\(\int \frac {x^4}{(1-a^2 x^2)^3 \text {arctanh}(a x)^2} \, dx\) [330]

Optimal result

Integrand size = 22, antiderivative size = 53 \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\frac {x^4}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}-\frac {\text {Shi}(2 \text {arctanh}(a x))}{a^5}+\frac {\text {Shi}(4 \text {arctanh}(a x))}{2 a^5} \] Output:

-x^4/a/(-a^2*x^2+1)^2/arctanh(a*x)-Shi(2*arctanh(a*x))/a^5+1/2*Shi(4*arcta 
nh(a*x))/a^5
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\frac {-\frac {2 a^4 x^4}{\left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)}-2 \text {Shi}(2 \text {arctanh}(a x))+\text {Shi}(4 \text {arctanh}(a x))}{2 a^5} \] Input:

Integrate[x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]
 

Output:

((-2*a^4*x^4)/((-1 + a^2*x^2)^2*ArcTanh[a*x]) - 2*SinhIntegral[2*ArcTanh[a 
*x]] + SinhIntegral[4*ArcTanh[a*x]])/(2*a^5)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6568, 6596, 5971, 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.

\(\displaystyle \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx\)

\(\Big \downarrow \) 6568

\(\displaystyle \frac {4 \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx}{a}-\frac {x^4}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\)

\(\Big \downarrow \) 6596

\(\displaystyle \frac {4 \int \frac {a^3 x^3}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^5}-\frac {x^4}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {4 \int \left (\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}-\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^5}-\frac {x^4}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 \left (\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))-\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))\right )}{a^5}-\frac {x^4}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\)

Input:

Int[x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]
 

Output:

-(x^4/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (4*(-1/4*SinhIntegral[2*ArcTanh[ 
a*x]] + SinhIntegral[4*ArcTanh[a*x]]/8))/a^5
 

Defintions of rubi rules used

rule 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6568

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]
 

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 = 1.51 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(\frac {-\frac {3}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}-\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{2}}{a^{5}}\)	\(62\)
default	\(\frac {-\frac {3}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}-\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{2}}{a^{5}}\)	\(62\)

Input:

int(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^5*(-3/8/arctanh(a*x)+1/2/arctanh(a*x)*cosh(2*arctanh(a*x))-Shi(2*arcta 
nh(a*x))-1/8/arctanh(a*x)*cosh(4*arctanh(a*x))+1/2*Shi(4*arctanh(a*x)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (50) = 100\).

Time = 0.08 (sec) , antiderivative size = 232, normalized size of antiderivative = 4.38 \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\frac {8 \, a^{4} x^{4} - {\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{4 \, {\left (a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )} \] Input:

integrate(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="fricas")
 

Output:

-1/4*(8*a^4*x^4 - ((a^4*x^4 - 2*a^2*x^2 + 1)*log_integral((a^2*x^2 + 2*a*x 
 + 1)/(a^2*x^2 - 2*a*x + 1)) - (a^4*x^4 - 2*a^2*x^2 + 1)*log_integral((a^2 
*x^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x + 1)) - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log 
_integral(-(a*x + 1)/(a*x - 1)) + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log_integral 
(-(a*x - 1)/(a*x + 1)))*log(-(a*x + 1)/(a*x - 1)))/((a^9*x^4 - 2*a^7*x^2 + 
 a^5)*log(-(a*x + 1)/(a*x - 1)))
 

Sympy [F]

\[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=- \int \frac {x^{4}}{a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/(-a**2*x**2+1)**3/atanh(a*x)**2,x)
 

Output:

-Integral(x**4/(a**6*x**6*atanh(a*x)**2 - 3*a**4*x**4*atanh(a*x)**2 + 3*a* 
*2*x**2*atanh(a*x)**2 - atanh(a*x)**2), x)
 

Maxima [F]

\[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{4}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="maxima")
 

Output:

-2*x^4/((a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1) - (a^5*x^4 - 2*a^3*x^2 + a) 
*log(-a*x + 1)) + 8*integrate(-x^3/((a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)* 
log(a*x + 1) - (a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log(-a*x + 1)), x)
 

Giac [F]

\[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{4}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="giac")
 

Output:

integrate(-x^4/((a^2*x^2 - 1)^3*arctanh(a*x)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\int \frac {x^4}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:

int(-x^4/(atanh(a*x)^2*(a^2*x^2 - 1)^3),x)
 

Output:

-int(x^4/(atanh(a*x)^2*(a^2*x^2 - 1)^3), x)
 

Reduce [F]

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

int(x^4/(-a^2*x^2+1)^3/atanh(a*x)^2,x)
                                                                                    
                                                                                    
 

Output:

( - 4*atanh(a*x)*int(x**3/(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)*a**6*x**2 + 4*atanh(a*x)*int(x**3 
/(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)*a**4 + 4*atanh(a*x)*int(x/(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)*a**4*x**2 - 4*ata 
nh(a*x)*int(x/(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)*a**2 - atanh(a*x)*int(1/(atanh(a*x)**2*a**6*x 
**6 - 3*atanh(a*x)**2*a**4*x**4 + 3*atanh(a*x)**2*a**2*x**2 - atanh(a*x)** 
2),x)*a**3*x**2 + atanh(a*x)*int(1/(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x) 
**2*a**4*x**4 + 3*atanh(a*x)**2*a**2*x**2 - atanh(a*x)**2),x)*a - a**2*x** 
2 - 1)/(atanh(a*x)*a**5*(a**2*x**2 - 1))