\(\int x (a+b \text {arctanh}(c x^{3/2})) \, dx\) [214]

Optimal result

Integrand size = 14, antiderivative size = 160 \[ \int x \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {3 b \sqrt {x}}{2 c}+\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {b \text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{2 c^{4/3}}+\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {b \text {arctanh}\left (\frac {\sqrt [3]{c} \sqrt {x}}{1+c^{2/3} x}\right )}{4 c^{4/3}} \] Output:

3/2*b*x^(1/2)/c+1/4*3^(1/2)*b*arctan(1/3*(1-2*c^(1/3)*x^(1/2))*3^(1/2))/c^ 
(4/3)-1/4*3^(1/2)*b*arctan(1/3*(1+2*c^(1/3)*x^(1/2))*3^(1/2))/c^(4/3)-1/2* 
b*arctanh(c^(1/3)*x^(1/2))/c^(4/3)+1/2*x^2*(a+b*arctanh(c*x^(3/2)))-1/4*b* 
arctanh(c^(1/3)*x^(1/2)/(1+c^(2/3)*x))/c^(4/3)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.39 \[ \int x \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {3 b \sqrt {x}}{2 c}+\frac {a x^2}{2}-\frac {\sqrt {3} b \arctan \left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}+\frac {1}{2} b x^2 \text {arctanh}\left (c x^{3/2}\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}\right )}{4 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}\right )}{4 c^{4/3}}+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}} \] Input:

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

Output:

(3*b*Sqrt[x])/(2*c) + (a*x^2)/2 - (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*Sqrt[x 
])/Sqrt[3]])/(4*c^(4/3)) - (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[ 
3]])/(4*c^(4/3)) + (b*x^2*ArcTanh[c*x^(3/2)])/2 + (b*Log[1 - c^(1/3)*Sqrt[ 
x]])/(4*c^(4/3)) - (b*Log[1 + c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + (b*Log[1 - c 
^(1/3)*Sqrt[x] + c^(2/3)*x])/(8*c^(4/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^ 
(2/3)*x])/(8*c^(4/3))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6452, 843, 851, 754, 27, 219, 1142, 25, 27, 1082, 217, 1103}

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.

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

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \int \frac {x^{5/2}}{1-c^2 x^3}dx\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {\int \frac {1}{\sqrt {x} \left (1-c^2 x^3\right )}dx}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \int \frac {1}{1-c^2 x^3}d\sqrt {x}}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 754

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{3} \int \frac {1}{1-c^{2/3} x}d\sqrt {x}+\frac {1}{3} \int \frac {2-\sqrt [3]{c} \sqrt {x}}{2 \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}+\frac {1}{3} \int \frac {\sqrt [3]{c} \sqrt {x}+2}{2 \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{3} \int \frac {1}{1-c^{2/3} x}d\sqrt {x}+\frac {1}{6} \int \frac {2-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{6} \int \frac {\sqrt [3]{c} \sqrt {x}+2}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{6} \int \frac {2-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{6} \int \frac {\sqrt [3]{c} \sqrt {x}+2}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int -\frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int \frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {3 \int \frac {1}{-x-3}d\left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{\sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {3 \int \frac {1}{-x-3}d\left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{\sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{4} b c \left (\frac {2 \left (\frac {1}{6} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}+\frac {\log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 \sqrt [3]{c}}\right )}{c^2}-\frac {2 \sqrt {x}}{c^2}\right )\)

Input:

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

Output:

(x^2*(a + b*ArcTanh[c*x^(3/2)]))/2 - (3*b*c*((-2*Sqrt[x])/c^2 + (2*(ArcTan 
h[c^(1/3)*Sqrt[x]]/(3*c^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x] 
)/Sqrt[3]])/c^(1/3)) - Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x]/(2*c^(1/3)))/6 
 + ((Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/c^(1/3) + Log[1 + c^ 
(1/3)*Sqrt[x] + c^(2/3)*x]/(2*c^(1/3)))/6))/c^2))/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 217

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

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 754

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a 
/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* 
Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 
*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) 
 Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 2)/4}], x]] / 
; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
 

rule 843

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

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, 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]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {a \,x^{2}}{2}+\frac {x^{2} b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {3 b \sqrt {x}}{2 c}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(194\)
default	\(\frac {a \,x^{2}}{2}+\frac {x^{2} b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {3 b \sqrt {x}}{2 c}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(194\)
parts	\(\frac {a \,x^{2}}{2}+\frac {x^{2} b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {3 b \sqrt {x}}{2 c}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}\)	\(194\)

Input:

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

Output:

