\(\int \frac {\text {arctanh}(a x)}{(1-a^2 x^2)^{9/2}} \, dx\) [464]

Optimal result

Integrand size = 19, antiderivative size = 177 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16}{35 a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \text {arctanh}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \text {arctanh}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \text {arctanh}(a x)}{35 \sqrt {1-a^2 x^2}} \] Output:

-1/49/a/(-a^2*x^2+1)^(7/2)-6/175/a/(-a^2*x^2+1)^(5/2)-8/105/a/(-a^2*x^2+1) 
^(3/2)-16/35/a/(-a^2*x^2+1)^(1/2)+1/7*x*arctanh(a*x)/(-a^2*x^2+1)^(7/2)+6/ 
35*x*arctanh(a*x)/(-a^2*x^2+1)^(5/2)+8/35*x*arctanh(a*x)/(-a^2*x^2+1)^(3/2 
)+16/35*x*arctanh(a*x)/(-a^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.46 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\frac {-2161+5726 a^2 x^2-5320 a^4 x^4+1680 a^6 x^6-105 a x \left (-35+70 a^2 x^2-56 a^4 x^4+16 a^6 x^6\right ) \text {arctanh}(a x)}{3675 a \left (1-a^2 x^2\right )^{7/2}} \] Input:

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

Output:

(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6 - 105*a*x*(-35 + 70*a^ 
2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[a*x])/(3675*a*(1 - a^2*x^2)^(7/2) 
)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6522, 6522, 6522, 6520}

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.

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

\(\Big \downarrow \) 6522

\(\displaystyle \frac {6}{7} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 6522

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 6522

\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 6520

\(\displaystyle \frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\)

Input:

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

Output:

-1/49*1/(a*(1 - a^2*x^2)^(7/2)) + (x*ArcTanh[a*x])/(7*(1 - a^2*x^2)^(7/2)) 
 + (6*(-1/25*1/(a*(1 - a^2*x^2)^(5/2)) + (x*ArcTanh[a*x])/(5*(1 - a^2*x^2) 
^(5/2)) + (4*(-1/9*1/(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x])/(3*(1 - a^ 
2*x^2)^(3/2)) + (2*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - 
 a^2*x^2]))/3))/5))/7
 

Defintions of rubi rules used

rule 6520

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb 
ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* 
Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
 

rule 6522

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo 
l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + 
 e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 
2*d*(q + 1))   Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre 
eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
 

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.56

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (1680 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}-1680 a^{6} x^{6}-5880 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}+5320 a^{4} x^{4}+7350 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-5726 a^{2} x^{2}-3675 a x \,\operatorname {arctanh}\left (a x \right )+2161\right )}{3675 a \left (a^{2} x^{2}-1\right )^{4}}\)	\(99\)
orering	\(\frac {\left (\frac {32}{7} a^{8} x^{9}-\frac {96}{5} a^{6} x^{7}+\frac {5364}{175} a^{4} x^{5}-\frac {27336}{1225} a^{2} x^{3}+\frac {7708}{1225} x \right ) \operatorname {arctanh}\left (a x \right )}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {\left (1680 a^{6} x^{6}-5320 a^{4} x^{4}+5726 a^{2} x^{2}-2161\right ) \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {a}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}+\frac {9 \,\operatorname {arctanh}\left (a x \right ) a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}\right )}{3675 a^{2}}\)	\(137\)

Input:

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

Output:

-1/3675/a*(-a^2*x^2+1)^(1/2)*(1680*arctanh(a*x)*a^7*x^7-1680*a^6*x^6-5880* 
arctanh(a*x)*a^5*x^5+5320*a^4*x^4+7350*a^3*x^3*arctanh(a*x)-5726*a^2*x^2-3 
675*a*x*arctanh(a*x)+2161)/(a^2*x^2-1)^4
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\frac {{\left (3360 \, a^{6} x^{6} - 10640 \, a^{4} x^{4} + 11452 \, a^{2} x^{2} - 105 \, {\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 4322\right )} \sqrt {-a^{2} x^{2} + 1}}{7350 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \] Input:

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

Output:

1/7350*(3360*a^6*x^6 - 10640*a^4*x^4 + 11452*a^2*x^2 - 105*(16*a^7*x^7 - 5 
6*a^5*x^5 + 70*a^3*x^3 - 35*a*x)*log(-(a*x + 1)/(a*x - 1)) - 4322)*sqrt(-a 
^2*x^2 + 1)/(a^9*x^8 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + a)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.79 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {1}{3675} \, a {\left (\frac {1680}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {280}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {126}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}} + \frac {75}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} a^{2}}\right )} + \frac {1}{35} \, {\left (\frac {16 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {6 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} + \frac {5 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

-1/3675*a*(1680/(sqrt(-a^2*x^2 + 1)*a^2) + 280/((-a^2*x^2 + 1)^(3/2)*a^2) 
+ 126/((-a^2*x^2 + 1)^(5/2)*a^2) + 75/((-a^2*x^2 + 1)^(7/2)*a^2)) + 1/35*( 
16*x/sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^2 + 1)^(3/2) + 6*x/(-a^2*x^2 + 1)^(5 
/2) + 5*x/(-a^2*x^2 + 1)^(7/2))*arctanh(a*x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.78 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (2 \, {\left (4 \, {\left (2 \, a^{6} x^{2} - 7 \, a^{4}\right )} x^{2} + 35 \, a^{2}\right )} x^{2} - 35\right )} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{70 \, {\left (a^{2} x^{2} - 1\right )}^{4}} - \frac {126 \, a^{2} x^{2} + 1680 \, {\left (a^{2} x^{2} - 1\right )}^{3} - 280 \, {\left (a^{2} x^{2} - 1\right )}^{2} - 201}{3675 \, {\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1} a} \] Input:

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

Output:

-1/70*sqrt(-a^2*x^2 + 1)*(2*(4*(2*a^6*x^2 - 7*a^4)*x^2 + 35*a^2)*x^2 - 35) 
*x*log(-(a*x + 1)/(a*x - 1))/(a^2*x^2 - 1)^4 - 1/3675*(126*a^2*x^2 + 1680* 
(a^2*x^2 - 1)^3 - 280*(a^2*x^2 - 1)^2 - 201)/((a^2*x^2 - 1)^3*sqrt(-a^2*x^ 
2 + 1)*a)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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