\(\int \frac {1}{(1-a^2 x^2)^{9/2} \text {arctanh}(a x)^2} \, dx\) [490]

Optimal result

Integrand size = 21, antiderivative size = 80 \[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \text {arctanh}(a x)}+\frac {35 \text {Shi}(\text {arctanh}(a x))}{64 a}+\frac {63 \text {Shi}(3 \text {arctanh}(a x))}{64 a}+\frac {35 \text {Shi}(5 \text {arctanh}(a x))}{64 a}+\frac {7 \text {Shi}(7 \text {arctanh}(a x))}{64 a} \] Output:

-1/a/(-a^2*x^2+1)^(7/2)/arctanh(a*x)+35/64*Shi(arctanh(a*x))/a+63/64*Shi(3 
*arctanh(a*x))/a+35/64*Shi(5*arctanh(a*x))/a+7/64*Shi(7*arctanh(a*x))/a
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=\frac {-\frac {64}{\left (1-a^2 x^2\right )^{7/2} \text {arctanh}(a x)}+35 \text {Shi}(\text {arctanh}(a x))+63 \text {Shi}(3 \text {arctanh}(a x))+35 \text {Shi}(5 \text {arctanh}(a x))+7 \text {Shi}(7 \text {arctanh}(a x))}{64 a} \] Input:

Integrate[1/((1 - a^2*x^2)^(9/2)*ArcTanh[a*x]^2),x]
 

Output:

(-64/((1 - a^2*x^2)^(7/2)*ArcTanh[a*x]) + 35*SinhIntegral[ArcTanh[a*x]] + 
63*SinhIntegral[3*ArcTanh[a*x]] + 35*SinhIntegral[5*ArcTanh[a*x]] + 7*Sinh 
Integral[7*ArcTanh[a*x]])/(64*a)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6528, 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.

\(\displaystyle \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx\)

\(\Big \downarrow \) 6528

\(\displaystyle 7 a \int \frac {x}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \text {arctanh}(a x)}\)

\(\Big \downarrow \) 6596

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {7 \int \left (\frac {5 a x}{64 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}+\frac {9 \sinh (3 \text {arctanh}(a x))}{64 \text {arctanh}(a x)}+\frac {5 \sinh (5 \text {arctanh}(a x))}{64 \text {arctanh}(a x)}+\frac {\sinh (7 \text {arctanh}(a x))}{64 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \text {arctanh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {7 \left (\frac {5}{64} \text {Shi}(\text {arctanh}(a x))+\frac {9}{64} \text {Shi}(3 \text {arctanh}(a x))+\frac {5}{64} \text {Shi}(5 \text {arctanh}(a x))+\frac {1}{64} \text {Shi}(7 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \text {arctanh}(a x)}\)

Input:

Int[1/((1 - a^2*x^2)^(9/2)*ArcTanh[a*x]^2),x]
 

Output:

-(1/(a*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x])) + (7*((5*SinhIntegral[ArcTanh[a* 
x]])/64 + (9*SinhIntegral[3*ArcTanh[a*x]])/64 + (5*SinhIntegral[5*ArcTanh[ 
a*x]])/64 + SinhIntegral[7*ArcTanh[a*x]]/64))/a
 

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 6528

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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(231\) vs. \(2(70)=140\).

Time = 1.15 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.90

method	result	size
default	\(\frac {35 \,\operatorname {arctanh}\left (a x \right ) \operatorname {Shi}\left (\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}+63 \,\operatorname {arctanh}\left (a x \right ) \operatorname {Shi}\left (3 \,\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}+35 \,\operatorname {arctanh}\left (a x \right ) \operatorname {Shi}\left (5 \,\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}+7 \,\operatorname {arctanh}\left (a x \right ) \operatorname {Shi}\left (7 \,\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}-21 \cosh \left (3 \,\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}-7 \cosh \left (5 \,\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}-\cosh \left (7 \,\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}-35 \,\operatorname {Shi}\left (\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right )-63 \,\operatorname {Shi}\left (3 \,\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right )-35 \,\operatorname {Shi}\left (5 \,\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right )-7 \,\operatorname {Shi}\left (7 \,\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cosh \left (3 \,\operatorname {arctanh}\left (a x \right )\right )+7 \cosh \left (5 \,\operatorname {arctanh}\left (a x \right )\right )+\cosh \left (7 \,\operatorname {arctanh}\left (a x \right )\right )}{64 a \,\operatorname {arctanh}\left (a x \right ) \left (a^{2} x^{2}-1\right )}\)	\(232\)

Input:

int(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/64/a*(35*arctanh(a*x)*Shi(arctanh(a*x))*a^2*x^2+63*arctanh(a*x)*Shi(3*ar 
ctanh(a*x))*a^2*x^2+35*arctanh(a*x)*Shi(5*arctanh(a*x))*a^2*x^2+7*arctanh( 
a*x)*Shi(7*arctanh(a*x))*a^2*x^2-21*cosh(3*arctanh(a*x))*a^2*x^2-7*cosh(5* 
arctanh(a*x))*a^2*x^2-cosh(7*arctanh(a*x))*a^2*x^2-35*Shi(arctanh(a*x))*ar 
ctanh(a*x)-63*Shi(3*arctanh(a*x))*arctanh(a*x)-35*Shi(5*arctanh(a*x))*arct 
anh(a*x)-7*Shi(7*arctanh(a*x))*arctanh(a*x)+35*(-a^2*x^2+1)^(1/2)+21*cosh( 
3*arctanh(a*x))+7*cosh(5*arctanh(a*x))+cosh(7*arctanh(a*x)))/arctanh(a*x)/ 
(a^2*x^2-1)
 

Fricas [F]

\[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=\int { \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^2,x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)/((a^10*x^10 - 5*a^8*x^8 + 10*a^6*x^6 - 10*a^4 
*x^4 + 5*a^2*x^2 - 1)*arctanh(a*x)^2), x)
 

Sympy [F]

\[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=\int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \] Input:

integrate(1/(-a**2*x**2+1)**(9/2)/atanh(a*x)**2,x)
 

Output:

Integral(1/((-(a*x - 1)*(a*x + 1))**(9/2)*atanh(a*x)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=\int { \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^2,x, algorithm="maxima")
 

Output:

integrate(1/((-a^2*x^2 + 1)^(9/2)*arctanh(a*x)^2), x)
 

Giac [F]

\[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=\int { \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^2,x, algorithm="giac")
 

Output:

integrate(1/((-a^2*x^2 + 1)^(9/2)*arctanh(a*x)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=\int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \] Input:

int(1/(atanh(a*x)^2*(1 - a^2*x^2)^(9/2)),x)
 

Output:

int(1/(atanh(a*x)^2*(1 - a^2*x^2)^(9/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)^2} \, dx=\int \frac {1}{\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{2} a^{8} x^{8}-4 \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}+6 \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}-4 \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{2}}d x \] Input:

int(1/(-a^2*x^2+1)^(9/2)/atanh(a*x)^2,x)
 

Output:

int(1/(sqrt( - a**2*x**2 + 1)*atanh(a*x)**2*a**8*x**8 - 4*sqrt( - a**2*x** 
2 + 1)*atanh(a*x)**2*a**6*x**6 + 6*sqrt( - a**2*x**2 + 1)*atanh(a*x)**2*a* 
*4*x**4 - 4*sqrt( - a**2*x**2 + 1)*atanh(a*x)**2*a**2*x**2 + sqrt( - a**2* 
x**2 + 1)*atanh(a*x)**2),x)