*x^2*PolyL og[2, E^(a + b*x)] + 6*b*x*PolyLog[3, -E^(a + b*x)] - 6*b*x*PolyLog[3, E^( a + b*x)] - 6*PolyLog[4, -E^(a + b*x)] + 6*PolyLog[4, E^(a + b*x)])/(6*b^3 )
Time = 0.55 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6767, 3011, 7163, 2720, 7143}
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.
| \(\displaystyle \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx\) |
| \(\Big \downarrow \) 6767 |
| \(\displaystyle \frac {1}{2} \int x^2 \log \left (1+e^{a+b x}\right )dx-\frac {1}{2} \int x^2 \log \left (1-e^{a+b x}\right )dx\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )+\frac {1}{2} \left (\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}-\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (3,-e^{a+b x}\right )dx}{b}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )+\frac {1}{2} \left (\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (3,e^{a+b x}\right )dx}{b}\right )}{b}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,-e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )+\frac {1}{2} \left (\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )+\frac {1}{2} \left (\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^2}\right )}{b}\right )\) |
(-((x^2*PolyLog[2, -E^(a + b*x)])/b) + (2*((x*PolyLog[3, -E^(a + b*x)])/b - PolyLog[4, -E^(a + b*x)]/b^2))/b)/2 + ((x^2*PolyLog[2, E^(a + b*x)])/b - (2*((x*PolyLog[3, E^(a + b*x)])/b - PolyLog[4, E^(a + b*x)]/b^2))/b)/2
3.4.51.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[ArcTanh[(a_.) + (b_.)*(f_)^((c_.) + (d_.)*(x_))]*(x_)^(m_.), x_Symbol] :> Simp[1/2 Int[x^m*Log[1 + a + b*f^(c + d*x)], x], x] - Simp[1/2 Int[x ^m*Log[1 - a - b*f^(c + d*x)], x], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[ m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(91)=182\).
Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.95
| method | result | size |
| risch | \(\frac {x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2 b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x \,a^{2}}{2 b^{2}}-\frac {a^{3} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2 b^{3}}-\frac {x \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {a^{2} \operatorname {dilog}\left ({\mathrm e}^{b x +a}\right )}{2 b^{3}}+\frac {\operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2 b}+\frac {x \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{2 b^{3}}-\frac {\operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(197\) |
| default | \(\frac {x^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{3}-\frac {\frac {\left (b x +a \right )^{3} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}-3 \left (b x +a \right ) \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )+3 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{3} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}+3 \left (b x +a \right ) \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )-3 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )-a^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\left (b x +a \right ) \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\left (b x +a \right ) \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}-\frac {\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}\right )}{3 b^{3}}\) | \(315\) |
| parts | \(\frac {x^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{3}-\frac {\frac {\left (b x +a \right )^{3} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}-3 \left (b x +a \right ) \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )+3 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{3} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\frac {3 \left (b x +a \right )^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}+3 \left (b x +a \right ) \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )-3 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )-a^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\left (b x +a \right ) \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\left (b x +a \right ) \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}-\frac {\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}\right )}{3 b^{3}}\) | \(315\) |
1/2*x^2*polylog(2,exp(b*x+a))/b-1/2/b^2*ln(1-exp(b*x+a))*x*a^2-1/2/b^3*a^3 *ln(1-exp(b*x+a))-x*polylog(3,exp(b*x+a))/b^2-1/2/b^3*polylog(2,exp(b*x+a) )*a^2-1/2/b^3*a^2*dilog(exp(b*x+a))+polylog(4,exp(b*x+a))/b^3-1/2*x^2*poly log(2,-exp(b*x+a))/b+x*polylog(3,-exp(b*x+a))/b^2+1/2/b^3*polylog(2,-exp(b *x+a))*a^2-1/2/b^3*dilog(exp(b*x+a)+1)*a^2-polylog(4,-exp(b*x+a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (89) = 178\).
Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.46 \[ \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx=\frac {b^{3} x^{3} \log \left (-\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b^{3} x^{3} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 3 \, b^{2} x^{2} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - a^{3} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 6 \, b x {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, b x {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\rm polylog}\left (4, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, {\rm polylog}\left (4, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{6 \, b^{3}} \]
1/6*(b^3*x^3*log(-(cosh(b*x + a) + sinh(b*x + a) + 1)/(cosh(b*x + a) + sin h(b*x + a) - 1)) - b^3*x^3*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 3*b^2* x^2*dilog(cosh(b*x + a) + sinh(b*x + a)) - 3*b^2*x^2*dilog(-cosh(b*x + a) - sinh(b*x + a)) - a^3*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 6*b*x*poly log(3, cosh(b*x + a) + sinh(b*x + a)) + 6*b*x*polylog(3, -cosh(b*x + a) - sinh(b*x + a)) + (b^3*x^3 + a^3)*log(-cosh(b*x + a) - sinh(b*x + a) + 1) + 6*polylog(4, cosh(b*x + a) + sinh(b*x + a)) - 6*polylog(4, -cosh(b*x + a) - sinh(b*x + a)))/b^3
\[ \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx=\int x^{2} \operatorname {atanh}{\left (e^{a} e^{b x} \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.41 \[ \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {artanh}\left (e^{\left (b x + a\right )}\right ) - \frac {1}{6} \, b {\left (\frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} - \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}}\right )} \]
1/3*x^3*arctanh(e^(b*x + a)) - 1/6*b*((b^3*x^3*log(e^(b*x + a) + 1) + 3*b^ 2*x^2*dilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x + a)))/b^4 - (b^3*x^3*log(-e^(b*x + a) + 1) + 3*b^2*x^2*dilog(e^(b *x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4, e^(b*x + a)))/b^4)
\[ \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx=\int { x^{2} \operatorname {artanh}\left (e^{\left (b x + a\right )}\right ) \,d x } \]
Timed out. \[ \int x^2 \text {arctanh}\left (e^{a+b x}\right ) \, dx=\int x^2\,\mathrm {atanh}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \]