3.203 \(\int x^2 (a+b \tanh ^{-1}(c \sqrt{x}))^3 \, dx\)

Optimal. Leaf size=304 \[ -\frac{23 b^3 \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{15 c^6}+\frac{b^2 x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{10 c^2}+\frac{8 b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{15 c^4}-\frac{46 b^2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{15 c^6}+\frac{b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{3 c^3}+\frac{b \sqrt{x} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{c^5}+\frac{23 b \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{15 c^6}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{3 c^6}+\frac{b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{5 c}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3+\frac{b^3 x^{3/2}}{30 c^3}+\frac{19 b^3 \sqrt{x}}{30 c^5}-\frac{19 b^3 \tanh ^{-1}\left (c \sqrt{x}\right )}{30 c^6} \]

[Out]

(19*b^3*Sqrt[x])/(30*c^5) + (b^3*x^(3/2))/(30*c^3) - (19*b^3*ArcTanh[c*Sqrt[x]])/(30*c^6) + (8*b^2*x*(a + b*Ar
cTanh[c*Sqrt[x]]))/(15*c^4) + (b^2*x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(10*c^2) + (23*b*(a + b*ArcTanh[c*Sqrt[x]])
^2)/(15*c^6) + (b*Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2)/c^5 + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(3*c^3)
+ (b*x^(5/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(5*c) - (a + b*ArcTanh[c*Sqrt[x]])^3/(3*c^6) + (x^3*(a + b*ArcTanh
[c*Sqrt[x]])^3)/3 - (46*b^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/(15*c^6) - (23*b^3*PolyLog[2, 1
- 2/(1 - c*Sqrt[x])])/(15*c^6)

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Rubi [F]  time = 0.0237346, 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 = {} \[ \int x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx &=\int x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx\\ \end{align*}

Mathematica [A]  time = 0.789276, size = 351, normalized size = 1.15 \[ \frac{46 b^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (30 a^2 c^6 x^3+4 a b c \sqrt{x} \left (3 c^4 x^2+5 c^2 x+15\right )+b^2 \left (3 c^4 x^2+16 c^2 x-19\right )-92 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )\right )+6 a^2 b c^5 x^{5/2}+10 a^2 b c^3 x^{3/2}+30 a^2 b c \sqrt{x}+15 a^2 b \log \left (1-c \sqrt{x}\right )-15 a^2 b \log \left (c \sqrt{x}+1\right )+10 a^3 c^6 x^3+3 a b^2 c^4 x^2+2 b^2 \tanh ^{-1}\left (c \sqrt{x}\right )^2 \left (15 a \left (c^6 x^3-1\right )+b \left (3 c^5 x^{5/2}+5 c^3 x^{3/2}+15 c \sqrt{x}-23\right )\right )+16 a b^2 c^2 x+46 a b^2 \log \left (1-c^2 x\right )-19 a b^2+b^3 c^3 x^{3/2}+10 b^3 \left (c^6 x^3-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^3+19 b^3 c \sqrt{x}}{30 c^6} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-19*a*b^2 + 30*a^2*b*c*Sqrt[x] + 19*b^3*c*Sqrt[x] + 16*a*b^2*c^2*x + 10*a^2*b*c^3*x^(3/2) + b^3*c^3*x^(3/2) +
3*a*b^2*c^4*x^2 + 6*a^2*b*c^5*x^(5/2) + 10*a^3*c^6*x^3 + 2*b^2*(b*(-23 + 15*c*Sqrt[x] + 5*c^3*x^(3/2) + 3*c^5
*x^(5/2)) + 15*a*(-1 + c^6*x^3))*ArcTanh[c*Sqrt[x]]^2 + 10*b^3*(-1 + c^6*x^3)*ArcTanh[c*Sqrt[x]]^3 + b*ArcTanh
[c*Sqrt[x]]*(30*a^2*c^6*x^3 + 4*a*b*c*Sqrt[x]*(15 + 5*c^2*x + 3*c^4*x^2) + b^2*(-19 + 16*c^2*x + 3*c^4*x^2) -
92*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 15*a^2*b*Log[1 - c*Sqrt[x]] - 15*a^2*b*Log[1 + c*Sqrt[x]] + 46*a*
b^2*Log[1 - c^2*x] + 46*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(30*c^6)

