\(\int x^3 (a+b \text {arctanh}(c x^{3/2})) \, dx\) [212]

Optimal result

Integrand size = 16, antiderivative size = 160 \[ \int x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {3 b x^{5/2}}{20 c}-\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}-\frac {b \text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {b \text {arctanh}\left (\frac {\sqrt [3]{c} \sqrt {x}}{1+c^{2/3} x}\right )}{8 c^{8/3}} \] Output:

3/20*b*x^(5/2)/c-1/8*3^(1/2)*b*arctan(1/3*(1-2*c^(1/3)*x^(1/2))*3^(1/2))/c 
^(8/3)+1/8*3^(1/2)*b*arctan(1/3*(1+2*c^(1/3)*x^(1/2))*3^(1/2))/c^(8/3)-1/4 
*b*arctanh(c^(1/3)*x^(1/2))/c^(8/3)+1/4*x^4*(a+b*arctanh(c*x^(3/2)))-1/8*b 
*arctanh(c^(1/3)*x^(1/2)/(1+c^(2/3)*x))/c^(8/3)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.39 \[ \int x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {3 b x^{5/2}}{20 c}+\frac {a x^4}{4}+\frac {\sqrt {3} b \arctan \left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {1}{4} b x^4 \text {arctanh}\left (c x^{3/2}\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}} \] Input:

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

Output:

(3*b*x^(5/2))/(20*c) + (a*x^4)/4 + (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*Sqrt[ 
x])/Sqrt[3]])/(8*c^(8/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt 
[3]])/(8*c^(8/3)) + (b*x^4*ArcTanh[c*x^(3/2)])/4 + (b*Log[1 - c^(1/3)*Sqrt 
[x]])/(8*c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + (b*Log[1 - 
c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + 
c^(2/3)*x])/(16*c^(8/3))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.32, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6452, 843, 851, 825, 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.

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

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \int \frac {x^{9/2}}{1-c^2 x^3}dx\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {\int \frac {x^{3/2}}{1-c^2 x^3}dx}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \int \frac {x^2}{1-c^2 x^3}d\sqrt {x}}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 825

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (\frac {\int \frac {1}{1-c^{2/3} x}d\sqrt {x}}{3 c^{4/3}}+\frac {\int -\frac {\sqrt [3]{c} \sqrt {x}+1}{2 \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}}{3 c^{4/3}}+\frac {\int -\frac {1-\sqrt [3]{c} \sqrt {x}}{2 \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}}{3 c^{4/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (\frac {\int \frac {1}{1-c^{2/3} x}d\sqrt {x}}{3 c^{4/3}}-\frac {\int \frac {\sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (-\frac {\int \frac {\sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (-\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {-\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {3 \int \frac {1}{-x-3}d\left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (-\frac {-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )-\frac {3}{8} b c \left (\frac {2 \left (-\frac {\frac {\log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\)

Input:

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

Output:

(x^4*(a + b*ArcTanh[c*x^(3/2)]))/4 - (3*b*c*((-2*x^(5/2))/(5*c^2) + (2*(Ar 
cTanh[c^(1/3)*Sqrt[x]]/(3*c^(5/3)) - (-((Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqr 
t[x])/Sqrt[3]])/c^(1/3)) + Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x]/(2*c^(1/3) 
))/(6*c^(4/3)) - ((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*c^(4/3))))/c^2))/ 
8
 

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 825

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k 
*m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + 
s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 
 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m))   Int[1/ 
(r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], 
x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 
] && 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 = 0.18 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {x^{4} a}{4}+\frac {b \,x^{4} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{4}+\frac {3 b \,x^{\frac {5}{2}}}{20 c}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}\)	\(194\)
default	\(\frac {x^{4} a}{4}+\frac {b \,x^{4} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{4}+\frac {3 b \,x^{\frac {5}{2}}}{20 c}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}\)	\(194\)
parts	\(\frac {x^{4} a}{4}+\frac {b \,x^{4} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{4}+\frac {3 b \,x^{\frac {5}{2}}}{20 c}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}\)	\(194\)

Input:

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

Output:

