\(\int \frac {(a+b \text {arctanh}(c x^2))^2}{x^4} \, dx\) [75]

Optimal result

Integrand size = 16, antiderivative size = 1102 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx =\text {Too large to display} \] Output:

-1/12*b^2*ln(c*x^2+1)^2/x^3+4/3*b^2*c^(3/2)*arctan(c^(1/2)*x)-1/3*b^2*c^(3 
/2)*polylog(2,1-2/(1+c^(1/2)*x))-1/3*b^2*c^(3/2)*polylog(2,1-2/(1-c^(1/2)* 
x))+4/3*b^2*c^(3/2)*arctanh(c^(1/2)*x)+1/3*b^2*c^(3/2)*arctanh(c^(1/2)*x)^ 
2+1/6*b^2*c^(3/2)*polylog(2,1-2*c^(1/2)*(1+(-c)^(1/2)*x)/((-c)^(1/2)+c^(1/ 
2))/(1+c^(1/2)*x))+1/6*b^2*c^(3/2)*polylog(2,1+2*c^(1/2)*(1-(-c)^(1/2)*x)/ 
((-c)^(1/2)-c^(1/2))/(1+c^(1/2)*x))-1/3*b^2*c^(3/2)*arctan(c^(1/2)*x)*ln(c 
*x^2+1)-1/3*b^2*c^(3/2)*arctanh(c^(1/2)*x)*ln(2*c^(1/2)*(1+(-c)^(1/2)*x)/( 
(-c)^(1/2)+c^(1/2))/(1+c^(1/2)*x))-1/3*I*b^2*c^(3/2)*arctan(c^(1/2)*x)^2-1 
/3*I*b^2*c^(3/2)*polylog(2,1-2/(1+I*c^(1/2)*x))-1/3*I*b^2*c^(3/2)*polylog( 
2,1-2/(1-I*c^(1/2)*x))-1/3*a*b*ln(c*x^2+1)/x^3-2/3*b^2*c*ln(c*x^2+1)/x+1/6 
*b^2*ln(-c*x^2+1)*ln(c*x^2+1)/x^3+1/3*b^2*c*ln(-c*x^2+1)/x-1/3*b*c*(2*a-b* 
ln(-c*x^2+1))/x-2/3*a*b*c/x-1/12*(2*a-b*ln(-c*x^2+1))^2/x^3+1/6*I*b^2*c^(3 
/2)*polylog(2,1+(-1+I)*(1+c^(1/2)*x)/(1-I*c^(1/2)*x))+1/6*I*b^2*c^(3/2)*po 
lylog(2,1-(1+I)*(1-c^(1/2)*x)/(1-I*c^(1/2)*x))+1/3*b*c^(3/2)*arctanh(c^(1/ 
2)*x)*(2*a-b*ln(-c*x^2+1))+1/3*b^2*c^(3/2)*arctan(c^(1/2)*x)*ln(-c*x^2+1)- 
1/3*b^2*c^(3/2)*arctanh(c^(1/2)*x)*ln(-2*c^(1/2)*(1-(-c)^(1/2)*x)/((-c)^(1 
/2)-c^(1/2))/(1+c^(1/2)*x))+2/3*b^2*c^(3/2)*arctanh(c^(1/2)*x)*ln(2/(1+c^( 
1/2)*x))-2/3*b^2*c^(3/2)*arctanh(c^(1/2)*x)*ln(2/(1-c^(1/2)*x))-1/3*b^2*c^ 
(3/2)*arctan(c^(1/2)*x)*ln((1+I)*(1-c^(1/2)*x)/(1-I*c^(1/2)*x))-1/3*b^2*c^ 
(3/2)*arctan(c^(1/2)*x)*ln((1-I)*(1+c^(1/2)*x)/(1-I*c^(1/2)*x))-2/3*b^2...
 

Mathematica [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx=\int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx \] Input:

Integrate[(a + b*ArcTanh[c*x^2])^2/x^4,x]
 

Output:

Integrate[(a + b*ArcTanh[c*x^2])^2/x^4, x]
 

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 1102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6456, 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.

