\(\int \frac {1}{(1-a^2 x^2)^{9/2} \text {arctanh}(a x)} \, dx\) [484]

Optimal result

Integrand size = 21, antiderivative size = 55 \[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)} \, dx=\frac {35 \text {Chi}(\text {arctanh}(a x))}{64 a}+\frac {21 \text {Chi}(3 \text {arctanh}(a x))}{64 a}+\frac {7 \text {Chi}(5 \text {arctanh}(a x))}{64 a}+\frac {\text {Chi}(7 \text {arctanh}(a x))}{64 a} \] Output:

35/64*Chi(arctanh(a*x))/a+21/64*Chi(3*arctanh(a*x))/a+7/64*Chi(5*arctanh(a 
*x))/a+1/64*Chi(7*arctanh(a*x))/a
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \text {arctanh}(a x)} \, dx=-\frac {-35 \text {Chi}(\text {arctanh}(a x))-21 \text {Chi}(3 \text {arctanh}(a x))-7 \text {Chi}(5 \text {arctanh}(a x))-\text {Chi}(7 \text {arctanh}(a x))}{64 a} \] Input:

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

Output:

-1/64*(-35*CoshIntegral[ArcTanh[a*x]] - 21*CoshIntegral[3*ArcTanh[a*x]] - 
7*CoshIntegral[5*ArcTanh[a*x]] - CoshIntegral[7*ArcTanh[a*x]])/a
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.

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

\(\Big \downarrow \) 6530

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^7}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {21 \cosh (3 \text {arctanh}(a x))}{64 \text {arctanh}(a x)}+\frac {7 \cosh (5 \text {arctanh}(a x))}{64 \text {arctanh}(a x)}+\frac {\cosh (7 \text {arctanh}(a x))}{64 \text {arctanh}(a x)}+\frac {35}{64 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {35}{64} \text {Chi}(\text {arctanh}(a x))+\frac {21}{64} \text {Chi}(3 \text {arctanh}(a x))+\frac {7}{64} \text {Chi}(5 \text {arctanh}(a x))+\frac {1}{64} \text {Chi}(7 \text {arctanh}(a x))}{a}\)

Input:

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

Output:

((35*CoshIntegral[ArcTanh[a*x]])/64 + (21*CoshIntegral[3*ArcTanh[a*x]])/64 
 + (7*CoshIntegral[5*ArcTanh[a*x]])/64 + CoshIntegral[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 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 6530

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])
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71

method	result	size
default	\(\frac {35 \,\operatorname {Chi}\left (\operatorname {arctanh}\left (a x \right )\right )+21 \,\operatorname {Chi}\left (3 \,\operatorname {arctanh}\left (a x \right )\right )+7 \,\operatorname {Chi}\left (5 \,\operatorname {arctanh}\left (a x \right )\right )+\operatorname {Chi}\left (7 \,\operatorname {arctanh}\left (a x \right )\right )}{64 a}\)	\(39\)

Input:

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

Output:

1/64*(35*Chi(arctanh(a*x))+21*Chi(3*arctanh(a*x))+7*Chi(5*arctanh(a*x))+Ch 
i(7*arctanh(a*x)))/a
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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