Integrand size = 10, antiderivative size = 71 \[ \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx=-\frac {x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b}+\frac {x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b}+\frac {\operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{2 b^2}-\frac {\operatorname {PolyLog}\left (3,e^{a+b x}\right )}{2 b^2} \]
-1/2*x*polylog(2,-exp(b*x+a))/b+1/2*x*polylog(2,exp(b*x+a))/b+1/2*polylog( 3,-exp(b*x+a))/b^2-1/2*polylog(3,exp(b*x+a))/b^2
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.59 \[ \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx=\frac {2 b^2 x^2 \text {arctanh}\left (e^{a+b x}\right )+b^2 x^2 \log \left (1-e^{a+b x}\right )-b^2 x^2 \log \left (1+e^{a+b x}\right )-2 b x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+2 b x \operatorname {PolyLog}\left (2,e^{a+b x}\right )+2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{4 b^2} \]
(2*b^2*x^2*ArcTanh[E^(a + b*x)] + b^2*x^2*Log[1 - E^(a + b*x)] - b^2*x^2*L og[1 + E^(a + b*x)] - 2*b*x*PolyLog[2, -E^(a + b*x)] + 2*b*x*PolyLog[2, E^ (a + b*x)] + 2*PolyLog[3, -E^(a + b*x)] - 2*PolyLog[3, E^(a + b*x)])/(4*b^ 2)
Time = 0.38 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6767, 3011, 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.
\(\displaystyle \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx\) |
\(\Big \downarrow \) 6767 |
\(\displaystyle \frac {1}{2} \int x \log \left (1+e^{a+b x}\right )dx-\frac {1}{2} \int x \log \left (1-e^{a+b x}\right )dx\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )+\frac {1}{2} \left (\frac {x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{2} \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )+\frac {1}{2} \left (\frac {x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,e^{a+b x}\right )de^{a+b x}}{b^2}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )+\frac {1}{2} \left (\frac {x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^2}\right )\) |
(-((x*PolyLog[2, -E^(a + b*x)])/b) + PolyLog[3, -E^(a + b*x)]/b^2)/2 + ((x *PolyLog[2, E^(a + b*x)])/b - PolyLog[3, E^(a + b*x)]/b^2)/2
3.4.50.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(59)=118\).
Time = 0.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.18
method | result | size |
risch | \(\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a x}{2 b}+\frac {a^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2 b}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {a \operatorname {dilog}\left ({\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {\operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2 b}+\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}+1\right ) a}{2 b^{2}}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {\operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}\) | \(155\) |
default | \(\frac {x^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{2}-\frac {a^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )+\frac {\left (b x +a \right )^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\left (b x +a \right ) \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\left (b x +a \right ) \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )+a \left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )-a \left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )-a \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )+a \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}\) | \(178\) |
parts | \(\frac {x^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{2}-\frac {a^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )+\frac {\left (b x +a \right )^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\left (b x +a \right ) \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )-\operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\left (b x +a \right ) \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )+\operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )+a \left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )-a \left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )-a \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )+a \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}\) | \(178\) |
1/2/b*ln(1-exp(b*x+a))*a*x+1/2/b^2*a^2*ln(1-exp(b*x+a))+1/2*x*polylog(2,ex p(b*x+a))/b+1/2/b^2*polylog(2,exp(b*x+a))*a+1/2/b^2*a*dilog(exp(b*x+a))-1/ 2*polylog(3,exp(b*x+a))/b^2-1/2*x*polylog(2,-exp(b*x+a))/b+1/2/b^2*dilog(e xp(b*x+a)+1)*a-1/2/b^2*polylog(2,-exp(b*x+a))*a+1/2*polylog(3,-exp(b*x+a)) /b^2
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (57) = 114\).
Time = 0.25 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.80 \[ \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx=\frac {b^{2} x^{2} \log \left (-\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b^{2} x^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, b x {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, b x {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + a^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{4 \, b^{2}} \]
1/4*(b^2*x^2*log(-(cosh(b*x + a) + sinh(b*x + a) + 1)/(cosh(b*x + a) + sin h(b*x + a) - 1)) - b^2*x^2*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 2*b*x* dilog(cosh(b*x + a) + sinh(b*x + a)) - 2*b*x*dilog(-cosh(b*x + a) - sinh(b *x + a)) + a^2*log(cosh(b*x + a) + sinh(b*x + a) - 1) + (b^2*x^2 - a^2)*lo g(-cosh(b*x + a) - sinh(b*x + a) + 1) - 2*polylog(3, cosh(b*x + a) + sinh( b*x + a)) + 2*polylog(3, -cosh(b*x + a) - sinh(b*x + a)))/b^2
\[ \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx=\int x \operatorname {atanh}{\left (e^{a} e^{b x} \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.52 \[ \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {artanh}\left (e^{\left (b x + a\right )}\right ) - \frac {1}{4} \, b {\left (\frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} - \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}}\right )} \]
1/2*x^2*arctanh(e^(b*x + a)) - 1/4*b*((b^2*x^2*log(e^(b*x + a) + 1) + 2*b* x*dilog(-e^(b*x + a)) - 2*polylog(3, -e^(b*x + a)))/b^3 - (b^2*x^2*log(-e^ (b*x + a) + 1) + 2*b*x*dilog(e^(b*x + a)) - 2*polylog(3, e^(b*x + a)))/b^3 )
\[ \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx=\int { x \operatorname {artanh}\left (e^{\left (b x + a\right )}\right ) \,d x } \]
Timed out. \[ \int x \text {arctanh}\left (e^{a+b x}\right ) \, dx=\int x\,\mathrm {atanh}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \]