\(\int x (a+b \text {arctanh}(\frac {c}{x^2}))^2 \, dx\) [172]

Optimal result

Integrand size = 14, antiderivative size = 94 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx=\frac {1}{2} c \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2+\frac {1}{2} x^2 \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2-b c \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right ) \log \left (\frac {2 c}{c-x^2}\right )-\frac {1}{2} b^2 c \operatorname {PolyLog}\left (2,1-\frac {2 c}{c-x^2}\right ) \] Output:

1/2*c*(a+b*arccoth(x^2/c))^2+1/2*x^2*(a+b*arccoth(x^2/c))^2-b*c*(a+b*arcco 
th(x^2/c))*ln(2*c/(-x^2+c))-1/2*b^2*c*polylog(2,1-2*c/(-x^2+c))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14 \[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx=\frac {1}{2} \left (b^2 \left (-c+x^2\right ) \text {arctanh}\left (\frac {c}{x^2}\right )^2+2 b \text {arctanh}\left (\frac {c}{x^2}\right ) \left (a x^2-b c \log \left (1-e^{-2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )\right )+a \left (a x^2+b c \log \left (1-\frac {c^2}{x^4}\right )-2 b c \log \left (\frac {c}{x^2}\right )\right )+b^2 c \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (\frac {c}{x^2}\right )}\right )\right ) \] Input:

Integrate[x*(a + b*ArcTanh[c/x^2])^2,x]
 

Output:

(b^2*(-c + x^2)*ArcTanh[c/x^2]^2 + 2*b*ArcTanh[c/x^2]*(a*x^2 - b*c*Log[1 - 
 E^(-2*ArcTanh[c/x^2])]) + a*(a*x^2 + b*c*Log[1 - c^2/x^4] - 2*b*c*Log[c/x 
^2]) + b^2*c*PolyLog[2, E^(-2*ArcTanh[c/x^2])])/2
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6454, 6452, 6550, 6494, 2897}

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.

\(\displaystyle \int x \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx\)

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6550

\(\displaystyle \frac {1}{2} \left (x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2-2 b c \left (\int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )}{\frac {c}{x^2}+1}d\frac {1}{x^2}+\frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{2 b}\right )\right )\)

\(\Big \downarrow \) 6494

\(\displaystyle \frac {1}{2} \left (x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2-2 b c \left (-b c \int \frac {\log \left (2-\frac {2}{\frac {c}{x^2}+1}\right )}{1-\frac {c^2}{x^4}}d\frac {1}{x^2}+\frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{\frac {c}{x^2}+1}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 2897

\(\displaystyle \frac {1}{2} \left (x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2-2 b c \left (\frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{\frac {c}{x^2}+1}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{\frac {c}{x^2}+1}-1\right )\right )\right )\)

Input:

Int[x*(a + b*ArcTanh[c/x^2])^2,x]
 

Output:

(x^2*(a + b*ArcTanh[c/x^2])^2 - 2*b*c*((a + b*ArcTanh[c/x^2])^2/(2*b) + (a 
 + b*ArcTanh[c/x^2])*Log[2 - 2/(1 + c/x^2)] - (b*PolyLog[2, -1 + 2/(1 + c/ 
x^2)])/2))/2
 

Defintions of rubi rules used

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6454

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x 
], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl 
ify[(m + 1)/n]]
 

rule 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] 
/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c 
^2*d^2 - e^2, 0]
 

rule 6550

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.80 (sec) , antiderivative size = 889, normalized size of antiderivative = 9.46

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(889\)
default	\(\text {Expression too large to display}\)	\(889\)
parts	\(\text {Expression too large to display}\)	\(889\)
risch	\(\text {Expression too large to display}\)	\(4795\)

Input:

int(x*(a+b*arctanh(c/x^2))^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*a^2*x^2-b^2*(-1/2*x^2*arctanh(c/x^2)^2+2*c*(-1/4*arctanh(c/x^2)*ln(1+c 
/x^2)-1/4*arctanh(c/x^2)*ln(c/x^2-1)+ln(1/x)*arctanh(c/x^2)-1/2*c*(Sum(-1/ 
4*(ln(1/x-_alpha)*ln(c/x^2-1)-2*c*(1/2*ln(1/x-_alpha)*(ln((RootOf(_Z^2*c+2 
*_Z*_alpha*c-2,index=1)-1/x+_alpha)/RootOf(_Z^2*c+2*_Z*_alpha*c-2,index=1) 
)+ln((RootOf(_Z^2*c+2*_Z*_alpha*c-2,index=2)-1/x+_alpha)/RootOf(_Z^2*c+2*_ 
Z*_alpha*c-2,index=2)))/c+1/2*(dilog((RootOf(_Z^2*c+2*_Z*_alpha*c-2,index= 
1)-1/x+_alpha)/RootOf(_Z^2*c+2*_Z*_alpha*c-2,index=1))+dilog((RootOf(_Z^2* 
c+2*_Z*_alpha*c-2,index=2)-1/x+_alpha)/RootOf(_Z^2*c+2*_Z*_alpha*c-2,index 
=2)))/c))/c,_alpha=RootOf(_Z^2*c+1))+Sum(1/4*(ln(1/x-_alpha)*ln(c/x^2-1)-2 
*c*(1/4/_alpha/c*ln(1/x-_alpha)^2-1/2*_alpha*ln(1/x-_alpha)*ln(1/2*(1/x+_a 
lpha)/_alpha)-1/2*_alpha*dilog(1/2*(1/x+_alpha)/_alpha)))/c,_alpha=RootOf( 
_Z^2*c-1))+Sum(-1/4*(ln(1/x-_alpha)*ln(1+c/x^2)-2*c*(1/4/_alpha/c*ln(1/x-_ 
alpha)^2+1/2*_alpha*ln(1/x-_alpha)*ln(1/2*(1/x+_alpha)/_alpha)+1/2*_alpha* 
dilog(1/2*(1/x+_alpha)/_alpha)))/c,_alpha=RootOf(_Z^2*c+1))+Sum(1/4*(ln(1/ 
x-_alpha)*ln(1+c/x^2)-2*c*(1/2*ln(1/x-_alpha)*(ln((RootOf(_Z^2*c+2*_Z*_alp 
ha*c+2,index=1)-1/x+_alpha)/RootOf(_Z^2*c+2*_Z*_alpha*c+2,index=1))+ln((Ro 
otOf(_Z^2*c+2*_Z*_alpha*c+2,index=2)-1/x+_alpha)/RootOf(_Z^2*c+2*_Z*_alpha 
*c+2,index=2)))/c+1/2*(dilog((RootOf(_Z^2*c+2*_Z*_alpha*c+2,index=1)-1/x+_ 
alpha)/RootOf(_Z^2*c+2*_Z*_alpha*c+2,index=1))+dilog((RootOf(_Z^2*c+2*_Z*_ 
alpha*c+2,index=2)-1/x+_alpha)/RootOf(_Z^2*c+2*_Z*_alpha*c+2,index=2)))...
 

Fricas [F]

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

integrate(x*(a+b*arctanh(c/x^2))^2,x, algorithm="fricas")
 

Output:

integral(b^2*x*arctanh(c/x^2)^2 + 2*a*b*x*arctanh(c/x^2) + a^2*x, x)
 

Sympy [F]

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

integrate(x*(a+b*atanh(c/x**2))**2,x)
 

Output:

Integral(x*(a + b*atanh(c/x**2))**2, x)
 

Maxima [F]

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

integrate(x*(a+b*arctanh(c/x^2))^2,x, algorithm="maxima")
 

Output:

1/2*a^2*x^2 + 1/2*(2*x^2*arctanh(c/x^2) + c*log(x^4 - c^2))*a*b + 1/8*(x^2 
*log(x^2 + c)^2 - 2*(x^2 + c)*log(x^2 + c)*log(x^2 - c) + (x^2 - c)*log(x^ 
2 - c)^2 + 2*integrate(2*(3*c*x^3 + c^2*x)*log(x^2 + c)/(x^4 - c^2), x))*b 
^2
 

Giac [F]

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

integrate(x*(a+b*arctanh(c/x^2))^2,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c/x^2) + a)^2*x, x)
 

Mupad [F(-1)]

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

int(x*(a + b*atanh(c/x^2))^2,x)
 

Output:

int(x*(a + b*atanh(c/x^2))^2, x)
 

Reduce [F]

\[ \int x \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx=-\mathit {atanh} \left (\frac {c}{x^{2}}\right ) a b c +\mathit {atanh} \left (\frac {c}{x^{2}}\right ) a b \,x^{2}+\left (\int \mathit {atanh} \left (\frac {c}{x^{2}}\right )^{2} x d x \right ) b^{2}+\mathrm {log}\left (x^{2}+c \right ) a b c +\frac {a^{2} x^{2}}{2} \] Input:

int(x*(a+b*atanh(c/x^2))^2,x)
 

Output:

( - 2*atanh(c/x**2)*a*b*c + 2*atanh(c/x**2)*a*b*x**2 + 2*int(atanh(c/x**2) 
**2*x,x)*b**2 + 2*log(c + x**2)*a*b*c + a**2*x**2)/2