1/2*a*x^2+1/2*x^2*b*arctanh(c*x^(3/2))+3/2*b*x^(1/2)/c+1/4*b/c^2/(1/c)^(2/ 
3)*ln(x^(1/2)-(1/c)^(1/3))-1/8*b/c^2/(1/c)^(2/3)*ln(x+(1/c)^(1/3)*x^(1/2)+ 
(1/c)^(2/3))-1/4*b/c^2/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/ 
3)*x^(1/2)+1))-1/4*b/c^2/(1/c)^(2/3)*ln(x^(1/2)+(1/c)^(1/3))+1/8*b/c^2/(1/ 
c)^(2/3)*ln(x-(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))-1/4*b/c^2/(1/c)^(2/3)*3^(1/ 
2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)-1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 1488, normalized size of antiderivative = 9.30 \[ \int x \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/16*(8*a*c*x^2 - 2*((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3) 
*(I*sqrt(3) + 1) + 2*b)*c*log(1/2*((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + 
b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c - b*c + b*sqrt(x)) - 4*(2*(-1/128* 
b^3 + 1/128*(c^4 - 1)*b^3/c^4 + 1/128*b^3/c^4)^(1/3)*(I*sqrt(3) + 1) - b)* 
c*log((2*(-1/128*b^3 + 1/128*(c^4 - 1)*b^3/c^4 + 1/128*b^3/c^4)^(1/3)*(I*s 
qrt(3) + 1) - b)*c + b*c + b*sqrt(x)) + (((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^3 
/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c - 6*b*c - 2*sqrt(-3/4*((1/2 
)^(1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 
 + 3*((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1 
) + 2*b)*b - 3*b^2)*c)*log(-1/2*((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^ 
3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c + b*c + sqrt(-3/4*((1/2)^(1/3)*(b^3 
- (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^( 
1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b - 
3*b^2)*c + 2*b*sqrt(x)) + (((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^3/c^4 
)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c - 6*b*c + 2*sqrt(-3/4*((1/2)^(1/3)*(b^3 - 
 (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^(1 
/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b - 3 
*b^2)*c)*log(-1/2*((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^3/c^4 + b^3/c^4)^(1/3)*( 
I*sqrt(3) + 1) + 2*b)*c + b*c - sqrt(-3/4*((1/2)^(1/3)*(b^3 - (c^4 - 1)*b^ 
3/c^4 + b^3/c^4)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^(1/3)*(b^3 -...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.08 \[ \int x \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {1}{2} \, a x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) - c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {7}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {7}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} - \frac {12 \, \sqrt {x}}{c^{2}}\right )}\right )} b \] Input:

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

Output:

1/2*a*x^2 + 1/8*(4*x^2*arctanh(c*x^(3/2)) - c*(2*sqrt(3)*arctan(1/3*sqrt(3 
)*(2*c^(2/3)*sqrt(x) + c^(1/3))/c^(1/3))/c^(7/3) + 2*sqrt(3)*arctan(1/3*sq 
rt(3)*(2*c^(2/3)*sqrt(x) - c^(1/3))/c^(1/3))/c^(7/3) + log(c^(2/3)*x + c^( 
1/3)*sqrt(x) + 1)/c^(7/3) - log(c^(2/3)*x - c^(1/3)*sqrt(x) + 1)/c^(7/3) + 
 2*log((c^(1/3)*sqrt(x) + 1)/c^(1/3))/c^(7/3) - 2*log((c^(1/3)*sqrt(x) - 1 
)/c^(1/3))/c^(7/3) - 12*sqrt(x)/c^2))*b
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 15.60 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.54 \[ \int x \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {a\,x^2}{2}+\frac {3\,b\,\sqrt {x}}{2\,c}+\frac {b\,\ln \left (\frac {c^{1/3}\,\sqrt {x}-1}{c^{1/3}\,\sqrt {x}+1}\right )}{4\,c^{4/3}}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (\frac {b\,x^2}{2}-\frac {b\,c^2\,x^5}{2}\right )}{2\,c^2\,x^3-2}+\frac {b\,x^2\,\ln \left (c\,x^{3/2}+1\right )}{4}+\frac {b\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{4\,c^{4/3}}+\frac {\sqrt {2}\,b\,\ln \left (\frac {2\,\sqrt {2}-c^{1/3}\,\sqrt {x}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{5/2}\,1{}\mathrm {i}-\sqrt {2}\,c^{2/3}\,x+\sqrt {2}\,\sqrt {3}\,c^{2/3}\,x\,1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{8\,c^{4/3}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.73 \[ \int x \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}-1}{\sqrt {3}}\right ) b -2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}+1}{\sqrt {3}}\right ) b +4 c^{\frac {4}{3}} \mathit {atanh} \left (\sqrt {x}\, c x \right ) b \,x^{2}+2 \mathit {atanh} \left (\sqrt {x}\, c x \right ) b +12 \sqrt {x}\, c^{\frac {1}{3}} b +4 c^{\frac {4}{3}} a \,x^{2}-3 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}+c^{\frac {1}{3}}\right ) b +3 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}-c^{\frac {1}{3}}\right ) b}{8 c^{\frac {4}{3}}} \] Input:

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

Output:

( - 2*sqrt(3)*atan((2*sqrt(x)*c**(1/3) - 1)/sqrt(3))*b - 2*sqrt(3)*atan((2 
*sqrt(x)*c**(1/3) + 1)/sqrt(3))*b + 4*c**(1/3)*atanh(sqrt(x)*c*x)*b*c*x**2 
 + 2*atanh(sqrt(x)*c*x)*b + 12*sqrt(x)*c**(1/3)*b + 4*c**(1/3)*a*c*x**2 - 
3*log(sqrt(x)*c**(2/3) + c**(1/3))*b + 3*log(sqrt(x)*c**(2/3) - c**(1/3))* 
b)/(8*c**(1/3)*c)