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Maple [C]  time = 0.326, size = 1423, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*I/c^6*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))^2-19/30*b^3*arctanh(c*x^(1/2))/c^6+19/30*b^3*
x^(1/2)/c^5+1/3*b^3*x^3*arctanh(c*x^(1/2))^3+23/15/c^6*b^3*arctanh(c*x^(1/2))^2-46/15/c^6*b^3*dilog(1-I*(1+c*x
^(1/2))/(-c^2*x+1)^(1/2))-46/15/c^6*b^3*dilog(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-1/3/c^6*b^3*arctanh(c*x^(1/2
))^3+1/3*x^3*a^3+1/30*b^3*x^(3/2)/c^3+1/2*I/c^6*b^3*Pi*csgn(I/((1+c*x^(1/2))^2/(-c^2*x+1)+1))^2*arctanh(c*x^(1
/2))^2+1/4*I/c^6*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+1/4*I/c^6*b^3*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^3*arctanh(c*x^(1/2))^2-1/2*I/c^6*b^3*Pi*csgn(I/((1+c*x^(1
/2))^2/(-c^2*x+1)+1))^3*arctanh(c*x^(1/2))^2+a*b^2*x^3*arctanh(c*x^(1/2))^2+a^2*b*x^3*arctanh(c*x^(1/2))+1/10/
c^2*b^3*arctanh(c*x^(1/2))*x^2+8/15/c^4*b^3*arctanh(c*x^(1/2))*x+1/c^5*b^3*arctanh(c*x^(1/2))^2*x^(1/2)+1/5/c*
b^3*arctanh(c*x^(1/2))^2*x^(5/2)+1/3/c^3*b^3*arctanh(c*x^(1/2))^2*x^(3/2)+1/c^5*a^2*b*x^(1/2)+1/5/c*a^2*b*x^(5
/2)+1/3/c^3*a^2*b*x^(3/2)-46/15/c^6*b^3*arctanh(c*x^(1/2))*ln(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-46/15/c^6*b^
3*arctanh(c*x^(1/2))*ln(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+1/2/c^6*b^3*arctanh(c*x^(1/2))^2*ln(c*x^(1/2)-1)-1
/2/c^6*b^3*arctanh(c*x^(1/2))^2*ln(1+c*x^(1/2))+1/c^6*b^3*arctanh(c*x^(1/2))^2*ln((1+c*x^(1/2))/(-c^2*x+1)^(1/
2))+1/4/c^6*a*b^2*ln(c*x^(1/2)-1)^2+1/4/c^6*a*b^2*ln(1+c*x^(1/2))^2+23/15/c^6*a*b^2*ln(c*x^(1/2)-1)+23/15/c^6*
a*b^2*ln(1+c*x^(1/2))+1/2/c^6*a^2*b*ln(c*x^(1/2)-1)-1/2/c^6*a^2*b*ln(1+c*x^(1/2))+8/15*a*b^2*x/c^4-1/4*I/c^6*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*arc
tanh(c*x^(1/2))^2-2/3/c^6*b^3+1/4*I/c^6*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+1/4*I/c^6*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+1/2*I/c^6*b^3*Pi*csgn(I*(1+c*x^(1/2))/(-c^2
*x+1)^(1/2))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^2*arctanh(c*x^(1/2))^2-1/2/c^6*a*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(
1+c*x^(1/2))+1/2/c^6*a*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1/2+1/2*c*x^(1/2))+1/c^6*a*b^2*arctanh(c*x^(1/2))*ln(c*x^
(1/2)-1)-1/c^6*a*b^2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-1/2/c^6*a*b^2*ln(c*x^(1/2)-1)*ln(1/2+1/2*c*x^(1/2))+2/
5/c*a*b^2*arctanh(c*x^(1/2))*x^(5/2)+2/3/c^3*a*b^2*arctanh(c*x^(1/2))*x^(3/2)+2/c^5*a*b^2*x^(1/2)*arctanh(c*x^
(1/2))-1/2*I/c^6*b^3*Pi*arctanh(c*x^(1/2))^2+1/10/c^2*x^2*a*b^2