1/4*x^4*a+1/4*b*x^4*arctanh(c*x^(3/2))+3/20*b*x^(5/2)/c-1/8*b/c^3/(1/c)^(1 
/3)*ln(x^(1/2)+(1/c)^(1/3))+1/16*b/c^3/(1/c)^(1/3)*ln(x-(1/c)^(1/3)*x^(1/2 
)+(1/c)^(2/3))+1/8*b/c^3*3^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^( 
1/3)*x^(1/2)-1))+1/8*b/c^3/(1/c)^(1/3)*ln(x^(1/2)-(1/c)^(1/3))-1/16*b/c^3/ 
(1/c)^(1/3)*ln(x+(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))+1/8*b/c^3*3^(1/2)/(1/c)^ 
(1/3)*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 = 0.88 (sec) , antiderivative size = 2039, normalized size of antiderivative = 12.74 \[ \int x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/160*(40*a*c*x^4 + 24*b*x^(5/2) - 10*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^ 
8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c*log(-1/4*((1/2)^(1/3)*(b^3 - ( 
c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*c^5 + ((1/2)^(1 
/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b*c^5 
 - b^2*c^5 + b^2*sqrt(x)) - 20*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 
+ 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)*c*log((4*(-1/1024*b^3 + 1/102 
4*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqrt(3) + 1) - b)^2*c^5 + 2 
*(4*(-1/1024*b^3 + 1/1024*(c^8 - 1)*b^3/c^8 + 1/1024*b^3/c^8)^(1/3)*(I*sqr 
t(3) + 1) - b)*b*c^5 + b^2*c^5 + b^2*sqrt(x)) + 5*(((1/2)^(1/3)*(b^3 - (c^ 
8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*c - 6*b*c - 2*sqrt( 
-3/4*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1 
) + 2*b)^2 + 3*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*s 
qrt(3) + 1) + 2*b)*b - 3*b^2)*c)*log(1/4*((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3 
/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*c^5 - ((1/2)^(1/3)*(b^3 - ( 
c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b*c^5 + b^2*c^5 + 
 2*b^2*sqrt(x) + 1/2*(((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/ 
3)*(I*sqrt(3) + 1) + 2*b)*c^5 - 2*b*c^5)*sqrt(-3/4*((1/2)^(1/3)*(b^3 - (c^ 
8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^(1/3)* 
(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b - 3*b^2 
)) + 5*(((1/2)^(1/3)*(b^3 - (c^8 - 1)*b^3/c^8 + b^3/c^8)^(1/3)*(I*sqrt(...
 

Sympy [F(-1)]

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

integrate(x**3*(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 \[ \int x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {1}{4} \, a x^{4} + \frac {1}{80} \, {\left (20 \, x^{4} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + c {\left (\frac {12 \, x^{\frac {5}{2}}}{c^{2}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} - \frac {5 \, \log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {11}{3}}} + \frac {5 \, \log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {11}{3}}} - \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}}\right )}\right )} b \] Input:

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

Output:

1/4*a*x^4 + 1/80*(20*x^4*arctanh(c*x^(3/2)) + c*(12*x^(5/2)/c^2 + 10*sqrt( 
3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) + c^(1/3))/c^(1/3))/c^(11/3) + 10 
*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) - c^(1/3))/c^(1/3))/c^(11/3 
) - 5*log(c^(2/3)*x + c^(1/3)*sqrt(x) + 1)/c^(11/3) + 5*log(c^(2/3)*x - c^ 
(1/3)*sqrt(x) + 1)/c^(11/3) - 10*log((c^(1/3)*sqrt(x) + 1)/c^(1/3))/c^(11/ 
3) + 10*log((c^(1/3)*sqrt(x) - 1)/c^(1/3))/c^(11/3)))*b
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.42 \[ \int x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {1}{4} \, a x^{4} + \frac {1}{80} \, {\left (10 \, x^{4} \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right ) + c {\left (\frac {12 \, x^{\frac {5}{2}}}{c^{2}} - \frac {10 \, \sqrt {3} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{2} {\left | c \right |}^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {1}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{c^{5}} + \frac {5 \, {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{2} {\left | c \right |}^{\frac {4}{3}} \log \left (x + \sqrt {x} \left (-\frac {1}{c}\right )^{\frac {1}{3}} + \left (-\frac {1}{c}\right )^{\frac {2}{3}}\right )}{c^{5}} - \frac {10 \, \left (-\frac {1}{c}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {1}{c}\right )^{\frac {1}{3}} \right |}\right )}{c^{3}} + \frac {10 \, \sqrt {3} {\left | c \right |}^{\frac {4}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} c^{\frac {1}{3}} {\left (2 \, \sqrt {x} + \frac {1}{c^{\frac {1}{3}}}\right )}\right )}{c^{5}} - \frac {5 \, {\left | c \right |}^{\frac {4}{3}} \log \left (x + \frac {\sqrt {x}}{c^{\frac {1}{3}}} + \frac {1}{c^{\frac {2}{3}}}\right )}{c^{5}} + \frac {10 \, \log \left ({\left | \sqrt {x} - \frac {1}{c^{\frac {1}{3}}} \right |}\right )}{c^{\frac {11}{3}}}\right )}\right )} b \] Input:

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

Output:

1/4*a*x^4 + 1/80*(10*x^4*log(-(c*x^(3/2) + 1)/(c*x^(3/2) - 1)) + c*(12*x^( 
5/2)/c^2 - 10*sqrt(3)*(1/2*I*sqrt(3) + 1/2)^2*abs(c)^(4/3)*arctan(1/3*sqrt 
(3)*(2*sqrt(x) + (-1/c)^(1/3))/(-1/c)^(1/3))/c^5 + 5*(1/2*I*sqrt(3) + 1/2) 
^2*abs(c)^(4/3)*log(x + sqrt(x)*(-1/c)^(1/3) + (-1/c)^(2/3))/c^5 - 10*(-1/ 
c)^(2/3)*log(abs(sqrt(x) - (-1/c)^(1/3)))/c^3 + 10*sqrt(3)*abs(c)^(4/3)*ar 
ctan(1/3*sqrt(3)*c^(1/3)*(2*sqrt(x) + 1/c^(1/3)))/c^5 - 5*abs(c)^(4/3)*log 
(x + sqrt(x)/c^(1/3) + 1/c^(2/3))/c^5 + 10*log(abs(sqrt(x) - 1/c^(1/3)))/c 
^(11/3)))*b
 

Mupad [B] (verification not implemented)

Time = 16.65 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.44 \[ \int x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {a\,x^4}{4}+\frac {3\,b\,x^{5/2}}{20\,c}+\frac {b\,\ln \left (\frac {c^{1/3}\,\sqrt {x}-1}{c^{1/3}\,\sqrt {x}+1}\right )}{8\,c^{8/3}}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (\frac {b\,x^4}{4}-\frac {b\,c^2\,x^7}{4}\right )}{2\,c^2\,x^3-2}+\frac {b\,x^4\,\ln \left (c\,x^{3/2}+1\right )}{8}+\frac {b\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{8\,c^{8/3}}+\frac {\sqrt {2}\,b\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{16\,c^{8/3}} \] Input:

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

Output:

(a*x^4)/4 + (3*b*x^(5/2))/(20*c) + (b*log((c^(1/3)*x^(1/2) - 1)/(c^(1/3)*x 
^(1/2) + 1)))/(8*c^(8/3)) + (log(1 - c*x^(3/2))*((b*x^4)/4 - (b*c^2*x^7)/4 
))/(2*c^2*x^3 - 2) + (b*x^4*log(c*x^(3/2) + 1))/8 + (b*log((3^(1/2) + c^(2 
/3)*x*1i - c^(1/3)*x^(1/2)*4i - 3^(1/2)*c^(2/3)*x + 1i)/(3^(1/2)*1i + 2*c^ 
(2/3)*x + 1))*((3^(1/2)*1i)/2 - 1/2)^(1/2))/(8*c^(8/3)) + (2^(1/2)*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 + 1)^(1/2)*1i)/(16*c^(8/3))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75 \[ \int x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {10 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}-1}{\sqrt {3}}\right ) b +10 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}+1}{\sqrt {3}}\right ) b +20 c^{\frac {8}{3}} \mathit {atanh} \left (\sqrt {x}\, c x \right ) b \,x^{4}+10 \mathit {atanh} \left (\sqrt {x}\, c x \right ) b +12 \sqrt {x}\, c^{\frac {5}{3}} b \,x^{2}+20 c^{\frac {8}{3}} a \,x^{4}-15 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}+c^{\frac {1}{3}}\right ) b +15 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}-c^{\frac {1}{3}}\right ) b}{80 c^{\frac {8}{3}}} \] Input:

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

Output:

(10*sqrt(3)*atan((2*sqrt(x)*c**(1/3) - 1)/sqrt(3))*b + 10*sqrt(3)*atan((2* 
sqrt(x)*c**(1/3) + 1)/sqrt(3))*b + 20*c**(2/3)*atanh(sqrt(x)*c*x)*b*c**2*x 
**4 + 10*atanh(sqrt(x)*c*x)*b + 12*sqrt(x)*c**(2/3)*b*c*x**2 + 20*c**(2/3) 
*a*c**2*x**4 - 15*log(sqrt(x)*c**(2/3) + c**(1/3))*b + 15*log(sqrt(x)*c**( 
2/3) - c**(1/3))*b)/(80*c**(2/3)*c**2)