3.3.5 \(\int (a+b \text {arctanh}(c \sqrt {x}))^3 \, dx\) [205]

3.3.5.1 Optimal result

Integrand size = 14, antiderivative size = 142 \[ \int \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {3 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{c^2}+\frac {3 b \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{c^2}+x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {6 b^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^2}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^2} \]

output

3*b*(a+b*arctanh(c*x^(1/2)))^2/c^2-(a+b*arctanh(c*x^(1/2)))^3/c^2+x*(a+b*a 
rctanh(c*x^(1/2)))^3-6*b^2*(a+b*arctanh(c*x^(1/2)))*ln(2/(1-c*x^(1/2)))/c^ 
2-3*b^3*polylog(2,1-2/(1-c*x^(1/2)))/c^2+3*b*(a+b*arctanh(c*x^(1/2)))^2*x^ 
(1/2)/c
 

3.3.5.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.42 \[ \int \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {6 b^2 \left (-1+c \sqrt {x}\right ) \left (a+b+a c \sqrt {x}\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+2 b^3 \left (-1+c^2 x\right ) \text {arctanh}\left (c \sqrt {x}\right )^3+6 b \text {arctanh}\left (c \sqrt {x}\right ) \left (2 a b c \sqrt {x}+a^2 c^2 x-2 b^2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+a \left (6 a b c \sqrt {x}+2 a^2 c^2 x+3 a b \log \left (1-c \sqrt {x}\right )-3 a b \log \left (1+c \sqrt {x}\right )+6 b^2 \log \left (1-c^2 x\right )\right )+6 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )}{2 c^2} \]

input

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

output

(6*b^2*(-1 + c*Sqrt[x])*(a + b + a*c*Sqrt[x])*ArcTanh[c*Sqrt[x]]^2 + 2*b^3 
*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^3 + 6*b*ArcTanh[c*Sqrt[x]]*(2*a*b*c*Sqrt[ 
x] + a^2*c^2*x - 2*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + a*(6*a*b*c*Sq 
rt[x] + 2*a^2*c^2*x + 3*a*b*Log[1 - c*Sqrt[x]] - 3*a*b*Log[1 + c*Sqrt[x]] 
+ 6*b^2*Log[1 - c^2*x]) + 6*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(2 
*c^2)
 

3.3.5.3 Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6442, 6452, 6542, 6436, 6510, 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.

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

\(\Big \downarrow \) 6442

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

input

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

output

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

3.3.5.3.1 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 6442

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

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 6510

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

rule 6542

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

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]
 

3.3.5.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.19 (sec) , antiderivative size = 5673, normalized size of antiderivative = 39.95

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5673\)
default	\(\text {Expression too large to display}\)	\(5673\)
parts	\(\text {Expression too large to display}\)	\(5674\)

input

int((a+b*arctanh(c*x^(1/2)))^3,x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.3.5.5 Fricas [F]

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

input

integrate((a+b*arctanh(c*x^(1/2)))^3,x, algorithm="fricas")
 

output

integral(b^3*arctanh(c*sqrt(x))^3 + 3*a*b^2*arctanh(c*sqrt(x))^2 + 3*a^2*b 
*arctanh(c*sqrt(x)) + a^3, x)
 

3.3.5.6 Sympy [F]

\[ \int \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}\, dx \]

input

integrate((a+b*atanh(c*x**(1/2)))**3,x)
 

output

Integral((a + b*atanh(c*sqrt(x)))**3, x)
 

3.3.5.7 Maxima [F]

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

input

integrate((a+b*arctanh(c*x^(1/2)))^3,x, algorithm="maxima")
 

output

3/2*(c*(2*sqrt(x)/c^2 - log(c*sqrt(x) + 1)/c^3 + log(c*sqrt(x) - 1)/c^3) + 
 2*x*arctanh(c*sqrt(x)))*a^2*b + 3/4*(4*c*(2*sqrt(x)/c^2 - log(c*sqrt(x) + 
 1)/c^3 + log(c*sqrt(x) - 1)/c^3)*arctanh(c*sqrt(x)) + 4*x*arctanh(c*sqrt( 
x))^2 - (2*(log(c*sqrt(x) - 1) - 2)*log(c*sqrt(x) + 1) - log(c*sqrt(x) + 1 
)^2 - log(c*sqrt(x) - 1)^2 - 4*log(c*sqrt(x) - 1))/c^2)*a*b^2 + a^3*x - 1/ 
32*b^3*(((4*log(-c*sqrt(x) + 1)^3 - 6*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqr 
t(x) + 1) - 3)*(c*sqrt(x) - 1)^2 + 8*(log(-c*sqrt(x) + 1)^3 - 3*log(-c*sqr 
t(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1))/c^2 - 4*integrat 
e(log(c*sqrt(x) + 1)^3 - 3*log(c*sqrt(x) + 1)^2*log(-c*sqrt(x) + 1) + 3*lo 
g(c*sqrt(x) + 1)*log(-c*sqrt(x) + 1)^2, x))
 

3.3.5.8 Giac [F]

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

input

integrate((a+b*arctanh(c*x^(1/2)))^3,x, algorithm="giac")
 

output

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

3.3.5.9 Mupad [F(-1)]

Timed out. \[ \int \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3 \,d x \]

input

int((a + b*atanh(c*x^(1/2)))^3,x)
 

output

int((a + b*atanh(c*x^(1/2)))^3, x)