\(\int x (a+b \text {arctanh}(c x^2))^2 \, dx\) [67]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int x \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\frac {b^2 \left (-1+c x^2\right ) \text {arctanh}\left (c x^2\right )^2+2 b \text {arctanh}\left (c x^2\right ) \left (a c x^2-b \log \left (1+e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )+a \left (a c x^2+b \log \left (1-c^2 x^4\right )\right )+b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )}{2 c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6454, 6436, 6546, 6470, 2849, 2752}

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.

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

\(\Big \downarrow \) 6454

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6436

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTanh[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 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 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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 6546

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

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {c \,x^{2} a^{2}+b^{2} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{2} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )\right )+2 a b c \,x^{2} \operatorname {arctanh}\left (c \,x^{2}\right )+a b \ln \left (-c^{2} x^{4}+1\right )}{2 c}\)	\(134\)
default	\(\frac {c \,x^{2} a^{2}+b^{2} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{2} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )\right )+2 a b c \,x^{2} \operatorname {arctanh}\left (c \,x^{2}\right )+a b \ln \left (-c^{2} x^{4}+1\right )}{2 c}\)	\(134\)
parts	\(\frac {a^{2} x^{2}}{2}+\frac {b^{2} \left (\operatorname {arctanh}\left (c \,x^{2}\right )^{2} \left (c \,x^{2}-1\right )+2 \operatorname {arctanh}\left (c \,x^{2}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{2}\right ) \ln \left (1+\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )-\operatorname {polylog}\left (2, -\frac {\left (c \,x^{2}+1\right )^{2}}{-c^{2} x^{4}+1}\right )\right )}{2 c}+\frac {a b \left (c \,x^{2} \operatorname {arctanh}\left (c \,x^{2}\right )+\frac {\ln \left (-c^{2} x^{4}+1\right )}{2}\right )}{c}\)	\(136\)
risch	\(\frac {b^{2} \ln \left (c \,x^{2}+1\right )^{2}}{8 c}+\frac {b a \ln \left (c \,x^{2}+1\right ) x^{2}}{2}+\frac {b a \ln \left (c \,x^{2}+1\right )}{2 c}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right ) \ln \left (c \,x^{2}+1\right ) x^{2}}{4}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right ) \ln \left (c \,x^{2}+1\right )}{4 c}+\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{2}}{2}\right ) \ln \left (c \,x^{2}+1\right )}{2 c}-\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{2}}{2}\right ) \ln \left (\frac {c \,x^{2}}{2}+\frac {1}{2}\right )}{2 c}+\frac {a^{2} x^{2}}{2}-\frac {a b}{c}-\frac {\ln \left (-c \,x^{2}+1\right ) a b \,x^{2}}{2}+\frac {\ln \left (-c \,x^{2}+1\right ) a b}{2 c}+\frac {b^{2} \ln \left (c \,x^{2}+1\right )^{2} x^{2}}{8}+\frac {\ln \left (-c \,x^{2}+1\right )^{2} b^{2} x^{2}}{8}-\frac {\ln \left (-c \,x^{2}+1\right )^{2} b^{2}}{8 c}+\frac {\ln \left (-c \,x^{2}+1\right ) b^{2}}{2 c}-\frac {b^{2}}{2 c}-\frac {a^{2}}{2 c}-\frac {b^{2} \operatorname {dilog}\left (\frac {c \,x^{2}}{2}+\frac {1}{2}\right )}{2 c}-\frac {b^{2} \ln \left (c \,x^{2}-1\right )}{2 c}\)	\(320\)

Input:

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

Output:

1/2/c*(c*x^2*a^2+b^2*(arctanh(c*x^2)^2*(c*x^2-1)+2*arctanh(c*x^2)^2-2*arct 
anh(c*x^2)*ln(1+(c*x^2+1)^2/(-c^2*x^4+1))-polylog(2,-(c*x^2+1)^2/(-c^2*x^4 
+1)))+2*a*b*c*x^2*arctanh(c*x^2)+a*b*ln(-c^2*x^4+1))
 

Fricas [F]

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

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*a^2*x^2 + 1/8*(x^2*log(-c*x^2 + 1)^2 - c^2*(2*x^2/c^2 - log(c*x^2 + 1) 
/c^3 + log(c*x^2 - 1)/c^3) - 2*c*(x^2/c + log(c*x^2 - 1)/c^2)*log(-c*x^2 + 
 1) + 12*c*integrate(x^3*log(c*x^2 + 1)/(c^2*x^4 - 1), x) + (c*x^2*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 + (2 
*c*x^2 + log(c*x^2 - 1)^2 + 2*log(c*x^2 - 1))/c - log(c^2*x^4 - 1)/c + 4*i 
ntegrate(x*log(c*x^2 + 1)/(c^2*x^4 - 1), x))*b^2 + 1/2*(2*c*x^2*arctanh(c* 
x^2) + log(-c^2*x^4 + 1))*a*b/c
 

Giac [F]

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

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

Output:

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

Reduce [F]

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

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

Output:

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