Integrand size = 19, antiderivative size = 41 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=\frac {\text {Chi}(2 \text {arctanh}(a x))}{2 a}+\frac {\text {Chi}(4 \text {arctanh}(a x))}{8 a}+\frac {3 \log (\text {arctanh}(a x))}{8 a} \] Output:
1/2*Chi(2*arctanh(a*x))/a+1/8*Chi(4*arctanh(a*x))/a+3/8*ln(arctanh(a*x))/a
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=-\frac {-4 \text {Chi}(2 \text {arctanh}(a x))-\text {Chi}(4 \text {arctanh}(a x))-3 \log (\text {arctanh}(a x))}{8 a} \] Input:
Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]
Output:
-1/8*(-4*CoshIntegral[2*ArcTanh[a*x]] - CoshIntegral[4*ArcTanh[a*x]] - 3*L og[ArcTanh[a*x]])/a
Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6530, 3042, 3793, 2009}
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.
\(\displaystyle \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx\) |
\(\Big \downarrow \) 6530 |
\(\displaystyle \frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}+\frac {3}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a}\) |
Input:
Int[1/((1 - a^2*x^2)^3*ArcTanh[a*x]),x]
Output:
(CoshIntegral[2*ArcTanh[a*x]]/2 + CoshIntegral[4*ArcTanh[a*x]]/8 + (3*Log[ ArcTanh[a*x]])/8)/a
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x _Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && I LtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Time = 0.79 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {\frac {3 \ln \left (\operatorname {arctanh}\left (a x \right )\right )}{8}+\frac {\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2}+\frac {\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8}}{a}\) | \(31\) |
default | \(\frac {\frac {3 \ln \left (\operatorname {arctanh}\left (a x \right )\right )}{8}+\frac {\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2}+\frac {\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8}}{a}\) | \(31\) |
Input:
int(1/(-a^2*x^2+1)^3/arctanh(a*x),x,method=_RETURNVERBOSE)
Output:
1/a*(3/8*ln(arctanh(a*x))+1/2*Chi(2*arctanh(a*x))+1/8*Chi(4*arctanh(a*x)))
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (35) = 70\).
Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.88 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=\frac {6 \, \log \left (\log \left (-\frac {a x + 1}{a x - 1}\right )\right ) + \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 4 \, \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + 4 \, \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{16 \, a} \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x),x, algorithm="fricas")
Output:
1/16*(6*log(log(-(a*x + 1)/(a*x - 1))) + log_integral((a^2*x^2 + 2*a*x + 1 )/(a^2*x^2 - 2*a*x + 1)) + log_integral((a^2*x^2 - 2*a*x + 1)/(a^2*x^2 + 2 *a*x + 1)) + 4*log_integral(-(a*x + 1)/(a*x - 1)) + 4*log_integral(-(a*x - 1)/(a*x + 1)))/a
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=- \int \frac {1}{a^{6} x^{6} \operatorname {atanh}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )} - \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:
integrate(1/(-a**2*x**2+1)**3/atanh(a*x),x)
Output:
-Integral(1/(a**6*x**6*atanh(a*x) - 3*a**4*x**4*atanh(a*x) + 3*a**2*x**2*a tanh(a*x) - atanh(a*x)), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )} \,d x } \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x),x, algorithm="maxima")
Output:
-integrate(1/((a^2*x^2 - 1)^3*arctanh(a*x)), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )} \,d x } \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x),x, algorithm="giac")
Output:
integrate(-1/((a^2*x^2 - 1)^3*arctanh(a*x)), x)
Timed out. \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=-\int \frac {1}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:
int(-1/(atanh(a*x)*(a^2*x^2 - 1)^3),x)
Output:
-int(1/(atanh(a*x)*(a^2*x^2 - 1)^3), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)} \, dx=-\left (\int \frac {1}{\mathit {atanh} \left (a x \right ) a^{6} x^{6}-3 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+3 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-\mathit {atanh} \left (a x \right )}d x \right ) \] Input:
int(1/(-a^2*x^2+1)^3/atanh(a*x),x)
Output:
- int(1/(atanh(a*x)*a**6*x**6 - 3*atanh(a*x)*a**4*x**4 + 3*atanh(a*x)*a** 2*x**2 - atanh(a*x)),x)