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Maxima [B]  time = 3.73239, size = 2132, normalized size = 7.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3 - 1/720*a*b^2*c*((20*c^5*x^3 + 39*c^3*x^2 + 138*c*x - 6*(10*c^5*x^3 + 12*c^4*x^(5/2) + 15*c^3*x^2
+ 20*c^2*x^(3/2) + 30*c*x + 60*sqrt(x))*log(c*sqrt(x) + 1))/c^6 - 222*log(c*sqrt(x) + 1)/c^7 - 222*log(c*sqrt(
x) - 1)/c^7) - 1/120*(60*x^3*log(c*sqrt(x) + 1) - c*((10*c^5*x^3 - 12*c^4*x^(5/2) + 15*c^3*x^2 - 20*c^2*x^(3/2
) + 30*c*x - 60*sqrt(x))/c^6 + 60*log(c*sqrt(x) + 1)/c^7))*a*b^2*log(-c*sqrt(x) + 1) + 1/120*(60*x^3*log(c*sqr
t(x) + 1) - c*((10*c^5*x^3 - 12*c^4*x^(5/2) + 15*c^3*x^2 - 20*c^2*x^(3/2) + 30*c*x - 60*sqrt(x))/c^6 + 60*log(
c*sqrt(x) + 1)/c^7))*a^2*b - 1/120*(60*x^3*log(-c*sqrt(x) + 1) - c*((10*c^5*x^3 + 12*c^4*x^(5/2) + 15*c^3*x^2
+ 20*c^2*x^(3/2) + 30*c*x + 60*sqrt(x))/c^6 + 60*log(c*sqrt(x) - 1)/c^7))*a^2*b + 1/7200*(100*(18*log(-c*sqrt(
x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^6 + 432*(25*log(-c*sqrt(x) + 1)^2 - 10*log(-c*sqrt(x) +
1) + 2)*(c*sqrt(x) - 1)^5 + 3375*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^4 + 40
00*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 13500*(2*log(-c*sqrt(x) + 1)^2 -
2*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^2 + 10800*(log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt(x) + 1) + 2)*(c*sq
rt(x) - 1))*a*b^2/c^6 - 1/864000*(1000*(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 + 1728*(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 + 16875*(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 + 80000*(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 + 135000*(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 + 216000*(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^6 + 23/15*(log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b^3/
c^6 - 8929/14400*b^3*log(c*sqrt(x) - 1)/c^6 + 1/120*(147*a*b^2 - 38*b^3)*log(c*sqrt(x) + 1)/c^6 + 1/864000*(10
00*(12*a*b^2*c^6 - b^3*c^6)*x^3 + 36000*(b^3*c^6*x^3 - b^3)*log(c*sqrt(x) + 1)^3 - 48*(660*a*b^2*c^5 + 91*b^3*
c^5)*x^(5/2) + 15*(4440*a*b^2*c^4 - 919*b^3*c^4)*x^2 + 14400*(15*a*b^2*c^6*x^3 + 3*b^3*c^5*x^(5/2) + 5*b^3*c^3
*x^(3/2) + 15*b^3*c*sqrt(x) - 15*a*b^2 + 23*b^3)*log(c*sqrt(x) + 1)^2 - 1800*(10*b^3*c^6*x^3 - 12*b^3*c^5*x^(5
/2) + 15*b^3*c^4*x^2 - 20*b^3*c^3*x^(3/2) + 30*b^3*c^2*x - 60*b^3*c*sqrt(x) + 37*b^3 - 60*(b^3*c^6*x^3 - b^3)*
log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 - 20*(6840*a*b^2*c^3 + 619*b^3*c^3)*x^(3/2) + 870*(360*a*b^2*c^2 - 1
61*b^3*c^2)*x - 7200*(10*a*b^2*c^6*x^3 - 12*a*b^2*c^5*x^(5/2) - 20*a*b^2*c^3*x^(3/2) - 60*a*b^2*c*sqrt(x) + 3*
(5*a*b^2*c^4 - 2*b^3*c^4)*x^2 + 2*(15*a*b^2*c^2 - 16*b^3*c^2)*x)*log(c*sqrt(x) + 1) + 60*(100*b^3*c^6*x^3 + 26
4*b^3*c^5*x^(5/2) - 165*b^3*c^4*x^2 + 1140*b^3*c^3*x^(3/2) - 1230*b^3*c^2*x + 8820*b^3*c*sqrt(x) - 1800*(b^3*c
^6*x^3 - b^3)*log(c*sqrt(x) + 1)^2 - 480*(3*b^3*c^5*x^(5/2) + 5*b^3*c^3*x^(3/2) + 15*b^3*c*sqrt(x) + 23*b^3)*l
og(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1) - 60*(17640*a*b^2*c + 4369*b^3*c)*sqrt(x))/c^6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{atanh}{\left (c \sqrt{x} \right )}\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*atanh(c*sqrt(x)))**3, x)

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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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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