\(\int \frac {(a+b \text {arctanh}(c x^2))^2}{x^3} \, dx\) [69]

Optimal result

Integrand size = 16, antiderivative size = 87 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=\frac {1}{2} c \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{2 x^2}+b c \left (a+b \text {arctanh}\left (c x^2\right )\right ) \log \left (2-\frac {2}{1+c x^2}\right )-\frac {1}{2} b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x^2}\right ) \] Output:

1/2*c*(a+b*arctanh(c*x^2))^2-1/2*(a+b*arctanh(c*x^2))^2/x^2+b*c*(a+b*arcta 
nh(c*x^2))*ln(2-2/(c*x^2+1))-1/2*b^2*c*polylog(2,-1+2/(c*x^2+1))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=-\frac {a^2}{2 x^2}+a b c \left (-\frac {\text {arctanh}\left (c x^2\right )}{c x^2}+\log \left (c x^2\right )-\frac {1}{2} \log \left (1-c^2 x^4\right )\right )+\frac {1}{2} b^2 c \left (\text {arctanh}\left (c x^2\right ) \left (\text {arctanh}\left (c x^2\right )-\frac {\text {arctanh}\left (c x^2\right )}{c x^2}+2 \log \left (1-e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right ) \] Input:

Integrate[(a + b*ArcTanh[c*x^2])^2/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, 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.

\(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx\)

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

Int[(a + b*ArcTanh[c*x^2])^2/x^3,x]
 

Output:

(-((a + b*ArcTanh[c*x^2])^2/x^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 = 0.86 (sec) , antiderivative size = 820, normalized size of antiderivative = 9.43

method	result	size
default	\(\text {Expression too large to display}\)	\(820\)
parts	\(\text {Expression too large to display}\)	\(820\)

Input:

int((a+b*arctanh(c*x^2))^2/x^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2/x^2*a^2+b^2*(-1/2/x^2*arctanh(c*x^2)^2+2*c*(ln(x)*arctanh(c*x^2)-1/4* 
arctanh(c*x^2)*ln(c*x^2+1)-1/4*arctanh(c*x^2)*ln(c*x^2-1)-1/2*c*(Sum(-1/4* 
(ln(x-_alpha)*ln(c*x^2-1)-2*c*(1/2*ln(x-_alpha)*(ln((RootOf(_Z^2*c+2*_Z*_a 
lpha*c-2,index=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)-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)-x+_alph 
a)/RootOf(_Z^2*c+2*_Z*_alpha*c-2,index=1))+dilog((RootOf(_Z^2*c+2*_Z*_alph 
a*c-2,index=2)-x+_alpha)/RootOf(_Z^2*c+2*_Z*_alpha*c-2,index=2)))/c))/c,_a 
lpha=RootOf(_Z^2*c+1))+Sum(1/4*(ln(x-_alpha)*ln(c*x^2-1)-2*c*(1/4/_alpha/c 
*ln(x-_alpha)^2-1/2*_alpha*ln(x-_alpha)*ln(1/2*(x+_alpha)/_alpha)-1/2*_alp 
ha*dilog(1/2*(x+_alpha)/_alpha)))/c,_alpha=RootOf(_Z^2*c-1))+Sum(-1/4*(ln( 
x-_alpha)*ln(c*x^2+1)-2*c*(1/4/_alpha/c*ln(x-_alpha)^2+1/2*_alpha*ln(x-_al 
pha)*ln(1/2*(x+_alpha)/_alpha)+1/2*_alpha*dilog(1/2*(x+_alpha)/_alpha)))/c 
,_alpha=RootOf(_Z^2*c+1))+Sum(1/4*(ln(x-_alpha)*ln(c*x^2+1)-2*c*(1/2*ln(x- 
_alpha)*(ln((RootOf(_Z^2*c+2*_Z*_alpha*c+2,index=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)-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)-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)-x+_alpha)/RootOf(_Z^2*c+2 
*_Z*_alpha*c+2,index=2)))/c))/c,_alpha=RootOf(_Z^2*c-1))+ln(x)*(ln(1+(-...
 

Fricas [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/x^3,x, algorithm="fricas")
 

Output:

integral((b^2*arctanh(c*x^2)^2 + 2*a*b*arctanh(c*x^2) + a^2)/x^3, x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{x^{3}}\, dx \] Input:

integrate((a+b*atanh(c*x**2))**2/x**3,x)
 

Output:

Integral((a + b*atanh(c*x**2))**2/x**3, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/x^3,x, algorithm="maxima")
 

Output:

-1/2*(c*(log(c^2*x^4 - 1) - log(x^4)) + 2*arctanh(c*x^2)/x^2)*a*b - 1/8*b^ 
2*(log(-c*x^2 + 1)^2/x^2 + 2*integrate(-((c*x^2 - 1)*log(c*x^2 + 1)^2 + 2* 
(c*x^2 - (c*x^2 - 1)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^5 - x^3), x)) - 
 1/2*a^2/x^2
 

Giac [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/x^3,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^2) + a)^2/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{x^3} \,d x \] Input:

int((a + b*atanh(c*x^2))^2/x^3,x)
 

Output:

int((a + b*atanh(c*x^2))^2/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^3} \, dx=\frac {-\mathit {atanh} \left (c \,x^{2}\right )^{2} b^{2}+2 \mathit {atanh} \left (c \,x^{2}\right ) a b c \,x^{2}-2 \mathit {atanh} \left (c \,x^{2}\right ) a b -4 \left (\int \frac {\mathit {atanh} \left (c \,x^{2}\right )}{c^{2} x^{5}-x}d x \right ) b^{2} c \,x^{2}-2 \,\mathrm {log}\left (c \,x^{2}+1\right ) a b c \,x^{2}+4 \,\mathrm {log}\left (x \right ) a b c \,x^{2}-a^{2}}{2 x^{2}} \] Input:

int((a+b*atanh(c*x^2))^2/x^3,x)
 

Output:

( - atanh(c*x**2)**2*b**2 + 2*atanh(c*x**2)*a*b*c*x**2 - 2*atanh(c*x**2)*a 
*b - 4*int(atanh(c*x**2)/(c**2*x**5 - x),x)*b**2*c*x**2 - 2*log(c*x**2 + 1 
)*a*b*c*x**2 + 4*log(x)*a*b*c*x**2 - a**2)/(2*x**2)