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

\(\Big \downarrow \) 6456

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcTanh[c*x^2])^2/x^4,x]
 

Output:

(-2*a*b*c)/(3*x) - (2*a*b*c^(3/2)*ArcTan[Sqrt[c]*x])/3 + (4*b^2*c^(3/2)*Ar 
cTan[Sqrt[c]*x])/3 - (I/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]^2 + (4*b^2*c^(3/2 
)*ArcTanh[Sqrt[c]*x])/3 + (b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]^2)/3 - (2*b^2*c^ 
(3/2)*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/3 + (2*b^2*c^(3/2)*ArcTan 
[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/3 - (b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*L 
og[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/3 - (2*b^2*c^(3/2)*ArcTan 
[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/3 + (2*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x 
]*Log[2/(1 + Sqrt[c]*x)])/3 - (b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt 
[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/3 - (b^2*c^ 
(3/2)*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqr 
t[c])*(1 + Sqrt[c]*x))])/3 - (b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[((1 - I)*( 
1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/3 + (b^2*c*Log[1 - c*x^2])/(3*x) + (b^ 
2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/3 - (b*c*(2*a - b*Log[1 - c*x^ 
2]))/(3*x) + (b*c^(3/2)*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/3 - ( 
2*a - b*Log[1 - c*x^2])^2/(12*x^3) - (a*b*Log[1 + c*x^2])/(3*x^3) - (2*b^2 
*c*Log[1 + c*x^2])/(3*x) - (b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/ 
3 + (b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/3 + (b^2*Log[1 - c*x^2 
]*Log[1 + c*x^2])/(6*x^3) - (b^2*Log[1 + c*x^2]^2)/(12*x^3) - (b^2*c^(3/2) 
*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/3 - (I/3)*b^2*c^(3/2)*PolyLog[2, 1 - 2 
/(1 - I*Sqrt[c]*x)] + (I/6)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(1 - Sq...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6456

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

Maple [F]

\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}^{2}}{x^{4}}d x\]

Input:

int((a+b*arctanh(c*x^2))^2/x^4,x)
 

Output:

int((a+b*arctanh(c*x^2))^2/x^4,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/x^4,x, algorithm="fricas")
 

Output:

integral((b^2*arctanh(c*x^2)^2 + 2*a*b*arctanh(c*x^2) + a^2)/x^4, x)
 

Sympy [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{x^{4}}\, dx \] Input:

integrate((a+b*atanh(c*x**2))**2/x**4,x)
 

Output:

Integral((a + b*atanh(c*x**2))**2/x**4, x)
 

Maxima [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/x^4,x, algorithm="maxima")
 

Output:

-1/3*((2*sqrt(c)*arctan(sqrt(c)*x) + sqrt(c)*log((c*x - sqrt(c))/(c*x + sq 
rt(c))) + 4/x)*c + 2*arctanh(c*x^2)/x^3)*a*b - 1/12*b^2*(log(-c*x^2 + 1)^2 
/x^3 + 3*integrate(-1/3*(3*(c*x^2 - 1)*log(c*x^2 + 1)^2 + 2*(2*c*x^2 - 3*( 
c*x^2 - 1)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^6 - x^4), x)) - 1/3*a^2/x 
^3
 

Giac [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/x^4,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^2) + a)^2/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{x^4} \,d x \] Input:

int((a + b*atanh(c*x^2))^2/x^4,x)
 

Output:

int((a + b*atanh(c*x^2))^2/x^4, x)
 

Reduce [F]

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

int((a+b*atanh(c*x^2))^2/x^4,x)
 

Output:

( - 2*sqrt(c)*atan((c*x)/sqrt(c))*a*b*c*x**3 - 2*sqrt(c)*atanh(c*x**2)*a*b 
*c*x**3 - 2*atanh(c*x**2)*a*b - 2*sqrt(c)*log(sqrt(c)*x - 1)*a*b*c*x**3 + 
sqrt(c)*log(c*x**2 + 1)*a*b*c*x**3 + 3*int(atanh(c*x**2)**2/x**4,x)*b**2*x 
**3 - a**2 - 4*a*b*c*x**2)/(3*x**3)