3.202 \(\int x^3 (a+b \tanh ^{-1}(c \sqrt{x}))^3 \, dx\)

Optimal. Leaf size=374 \[ -\frac{44 b^3 \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{35 c^8}+\frac{b^2 x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{28 c^2}+\frac{9 b^2 x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{70 c^4}+\frac{71 b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{140 c^6}-\frac{88 b^2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{35 c^8}+\frac{3 b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{20 c^3}+\frac{b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{4 c^5}+\frac{3 b \sqrt{x} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{4 c^7}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{4 c^8}+\frac{44 b \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{35 c^8}+\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3+\frac{3 b x^{7/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{28 c}+\frac{b^3 x^{5/2}}{140 c^3}+\frac{23 b^3 x^{3/2}}{420 c^5}+\frac{47 b^3 \sqrt{x}}{70 c^7}-\frac{47 b^3 \tanh ^{-1}\left (c \sqrt{x}\right )}{70 c^8} \]

[Out]

(47*b^3*Sqrt[x])/(70*c^7) + (23*b^3*x^(3/2))/(420*c^5) + (b^3*x^(5/2))/(140*c^3) - (47*b^3*ArcTanh[c*Sqrt[x]])
/(70*c^8) + (71*b^2*x*(a + b*ArcTanh[c*Sqrt[x]]))/(140*c^6) + (9*b^2*x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(70*c^4)
+ (b^2*x^3*(a + b*ArcTanh[c*Sqrt[x]]))/(28*c^2) + (44*b*(a + b*ArcTanh[c*Sqrt[x]])^2)/(35*c^8) + (3*b*Sqrt[x]*
(a + b*ArcTanh[c*Sqrt[x]])^2)/(4*c^7) + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(4*c^5) + (3*b*x^(5/2)*(a + b
*ArcTanh[c*Sqrt[x]])^2)/(20*c^3) + (3*b*x^(7/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(28*c) - (a + b*ArcTanh[c*Sqrt[x
]])^3/(4*c^8) + (x^4*(a + b*ArcTanh[c*Sqrt[x]])^3)/4 - (88*b^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x]
)])/(35*c^8) - (44*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(35*c^8)

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Rubi [F]  time = 0.0233636, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx &=\int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx\\ \end{align*}

Mathematica [A]  time = 1.1936, size = 418, normalized size = 1.12 \[ \frac{1056 b^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+6 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (105 a^2 c^8 x^4+2 a b c \sqrt{x} \left (15 c^6 x^3+21 c^4 x^2+35 c^2 x+105\right )+b^2 \left (5 c^6 x^3+18 c^4 x^2+71 c^2 x-94\right )-352 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )\right )+90 a^2 b c^7 x^{7/2}+126 a^2 b c^5 x^{5/2}+210 a^2 b c^3 x^{3/2}+630 a^2 b c \sqrt{x}+315 a^2 b \log \left (1-c \sqrt{x}\right )-315 a^2 b \log \left (c \sqrt{x}+1\right )+210 a^3 c^8 x^4+30 a b^2 c^6 x^3+108 a b^2 c^4 x^2+6 b^2 \tanh ^{-1}\left (c \sqrt{x}\right )^2 \left (105 a \left (c^8 x^4-1\right )+b \left (15 c^7 x^{7/2}+21 c^5 x^{5/2}+35 c^3 x^{3/2}+105 c \sqrt{x}-176\right )\right )+426 a b^2 c^2 x+1056 a b^2 \log \left (1-c^2 x\right )-564 a b^2+6 b^3 c^5 x^{5/2}+46 b^3 c^3 x^{3/2}+210 b^3 \left (c^8 x^4-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^3+564 b^3 c \sqrt{x}}{840 c^8} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-564*a*b^2 + 630*a^2*b*c*Sqrt[x] + 564*b^3*c*Sqrt[x] + 426*a*b^2*c^2*x + 210*a^2*b*c^3*x^(3/2) + 46*b^3*c^3*x
^(3/2) + 108*a*b^2*c^4*x^2 + 126*a^2*b*c^5*x^(5/2) + 6*b^3*c^5*x^(5/2) + 30*a*b^2*c^6*x^3 + 90*a^2*b*c^7*x^(7/
2) + 210*a^3*c^8*x^4 + 6*b^2*(b*(-176 + 105*c*Sqrt[x] + 35*c^3*x^(3/2) + 21*c^5*x^(5/2) + 15*c^7*x^(7/2)) + 10
5*a*(-1 + c^8*x^4))*ArcTanh[c*Sqrt[x]]^2 + 210*b^3*(-1 + c^8*x^4)*ArcTanh[c*Sqrt[x]]^3 + 6*b*ArcTanh[c*Sqrt[x]
]*(105*a^2*c^8*x^4 + b^2*(-94 + 71*c^2*x + 18*c^4*x^2 + 5*c^6*x^3) + 2*a*b*c*Sqrt[x]*(105 + 35*c^2*x + 21*c^4*
x^2 + 15*c^6*x^3) - 352*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 315*a^2*b*Log[1 - c*Sqrt[x]] - 315*a^2*b*Log
[1 + c*Sqrt[x]] + 1056*a*b^2*Log[1 - c^2*x] + 1056*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(840*c^8)

