\(\int \frac {x^6}{(1-a^2 x^2)^4 \text {arctanh}(a x)} \, dx\) [351]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=\frac {15 \text {Chi}(2 \text {arctanh}(a x))}{32 a^7}-\frac {3 \text {Chi}(4 \text {arctanh}(a x))}{16 a^7}+\frac {\text {Chi}(6 \text {arctanh}(a x))}{32 a^7}-\frac {5 \log (\text {arctanh}(a x))}{16 a^7} \] Output:

15/32*Chi(2*arctanh(a*x))/a^7-3/16*Chi(4*arctanh(a*x))/a^7+1/32*Chi(6*arct 
anh(a*x))/a^7-5/16*ln(arctanh(a*x))/a^7
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=\frac {15 \text {Chi}(2 \text {arctanh}(a x))-6 \text {Chi}(4 \text {arctanh}(a x))+\text {Chi}(6 \text {arctanh}(a x))-10 \log (\text {arctanh}(a x))}{32 a^7} \] Input:

Integrate[x^6/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]
 

Output:

(15*CoshIntegral[2*ArcTanh[a*x]] - 6*CoshIntegral[4*ArcTanh[a*x]] + CoshIn 
tegral[6*ArcTanh[a*x]] - 10*Log[ArcTanh[a*x]])/(32*a^7)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6596, 3042, 25, 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.

\(\displaystyle \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx\)

\(\Big \downarrow \) 6596

\(\displaystyle \frac {\int \frac {a^6 x^6}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -\frac {\sin (i \text {arctanh}(a x))^6}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^7}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {\sin (i \text {arctanh}(a x))^6}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^7}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {\int \left (-\frac {15 \cosh (2 \text {arctanh}(a x))}{32 \text {arctanh}(a x)}+\frac {3 \cosh (4 \text {arctanh}(a x))}{16 \text {arctanh}(a x)}-\frac {\cosh (6 \text {arctanh}(a x))}{32 \text {arctanh}(a x)}+\frac {5}{16 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {15}{32} \text {Chi}(2 \text {arctanh}(a x))-\frac {3}{16} \text {Chi}(4 \text {arctanh}(a x))+\frac {1}{32} \text {Chi}(6 \text {arctanh}(a x))-\frac {5}{16} \log (\text {arctanh}(a x))}{a^7}\)

Input:

Int[x^6/((1 - a^2*x^2)^4*ArcTanh[a*x]),x]
 

Output:

((15*CoshIntegral[2*ArcTanh[a*x]])/32 - (3*CoshIntegral[4*ArcTanh[a*x]])/1 
6 + CoshIntegral[6*ArcTanh[a*x]]/32 - (5*Log[ArcTanh[a*x]])/16)/a^7
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

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]))
 

rule 6596

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])
 

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {-\frac {5 \ln \left (\operatorname {arctanh}\left (a x \right )\right )}{16}+\frac {15 \,\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{32}-\frac {3 \,\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16}+\frac {\operatorname {Chi}\left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{32}}{a^{7}}\)	\(40\)
default	\(\frac {-\frac {5 \ln \left (\operatorname {arctanh}\left (a x \right )\right )}{16}+\frac {15 \,\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{32}-\frac {3 \,\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16}+\frac {\operatorname {Chi}\left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{32}}{a^{7}}\)	\(40\)

Input:

int(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x,method=_RETURNVERBOSE)
 

Output:

1/a^7*(-5/16*ln(arctanh(a*x))+15/32*Chi(2*arctanh(a*x))-3/16*Chi(4*arctanh 
(a*x))+1/32*Chi(6*arctanh(a*x)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (47) = 94\).

