Integrand size = 21, antiderivative size = 63 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {2 x}{\sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \] Output:
2*x/(-a^2*x^2+1)^(1/2)-2*arctanh(a*x)/a/(-a^2*x^2+1)^(1/2)+x*arctanh(a*x)^ 2/(-a^2*x^2+1)^(1/2)
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.60 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {2 a x-2 \text {arctanh}(a x)+a x \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}} \] Input:
Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2),x]
Output:
(2*a*x - 2*ArcTanh[a*x] + a*x*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2])
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6524, 208}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6524 |
\(\displaystyle 2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{a \sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {2 x}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2),x]
Output:
(2*x)/Sqrt[1 - a^2*x^2] - (2*ArcTanh[a*x])/(a*Sqrt[1 - a^2*x^2]) + (x*ArcT anh[a*x]^2)/Sqrt[1 - a^2*x^2]
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) ), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 *p*(p - 1) Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
Time = 0.44 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\operatorname {arctanh}\left (a x \right )^{2} a x +2 a x -2 \,\operatorname {arctanh}\left (a x \right )\right )}{a \left (a^{2} x^{2}-1\right )}\) | \(49\) |
orering | \(\frac {\left (-12 a^{4} x^{5}+11 a^{2} x^{3}+x \right ) \operatorname {arctanh}\left (a x \right )^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {\left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (8 a^{2} x^{2}+1\right ) \left (\frac {2 \,\operatorname {arctanh}\left (a x \right ) a}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{a^{2}}-\frac {x \left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (\frac {2 a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {15 \operatorname {arctanh}\left (a x \right )^{2} a^{4} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{a^{2}}\) | \(221\) |
Input:
int(arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/a*(-a^2*x^2+1)^(1/2)*(arctanh(a*x)^2*a*x+2*a*x-2*arctanh(a*x))/(a^2*x^2 -1)
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 8 \, a x - 4 \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )}}{4 \, {\left (a^{3} x^{2} - a\right )}} \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
-1/4*sqrt(-a^2*x^2 + 1)*(a*x*log(-(a*x + 1)/(a*x - 1))^2 + 8*a*x - 4*log(- (a*x + 1)/(a*x - 1)))/(a^3*x^2 - a)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(atanh(a*x)**2/(-a**2*x**2+1)**(3/2),x)
Output:
Integral(atanh(a*x)**2/(-(a*x - 1)*(a*x + 1))**(3/2), x)
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {x \operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {2 \, \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a} \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
x*arctanh(a*x)^2/sqrt(-a^2*x^2 + 1) + 2*x/sqrt(-a^2*x^2 + 1) - 2*arctanh(a *x)/(sqrt(-a^2*x^2 + 1)*a)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
integrate(arctanh(a*x)^2/(-a^2*x^2 + 1)^(3/2), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(atanh(a*x)^2/(1 - a^2*x^2)^(3/2),x)
Output:
int(atanh(a*x)^2/(1 - a^2*x^2)^(3/2), x)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x \right ) \] Input:
int(atanh(a*x)^2/(-a^2*x^2+1)^(3/2),x)
Output:
- int(atanh(a*x)**2/(sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1)),x)