\(\int (1-a^2 x^2)^{5/2} \text {arctanh}(a x) \, dx\) [459]

Optimal result

Integrand size = 19, antiderivative size = 233 \[ \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx=\frac {5 \sqrt {1-a^2 x^2}}{16 a}+\frac {5 \left (1-a^2 x^2\right )^{3/2}}{72 a}+\frac {\left (1-a^2 x^2\right )^{5/2}}{30 a}+\frac {5}{16} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {5}{24} x \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)-\frac {5 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{8 a}-\frac {5 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{16 a}+\frac {5 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{16 a} \] Output:

5/16*(-a^2*x^2+1)^(1/2)/a+5/72*(-a^2*x^2+1)^(3/2)/a+1/30*(-a^2*x^2+1)^(5/2 
)/a+5/16*x*(-a^2*x^2+1)^(1/2)*arctanh(a*x)+5/24*x*(-a^2*x^2+1)^(3/2)*arcta 
nh(a*x)+1/6*x*(-a^2*x^2+1)^(5/2)*arctanh(a*x)-5/8*arctan((-a*x+1)^(1/2)/(a 
*x+1)^(1/2))*arctanh(a*x)/a-5/16*I*polylog(2,-I*(-a*x+1)^(1/2)/(a*x+1)^(1/ 
2))/a+5/16*I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.96 \[ \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx=\frac {299 \sqrt {1-a^2 x^2}-98 a^2 x^2 \sqrt {1-a^2 x^2}+24 a^4 x^4 \sqrt {1-a^2 x^2}+495 a x \sqrt {1-a^2 x^2} \text {arctanh}(a x)-390 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+120 a^5 x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-225 i \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )+225 i \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-225 i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )+225 i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )}{720 a} \] Input:

Integrate[(1 - a^2*x^2)^(5/2)*ArcTanh[a*x],x]
 

Output:

(299*Sqrt[1 - a^2*x^2] - 98*a^2*x^2*Sqrt[1 - a^2*x^2] + 24*a^4*x^4*Sqrt[1 
- a^2*x^2] + 495*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 390*a^3*x^3*Sqrt[1 - 
 a^2*x^2]*ArcTanh[a*x] + 120*a^5*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (225 
*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (225*I)*ArcTanh[a*x]*Log[1 + 
I/E^ArcTanh[a*x]] - (225*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (225*I)*Poly 
Log[2, I/E^ArcTanh[a*x]])/(720*a)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6504, 6504, 6504, 6512}

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.

\(\displaystyle \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx\)

\(\Big \downarrow \) 6504

\(\displaystyle \frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right )^{5/2}}{30 a}\)

\(\Big \downarrow \) 6504

\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2} \text {arctanh}(a x)dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right )^{5/2}}{30 a}\)

\(\Big \downarrow \) 6504

\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {\sqrt {1-a^2 x^2}}{2 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right )^{5/2}}{30 a}\)

\(\Big \downarrow \) 6512

\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {\sqrt {1-a^2 x^2}}{2 a}+\frac {1}{2} \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)+\frac {\left (1-a^2 x^2\right )^{5/2}}{30 a}\)

Input:

Int[(1 - a^2*x^2)^(5/2)*ArcTanh[a*x],x]
 

Output:

(1 - a^2*x^2)^(5/2)/(30*a) + (x*(1 - a^2*x^2)^(5/2)*ArcTanh[a*x])/6 + (5*( 
(1 - a^2*x^2)^(3/2)/(12*a) + (x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/4 + (3*( 
Sqrt[1 - a^2*x^2]/(2*a) + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/2 + ((-2*ArcT 
an[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqrt 
[1 - a*x])/Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a* 
x]])/a)/2))/4))/6
 

Defintions of rubi rules used

rule 6504

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symb 
ol] :> Simp[b*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2)^q 
*((a + b*ArcTanh[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1))   Int[(d + e 
*x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && 
 EqQ[c^2*d + e, 0] && GtQ[q, 0]
 

rule 6512

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol 
] :> Simp[-2*(a + b*ArcTanh[c*x])*(ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*S 
qrt[d])), x] + (-Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/( 
c*Sqrt[d])), x] + Simp[I*b*(PolyLog[2, I*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/(c* 
Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 
0]
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\left (120 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}+24 a^{4} x^{4}-390 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-98 a^{2} x^{2}+495 a x \,\operatorname {arctanh}\left (a x \right )+299\right ) \sqrt {-a^{2} x^{2}+1}}{720 a}-\frac {5 i \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a}+\frac {5 i \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a}\)	\(193\)

Input:

int((-a^2*x^2+1)^(5/2)*arctanh(a*x),x,method=_RETURNVERBOSE)
 

Output:

1/720*(120*arctanh(a*x)*a^5*x^5+24*a^4*x^4-390*a^3*x^3*arctanh(a*x)-98*a^2 
*x^2+495*a*x*arctanh(a*x)+299)*(-a^2*x^2+1)^(1/2)/a-5/16*I/a*arctanh(a*x)* 
ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+5/16*I/a*arctanh(a*x)*ln(1-I*(a*x+1)/(- 
a^2*x^2+1)^(1/2))-5/16*I/a*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+5/16*I/a* 
dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

\[ \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right ) \,d x } \] Input:

integrate((-a^2*x^2+1)^(5/2)*arctanh(a*x),x, algorithm="fricas")
 

Output:

integral((a^4*x^4 - 2*a^2*x^2 + 1)*sqrt(-a^2*x^2 + 1)*arctanh(a*x), x)
 

Sympy [F]

\[ \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx=\int \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \operatorname {atanh}{\left (a x \right )}\, dx \] Input:

integrate((-a**2*x**2+1)**(5/2)*atanh(a*x),x)
 

Output:

Integral((-(a*x - 1)*(a*x + 1))**(5/2)*atanh(a*x), x)
 

Maxima [F]

\[ \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right ) \,d x } \] Input:

integrate((-a^2*x^2+1)^(5/2)*arctanh(a*x),x, algorithm="maxima")
 

Output:

integrate((-a^2*x^2 + 1)^(5/2)*arctanh(a*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*x^2+1)^(5/2)*arctanh(a*x),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x) \, dx=\int \mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{5/2} \,d x \] Input:

int(atanh(a*x)*(1 - a^2*x^2)^(5/2),x)
 

Output:

int(atanh(a*x)*(1 - a^2*x^2)^(5/2), x)
 

Reduce [F]

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

int((-a^2*x^2+1)^(5/2)*atanh(a*x),x)
 

Output:

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