Integrand size = 22, antiderivative size = 68 \[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=-\frac {x}{2 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}-\frac {1}{2 a^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}+\frac {\text {Shi}(\text {arctanh}(a x))}{2 a^2} \] Output:
-1/2*x/a/(-a^2*x^2+1)^(1/2)/arctanh(a*x)^2-1/2/a^2/(-a^2*x^2+1)^(1/2)/arct anh(a*x)+1/2*Shi(arctanh(a*x))/a^2
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.63 \[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=\frac {-\frac {a x+\text {arctanh}(a x)}{\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}+\text {Shi}(\text {arctanh}(a x))}{2 a^2} \] Input:
Integrate[x/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]
Output:
(-((a*x + ArcTanh[a*x])/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) + SinhIntegral [ArcTanh[a*x]])/(2*a^2)
Time = 0.57 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6568, 6528, 6596, 3042, 26, 3779}
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.
\(\displaystyle \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx\) |
\(\Big \downarrow \) 6568 |
\(\displaystyle \frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^2}dx}{2 a}-\frac {x}{2 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6528 |
\(\displaystyle \frac {a \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}dx-\frac {1}{a \sqrt {1-a^2 x^2} \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6596 |
\(\displaystyle \frac {\frac {\int \frac {a x}{\sqrt {1-a^2 x^2} \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \sqrt {1-a^2 x^2} \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x}{2 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}+\frac {-\frac {1}{a \sqrt {1-a^2 x^2} \text {arctanh}(a x)}+\frac {\int -\frac {i \sin (i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{2 a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {x}{2 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}+\frac {-\frac {1}{a \sqrt {1-a^2 x^2} \text {arctanh}(a x)}-\frac {i \int \frac {\sin (i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{2 a}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {\frac {\text {Shi}(\text {arctanh}(a x))}{a}-\frac {1}{a \sqrt {1-a^2 x^2} \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}\) |
Input:
Int[x/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3),x]
Output:
-1/2*x/(a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2) + (-(1/(a*Sqrt[1 - a^2*x^2]*Ar cTanh[a*x])) + SinhIntegral[ArcTanh[a*x]]/a)/(2*a)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + Simp[2*c*((q + 1)/(b*(p + 1))) Int[x*(d + e*x^2)^q*(a + b*A rcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Arc Tanh[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[f*(m/(b*c*(p + 1))) Int[(f *x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && Lt Q[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sinh[x]^ m/Cosh[x]^(m + 2*(q + 1))), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (In tegerQ[q] || GtQ[d, 0])
Time = 0.53 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.28
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (a x \right )^{2} \operatorname {Shi}\left (\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}-\operatorname {Shi}\left (\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right )^{2}+a x \sqrt {-a^{2} x^{2}+1}+\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )}{2 a^{2} \operatorname {arctanh}\left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}\) | \(87\) |
Input:
int(x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x,method=_RETURNVERBOSE)
Output:
1/2/a^2*(arctanh(a*x)^2*Shi(arctanh(a*x))*a^2*x^2-Shi(arctanh(a*x))*arctan h(a*x)^2+a*x*(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)*arctanh(a*x))/arctanh(a *x)^2/(a^2*x^2-1)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=\int { \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x, algorithm="fricas")
Output:
integral(sqrt(-a^2*x^2 + 1)*x/((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^3), x)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=\int \frac {x}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \] Input:
integrate(x/(-a**2*x**2+1)**(3/2)/atanh(a*x)**3,x)
Output:
Integral(x/((-(a*x - 1)*(a*x + 1))**(3/2)*atanh(a*x)**3), x)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=\int { \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x, algorithm="maxima")
Output:
integrate(x/((-a^2*x^2 + 1)^(3/2)*arctanh(a*x)^3), x)
Exception generated. \[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/(-a^2*x^2+1)^(3/2)/arctanh(a*x)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=\int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(x/(atanh(a*x)^3*(1 - a^2*x^2)^(3/2)),x)
Output:
int(x/(atanh(a*x)^3*(1 - a^2*x^2)^(3/2)), x)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^3} \, dx=-\left (\int \frac {x}{\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{3}}d x \right ) \] Input:
int(x/(-a^2*x^2+1)^(3/2)/atanh(a*x)^3,x)
Output:
- int(x/(sqrt( - a**2*x**2 + 1)*atanh(a*x)**3*a**2*x**2 - sqrt( - a**2*x* *2 + 1)*atanh(a*x)**3),x)