\(\int \frac {x^2 (a+b \text {arctanh}(c x))}{d+c d x} \, dx\) [44]

Optimal result

Integrand size = 20, antiderivative size = 145 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{d+c d x} \, dx=-\frac {a x}{c^2 d}+\frac {b x}{2 c^2 d}-\frac {b \text {arctanh}(c x)}{2 c^3 d}-\frac {b x \text {arctanh}(c x)}{c^2 d}+\frac {x^2 (a+b \text {arctanh}(c x))}{2 c d}-\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{c^3 d}-\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 c^3 d} \] Output:

-a*x/c^2/d+1/2*b*x/c^2/d-1/2*b*arctanh(c*x)/c^3/d-b*x*arctanh(c*x)/c^2/d+1 
/2*x^2*(a+b*arctanh(c*x))/c/d-(a+b*arctanh(c*x))*ln(2/(c*x+1))/c^3/d-1/2*b 
*ln(-c^2*x^2+1)/c^3/d+1/2*b*polylog(2,1-2/(c*x+1))/c^3/d
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{d+c d x} \, dx=\frac {-2 a c x+b c x+a c^2 x^2+b \text {arctanh}(c x) \left (-1-2 c x+c^2 x^2-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+2 a \log (1+c x)-b \log \left (1-c^2 x^2\right )+b \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{2 c^3 d} \] Input:

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

Output:

(-2*a*c*x + b*c*x + a*c^2*x^2 + b*ArcTanh[c*x]*(-1 - 2*c*x + c^2*x^2 - 2*L 
og[1 + E^(-2*ArcTanh[c*x])]) + 2*a*Log[1 + c*x] - b*Log[1 - c^2*x^2] + b*P 
olyLog[2, -E^(-2*ArcTanh[c*x])])/(2*c^3*d)
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6492, 27, 6452, 262, 219, 6492, 2009, 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.

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

\(\Big \downarrow \) 6492

\(\displaystyle \frac {\int x (a+b \text {arctanh}(c x))dx}{c d}-\frac {\int \frac {x (a+b \text {arctanh}(c x))}{d (c x+1)}dx}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x (a+b \text {arctanh}(c x))dx}{c d}-\frac {\int \frac {x (a+b \text {arctanh}(c x))}{c x+1}dx}{c d}\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c d}-\frac {\int \frac {x (a+b \text {arctanh}(c x))}{c x+1}dx}{c d}\)

\(\Big \downarrow \) 6492

\(\displaystyle \frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c d}-\frac {\frac {\int (a+b \text {arctanh}(c x))dx}{c}-\frac {\int \frac {a+b \text {arctanh}(c x)}{c x+1}dx}{c}}{c d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c d}-\frac {\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c}-\frac {\int \frac {a+b \text {arctanh}(c x)}{c x+1}dx}{c}}{c d}\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c d}-\frac {\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c}-\frac {b \int \frac {\log \left (\frac {2}{c x+1}\right )}{1-c^2 x^2}dx-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c}}{c}}{c d}\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c d}-\frac {\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c}-\frac {\frac {b \int \frac {\log \left (\frac {2}{c x+1}\right )}{1-\frac {2}{c x+1}}d\frac {1}{c x+1}}{c}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c}}{c}}{c d}\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c d}-\frac {\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c}-\frac {\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c}}{c}}{c d}\)

Input:

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

Output:

((x^2*(a + b*ArcTanh[c*x]))/2 - (b*c*(-(x/c^2) + ArcTanh[c*x]/c^3))/2)/(c* 
d) - ((a*x + b*x*ArcTanh[c*x] + (b*Log[1 - c^2*x^2])/(2*c))/c - (-(((a + b 
*ArcTanh[c*x])*Log[2/(1 + c*x)])/c) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c 
))/c)/(c*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 262

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

rule 2009

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

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 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 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 6492