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Maple [C]  time = 1.179, size = 1518, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/c^8*b^3*arctanh(c*x^(1/2))^3+1/4*b^3*x^4*arctanh(c*x^(1/2))^3-11/15/c^8*b^3-47/70*b^3*arctanh(c*x^(1/2))/
c^8+47/70*b^3*x^(1/2)/c^7+23/420*b^3*x^(3/2)/c^5+1/140*b^3*x^(5/2)/c^3+1/28*a*b^2*x^3/c^2+9/70/c^4*x^2*a*b^2+7
1/140/c^6*b^2*x*a+3/4/c^7*a^2*b*x^(1/2)+3/20/c^3*a^2*b*x^(5/2)+1/4/c^5*a^2*b*x^(3/2)+3/16/c^8*a*b^2*ln(c*x^(1/
2)-1)^2+3/16/c^8*a*b^2*ln(1+c*x^(1/2))^2+44/35/c^8*a*b^2*ln(c*x^(1/2)-1)+44/35/c^8*a*b^2*ln(1+c*x^(1/2))+3/8/c
^8*a^2*b*ln(c*x^(1/2)-1)-3/8/c^8*a^2*b*ln(1+c*x^(1/2))-88/35/c^8*b^3*arctanh(c*x^(1/2))*ln(1+I*(1+c*x^(1/2))/(
-c^2*x+1)^(1/2))-88/35/c^8*b^3*arctanh(c*x^(1/2))*ln(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/8/c^8*b^3*arctanh(c
*x^(1/2))^2*ln(c*x^(1/2)-1)-3/8/c^8*b^3*arctanh(c*x^(1/2))^2*ln(1+c*x^(1/2))+3/4/c^8*b^3*arctanh(c*x^(1/2))^2*
ln((1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/28/c*x^(7/2)*a^2*b+9/70/c^4*b^3*arctanh(c*x^(1/2))*x^2+71/140/c^6*b^3*arc
tanh(c*x^(1/2))*x+1/28/c^2*b^3*arctanh(c*x^(1/2))*x^3+3/4/c^7*b^3*arctanh(c*x^(1/2))^2*x^(1/2)+3/28/c*b^3*arct
anh(c*x^(1/2))^2*x^(7/2)+3/20/c^3*b^3*arctanh(c*x^(1/2))^2*x^(5/2)+1/4/c^5*b^3*arctanh(c*x^(1/2))^2*x^(3/2)+3/
4*a*b^2*x^4*arctanh(c*x^(1/2))^2+3/4*a^2*b*x^4*arctanh(c*x^(1/2))+1/4*x^4*a^3+3/8/c^8*a*b^2*ln(-1/2*c*x^(1/2)+
1/2)*ln(1/2+1/2*c*x^(1/2))+3/4/c^8*a*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-3/4/c^8*a*b^2*arctanh(c*x^(1/2))*l
n(1+c*x^(1/2))-3/8/c^8*a*b^2*ln(c*x^(1/2)-1)*ln(1/2+1/2*c*x^(1/2))+3/14/c*a*b^2*arctanh(c*x^(1/2))*x^(7/2)+3/1
0/c^3*a*b^2*arctanh(c*x^(1/2))*x^(5/2)+1/2/c^5*a*b^2*arctanh(c*x^(1/2))*x^(3/2)+3/2/c^7*a*b^2*x^(1/2)*arctanh(
c*x^(1/2))-3/8/c^8*a*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+c*x^(1/2))-3/8*I/c^8*b^3*Pi*arctanh(c*x^(1/2))^2-3/16*I/c
^8*b^3*Pi*csgn(I/((1+c*x^(1/2))^2/(-c^2*x+1)+1))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*csgn(I*(1+c*x^(1/2))^2/(c^2
*x-1)/((1+c*x^(1/2))^2/(-c^2*x+1)+1))*arctanh(c*x^(1/2))^2+44/35/c^8*b^3*arctanh(c*x^(1/2))^2-88/35/c^8*b^3*di
log(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-88/35/c^8*b^3*dilog(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/16*I/c^8*b^3
*Pi*csgn(I/((1+c*x^(1/2))^2/(-c^2*x+1)+1))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/((1+c*x^(1/2))^2/(-c^2*x+1)+1))^2*
arctanh(c*x^(1/2))^2-3/16*I/c^8*b^3*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/((1+
c*x^(1/2))^2/(-c^2*x+1)+1))^2*arctanh(c*x^(1/2))^2+3/8*I/c^8*b^3*Pi*csgn(I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))*csg
n(I*(1+c*x^(1/2))^2/(c^2*x-1))^2*arctanh(c*x^(1/2))^2+3/16*I/c^8*b^3*Pi*csgn(I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))
^2*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*arctanh(c*x^(1/2))^2+3/16*I/c^8*b^3*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^
3*arctanh(c*x^(1/2))^2+3/8*I/c^8*b^3*Pi*csgn(I/((1+c*x^(1/2))^2/(-c^2*x+1)+1))^2*arctanh(c*x^(1/2))^2-3/8*I/c^
8*b^3*Pi*csgn(I/((1+c*x^(1/2))^2/(-c^2*x+1)+1))^3*arctanh(c*x^(1/2))^2+3/16*I/c^8*b^3*Pi*csgn(I*(1+c*x^(1/2))^
2/(c^2*x-1)/((1+c*x^(1/2))^2/(-c^2*x+1)+1))^3*arctanh(c*x^(1/2))^2

________________________________________________________________________________________

Maxima [B]  time = 4.15179, size = 2662, normalized size = 7.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*x^4 - 1/26880*a*b^2*c*((315*c^7*x^4 + 500*c^5*x^3 + 1002*c^3*x^2 + 3684*c*x - 12*(105*c^7*x^4 + 120*c^
6*x^(7/2) + 140*c^5*x^3 + 168*c^4*x^(5/2) + 210*c^3*x^2 + 280*c^2*x^(3/2) + 420*c*x + 840*sqrt(x))*log(c*sqrt(
x) + 1))/c^8 - 6396*log(c*sqrt(x) + 1)/c^9 - 6396*log(c*sqrt(x) - 1)/c^9) - 1/2240*(840*x^4*log(c*sqrt(x) + 1)
- c*((105*c^7*x^4 - 120*c^6*x^(7/2) + 140*c^5*x^3 - 168*c^4*x^(5/2) + 210*c^3*x^2 - 280*c^2*x^(3/2) + 420*c*x
- 840*sqrt(x))/c^8 + 840*log(c*sqrt(x) + 1)/c^9))*a*b^2*log(-c*sqrt(x) + 1) + 1/2240*(840*x^4*log(c*sqrt(x) +
1) - c*((105*c^7*x^4 - 120*c^6*x^(7/2) + 140*c^5*x^3 - 168*c^4*x^(5/2) + 210*c^3*x^2 - 280*c^2*x^(3/2) + 420*
c*x - 840*sqrt(x))/c^8 + 840*log(c*sqrt(x) + 1)/c^9))*a^2*b - 1/2240*(840*x^4*log(-c*sqrt(x) + 1) - c*((105*c^
7*x^4 + 120*c^6*x^(7/2) + 140*c^5*x^3 + 168*c^4*x^(5/2) + 210*c^3*x^2 + 280*c^2*x^(3/2) + 420*c*x + 840*sqrt(x
))/c^8 + 840*log(c*sqrt(x) - 1)/c^9))*a^2*b + 1/1881600*(11025*(32*log(-c*sqrt(x) + 1)^2 - 8*log(-c*sqrt(x) +
1) + 1)*(c*sqrt(x) - 1)^8 + 57600*(49*log(-c*sqrt(x) + 1)^2 - 14*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^7 +
548800*(18*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^6 + 790272*(25*log(-c*sqrt(x) +
1)^2 - 10*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^5 + 3087000*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1
) + 1)*(c*sqrt(x) - 1)^4 + 2195200*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 4
939200*(2*log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^2 + 2822400*(log(-c*sqrt(x) + 1)^
2 - 2*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1))*a*b^2/c^8 - 1/3161088000*(385875*(256*log(-c*sqrt(x) + 1)^3 -
96*log(-c*sqrt(x) + 1)^2 + 24*log(-c*sqrt(x) + 1) - 3)*(c*sqrt(x) - 1)^8 + 2304000*(343*log(-c*sqrt(x) + 1)^3
- 147*log(-c*sqrt(x) + 1)^2 + 42*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1)^7 + 76832000*(36*log(-c*sqrt(x) + 1)
^3 - 18*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 1)*(c*sqrt(x) - 1)^6 + 44255232*(125*log(-c*sqrt(x) +
1)^3 - 75*log(-c*sqrt(x) + 1)^2 + 30*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1)^5 + 216090000*(32*log(-c*sqrt(x)
+ 1)^3 - 24*log(-c*sqrt(x) + 1)^2 + 12*log(-c*sqrt(x) + 1) - 3)*(c*sqrt(x) - 1)^4 + 614656000*(9*log(-c*sqrt(
x) + 1)^3 - 9*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 2)*(c*sqrt(x) - 1)^3 + 691488000*(4*log(-c*sqrt(
x) + 1)^3 - 6*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 3)*(c*sqrt(x) - 1)^2 + 790272000*(log(-c*sqrt(x)
+ 1)^3 - 3*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1))*b^3/c^8 + 44/35*(log(c*sqrt(x)
+ 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b^3/c^8 - 1881559/3763200*b^3*log(c*sqrt(x) - 1)
/c^8 + 1/2240*(2283*a*b^2 - 752*b^3)*log(c*sqrt(x) + 1)/c^8 + 1/3161088000*(1157625*(16*a*b^2*c^8 - b^3*c^8)*x
^4 - 27000*(1680*a*b^2*c^7 + 169*b^3*c^7)*x^(7/2) + 3500*(24528*a*b^2*c^6 - 3565*b^3*c^6)*x^3 + 98784000*(b^3*
c^8*x^4 - b^3)*log(c*sqrt(x) + 1)^3 - 168*(895440*a*b^2*c^5 + 44269*b^3*c^5)*x^(5/2) + 210*(1248240*a*b^2*c^4
- 334699*b^3*c^4)*x^2 + 5644800*(105*a*b^2*c^8*x^4 + 15*b^3*c^7*x^(7/2) + 21*b^3*c^5*x^(5/2) + 35*b^3*c^3*x^(3
/2) + 105*b^3*c*sqrt(x) - 105*a*b^2 + 176*b^3)*log(c*sqrt(x) + 1)^2 - 352800*(105*b^3*c^8*x^4 - 120*b^3*c^7*x^
(7/2) + 140*b^3*c^6*x^3 - 168*b^3*c^5*x^(5/2) + 210*b^3*c^4*x^2 - 280*b^3*c^3*x^(3/2) + 420*b^3*c^2*x - 840*b^
3*c*sqrt(x) + 533*b^3 - 840*(b^3*c^8*x^4 - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 - 280*(1718640*a*b^2
*c^3 + 2899*b^3*c^3)*x^(3/2) + 420*(2424240*a*b^2*c^2 - 1227199*b^3*c^2)*x - 1411200*(105*a*b^2*c^8*x^4 - 120*
a*b^2*c^7*x^(7/2) - 168*a*b^2*c^5*x^(5/2) - 280*a*b^2*c^3*x^(3/2) - 840*a*b^2*c*sqrt(x) + 20*(7*a*b^2*c^6 - 2*
b^3*c^6)*x^3 + 6*(35*a*b^2*c^4 - 24*b^3*c^4)*x^2 + 4*(105*a*b^2*c^2 - 142*b^3*c^2)*x)*log(c*sqrt(x) + 1) + 840
*(11025*b^3*c^8*x^4 + 27000*b^3*c^7*x^(7/2) - 16100*b^3*c^6*x^3 + 89544*b^3*c^5*x^(5/2) - 85890*b^3*c^4*x^2 +
286440*b^3*c^3*x^(3/2) - 348180*b^3*c^2*x + 1917720*b^3*c*sqrt(x) - 352800*(b^3*c^8*x^4 - b^3)*log(c*sqrt(x) +
1)^2 - 13440*(15*b^3*c^7*x^(7/2) + 21*b^3*c^5*x^(5/2) + 35*b^3*c^3*x^(3/2) + 105*b^3*c*sqrt(x) + 176*b^3)*log
(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1) - 840*(3835440*a*b^2*c + 618199*b^3*c)*sqrt(x))/c^8

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{3} \operatorname{artanh}\left (c \sqrt{x}\right )^{3} + 3 \, a b^{2} x^{3} \operatorname{artanh}\left (c \sqrt{x}\right )^{2} + 3 \, a^{2} b x^{3} \operatorname{artanh}\left (c \sqrt{x}\right ) + a^{3} x^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )}^{3} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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