Time = 0.09 (sec) , antiderivative size = 220, normalized size of antiderivative = 4.00 \[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=-\frac {20 \, \log \left (\log \left (-\frac {a x + 1}{a x - 1}\right )\right ) - \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 6 \, \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 6 \, \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) - 15 \, \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - 15 \, \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{64 \, a^{7}} \] Input:

integrate(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x, algorithm="fricas")
 

Output:

-1/64*(20*log(log(-(a*x + 1)/(a*x - 1))) - log_integral(-(a^3*x^3 + 3*a^2* 
x^2 + 3*a*x + 1)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) - log_integral(-(a^3*x 
^3 - 3*a^2*x^2 + 3*a*x - 1)/(a^3*x^3 + 3*a^2*x^2 + 3*a*x + 1)) + 6*log_int 
egral((a^2*x^2 + 2*a*x + 1)/(a^2*x^2 - 2*a*x + 1)) + 6*log_integral((a^2*x 
^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x + 1)) - 15*log_integral(-(a*x + 1)/(a*x - 
 1)) - 15*log_integral(-(a*x - 1)/(a*x + 1)))/a^7
 

Sympy [F]

\[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=\int \frac {x^{6}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:

integrate(x**6/(-a**2*x**2+1)**4/atanh(a*x),x)
 

Output:

Integral(x**6/((a*x - 1)**4*(a*x + 1)**4*atanh(a*x)), x)
 

Maxima [F]

\[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=\int { \frac {x^{6}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )} \,d x } \] Input:

integrate(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x, algorithm="maxima")
 

Output:

integrate(x^6/((a^2*x^2 - 1)^4*arctanh(a*x)), x)
 

Giac [F]

\[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=\int { \frac {x^{6}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )} \,d x } \] Input:

integrate(x^6/(-a^2*x^2+1)^4/arctanh(a*x),x, algorithm="giac")
 

Output:

integrate(x^6/((a^2*x^2 - 1)^4*arctanh(a*x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=\int \frac {x^6}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^4} \,d x \] Input:

int(x^6/(atanh(a*x)*(a^2*x^2 - 1)^4),x)
 

Output:

int(x^6/(atanh(a*x)*(a^2*x^2 - 1)^4), x)
 

Reduce [F]

\[ \int \frac {x^6}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)} \, dx=\frac {3 \left (\int \frac {x^{4}}{\mathit {atanh} \left (a x \right ) a^{8} x^{8}-4 \mathit {atanh} \left (a x \right ) a^{6} x^{6}+6 \mathit {atanh} \left (a x \right ) a^{4} x^{4}-4 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+\mathit {atanh} \left (a x \right )}d x \right ) a^{5}-3 \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right ) a^{8} x^{8}-4 \mathit {atanh} \left (a x \right ) a^{6} x^{6}+6 \mathit {atanh} \left (a x \right ) a^{4} x^{4}-4 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+\mathit {atanh} \left (a x \right )}d x \right ) a^{3}+\left (\int \frac {1}{\mathit {atanh} \left (a x \right ) a^{8} x^{8}-4 \mathit {atanh} \left (a x \right ) a^{6} x^{6}+6 \mathit {atanh} \left (a x \right ) a^{4} x^{4}-4 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+\mathit {atanh} \left (a x \right )}d x \right ) a -\mathrm {log}\left (\mathit {atanh} \left (a x \right )\right )}{a^{7}} \] Input:

int(x^6/(-a^2*x^2+1)^4/atanh(a*x),x)
 

Output:

(3*int(x**4/(atanh(a*x)*a**8*x**8 - 4*atanh(a*x)*a**6*x**6 + 6*atanh(a*x)* 
a**4*x**4 - 4*atanh(a*x)*a**2*x**2 + atanh(a*x)),x)*a**5 - 3*int(x**2/(ata 
nh(a*x)*a**8*x**8 - 4*atanh(a*x)*a**6*x**6 + 6*atanh(a*x)*a**4*x**4 - 4*at 
anh(a*x)*a**2*x**2 + atanh(a*x)),x)*a**3 + int(1/(atanh(a*x)*a**8*x**8 - 4 
*atanh(a*x)*a**6*x**6 + 6*atanh(a*x)*a**4*x**4 - 4*atanh(a*x)*a**2*x**2 + 
atanh(a*x)),x)*a - log(atanh(a*x)))/a**7