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {c^{2} x^{2}}{2}-c x +\ln \left (c x +1\right )\right )}{d}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}-\operatorname {arctanh}\left (c x \right ) c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {c x}{2}+\frac {1}{2}-\frac {3 \ln \left (c x +1\right )}{4}-\frac {\ln \left (c x -1\right )}{4}\right )}{d}}{c^{3}}\)	\(132\)
default	\(\frac {\frac {a \left (\frac {c^{2} x^{2}}{2}-c x +\ln \left (c x +1\right )\right )}{d}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}-\operatorname {arctanh}\left (c x \right ) c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {c x}{2}+\frac {1}{2}-\frac {3 \ln \left (c x +1\right )}{4}-\frac {\ln \left (c x -1\right )}{4}\right )}{d}}{c^{3}}\)	\(132\)
parts	\(\frac {a \left (\frac {\frac {1}{2} c \,x^{2}-x}{c^{2}}+\frac {\ln \left (c x +1\right )}{c^{3}}\right )}{d}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}-\operatorname {arctanh}\left (c x \right ) c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {c x}{2}+\frac {1}{2}-\frac {3 \ln \left (c x +1\right )}{4}-\frac {\ln \left (c x -1\right )}{4}\right )}{d \,c^{3}}\)	\(137\)
risch	\(\frac {b \ln \left (c x +1\right )^{2}}{4 c^{3} d}+\frac {b \left (\frac {1}{2} c \,x^{2}-x \right ) \ln \left (c x +1\right )}{2 c^{2} d}-\frac {b \ln \left (-c x +1\right ) x^{2}}{4 d c}+\frac {\ln \left (-c x +1\right ) b x}{2 d \,c^{2}}-\frac {b \ln \left (-c x +1\right )}{4 d \,c^{3}}+\frac {b x}{2 c^{2} d}+\frac {b}{8 c^{3} d}+\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d \,c^{3}}-\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d \,c^{3}}+\frac {b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d \,c^{3}}+\frac {a \,x^{2}}{2 d c}-\frac {a x}{c^{2} d}+\frac {a}{2 d \,c^{3}}+\frac {a \ln \left (-c x -1\right )}{d \,c^{3}}-\frac {3 b \ln \left (c x +1\right )}{4 c^{3} d}\)	\(238\)

Input:

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

Output:

1/c^3*(a/d*(1/2*c^2*x^2-c*x+ln(c*x+1))+b/d*(1/2*arctanh(c*x)*c^2*x^2-arcta 
nh(c*x)*c*x+arctanh(c*x)*ln(c*x+1)-1/4*ln(c*x+1)^2+1/2*(ln(c*x+1)-ln(1/2*c 
*x+1/2))*ln(-1/2*c*x+1/2)-1/2*dilog(1/2*c*x+1/2)+1/2*c*x+1/2-3/4*ln(c*x+1) 
-1/4*ln(c*x-1)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

(Integral(a*x**2/(c*x + 1), x) + Integral(b*x**2*atanh(c*x)/(c*x + 1), x)) 
/d
 

Maxima [F]

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

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

Output:

1/8*(c^3*(x^2/(c^4*d) + log(c^2*x^2 - 1)/(c^6*d)) + 8*c^3*integrate(1/2*x^ 
3*log(c*x + 1)/(c^4*d*x^2 - c^2*d), x) - c^2*(2*x/(c^4*d) - log(c*x + 1)/( 
c^5*d) + log(c*x - 1)/(c^5*d)) - 8*c^2*integrate(1/2*x^2*log(c*x + 1)/(c^4 
*d*x^2 - c^2*d), x) + 8*c*integrate(1/2*x*log(c*x + 1)/(c^4*d*x^2 - c^2*d) 
, x) - 2*(c^2*x^2 - 2*c*x + 2*log(c*x + 1))*log(-c*x + 1)/(c^3*d) - 2*log( 
2*c^4*d*x^2 - 2*c^2*d)/(c^3*d) + 8*integrate(1/2*log(c*x + 1)/(c^4*d*x^2 - 
 c^2*d), x))*b + 1/2*a*((c*x^2 - 2*x)/(c^2*d) + 2*log(c*x + 1)/(c^3*d))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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