3.204 \(\int x (a+b \tanh ^{-1}(c \sqrt{x}))^3 \, dx\)
Optimal. Leaf size=234 \[ -\frac{2 b^3 \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{c^4}+\frac{b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{4 b^2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}+\frac{3 b \sqrt{x} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{2 c^3}+\frac{2 b \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{c^4}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{2 c^4}+\frac{b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{2 c}+\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3+\frac{b^3 \sqrt{x}}{2 c^3}-\frac{b^3 \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^4} \]
[Out]
(b^3*Sqrt[x])/(2*c^3) - (b^3*ArcTanh[c*Sqrt[x]])/(2*c^4) + (b^2*x*(a + b*ArcTanh[c*Sqrt[x]]))/(2*c^2) + (2*b*(
a + b*ArcTanh[c*Sqrt[x]])^2)/c^4 + (3*b*Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*c^3) + (b*x^(3/2)*(a + b*ArcT
anh[c*Sqrt[x]])^2)/(2*c) - (a + b*ArcTanh[c*Sqrt[x]])^3/(2*c^4) + (x^2*(a + b*ArcTanh[c*Sqrt[x]])^3)/2 - (4*b^
2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^4 - (2*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^4
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Rubi [F] time = 0.0142464, 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 = {}
\[ \int x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
[In]
Int[x*(a + b*ArcTanh[c*Sqrt[x]])^3,x]
[Out]
Defer[Int][x*(a + b*ArcTanh[c*Sqrt[x]])^3, x]
Rubi steps
\begin{align*} \int x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx &=\int x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx\\ \end{align*}
Mathematica [A] time = 0.527349, size = 285, normalized size = 1.22 \[ \frac{8 b^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+2 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (3 a^2 c^4 x^2+2 a b c \sqrt{x} \left (c^2 x+3\right )+b^2 \left (c^2 x-1\right )-8 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )\right )+2 a^2 b c^3 x^{3/2}+6 a^2 b c \sqrt{x}+3 a^2 b \log \left (1-c \sqrt{x}\right )-3 a^2 b \log \left (c \sqrt{x}+1\right )+2 a^3 c^4 x^2+2 b^2 \tanh ^{-1}\left (c \sqrt{x}\right )^2 \left (3 a \left (c^4 x^2-1\right )+b \left (c^3 x^{3/2}+3 c \sqrt{x}-4\right )\right )+2 a b^2 c^2 x+8 a b^2 \log \left (1-c^2 x\right )-2 a b^2+2 b^3 \left (c^4 x^2-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^3+2 b^3 c \sqrt{x}}{4 c^4} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x*(a + b*ArcTanh[c*Sqrt[x]])^3,x]
[Out]
(-2*a*b^2 + 6*a^2*b*c*Sqrt[x] + 2*b^3*c*Sqrt[x] + 2*a*b^2*c^2*x + 2*a^2*b*c^3*x^(3/2) + 2*a^3*c^4*x^2 + 2*b^2*
(b*(-4 + 3*c*Sqrt[x] + c^3*x^(3/2)) + 3*a*(-1 + c^4*x^2))*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^4*x^2)*ArcTanh[
c*Sqrt[x]]^3 + 2*b*ArcTanh[c*Sqrt[x]]*(3*a^2*c^4*x^2 + b^2*(-1 + c^2*x) + 2*a*b*c*Sqrt[x]*(3 + c^2*x) - 8*b^2*
Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 3*a^2*b*Log[1 - c*Sqrt[x]] - 3*a^2*b*Log[1 + c*Sqrt[x]] + 8*a*b^2*Log[1
- c^2*x] + 8*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(4*c^4)
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Maple [C] time = 0.286, size = 1339, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*arctanh(c*x^(1/2)))^3,x)
[Out]
1/2*b^3*x^2*arctanh(c*x^(1/2))^3+2/c^4*b^3*arctanh(c*x^(1/2))^2-4/c^4*b^3*dilog(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(
1/2))-4/c^4*b^3*dilog(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-1/2/c^4*b^3*arctanh(c*x^(1/2))^3-3/8*I/c^4*b^3*Pi*cs
gn(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))^2-1/2*b^3*arctanh(c*x^(1/2))/c^4+1/2*b^3*x^(1/2)/c^3-3/4*I/c^4*b^
3*Pi*arctanh(c*x^(1/2))^2+1/c*a*b^2*arctanh(c*x^(1/2))*x^(3/2)+3/c^3*a*b^2*x^(1/2)*arctanh(c*x^(1/2))-3/4/c^4*
a*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+c*x^(1/2))+3/4/c^4*a*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1/2+1/2*c*x^(1/2))+3/2/c^
4*a*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-3/2/c^4*a*b^2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-3/4/c^4*a*b^2*ln(c
*x^(1/2)-1)*ln(1/2+1/2*c*x^(1/2))+1/2*a*b^2*x/c^2-3/4*I/c^4*b^3*Pi*csgn(I/((1+c*x^(1/2))^2/(-c^2*x+1)+1))^3*ar
ctanh(c*x^(1/2))^2+3/4*I/c^4*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^4*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^4*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))^2-4/c^4*b^3*arctanh(c*x^(1/2))*ln(1+I*(1+c*x^(1/2))/(-c
^2*x+1)^(1/2))-4/c^4*b^3*arctanh(c*x^(1/2))*ln(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/4/c^4*b^3*arctanh(c*x^(1/
2))^2*ln(c*x^(1/2)-1)-3/4/c^4*b^3*arctanh(c*x^(1/2))^2*ln(1+c*x^(1/2))+3/2/c^4*b^3*arctanh(c*x^(1/2))^2*ln((1+
c*x^(1/2))/(-c^2*x+1)^(1/2))+3/2*a*b^2*x^2*arctanh(c*x^(1/2))^2+3/2*a^2*b*x^2*arctanh(c*x^(1/2))+2/c^4*a*b^2*l
n(1+c*x^(1/2))+3/4/c^4*a^2*b*ln(c*x^(1/2)-1)-3/4/c^4*a^2*b*ln(1+c*x^(1/2))+3/8/c^4*a*b^2*ln(c*x^(1/2)-1)^2+3/8
/c^4*a*b^2*ln(1+c*x^(1/2))^2+2/c^4*a*b^2*ln(c*x^(1/2)-1)+1/2/c^2*b^3*arctanh(c*x^(1/2))*x+3/2/c^3*b^3*arctanh(
c*x^(1/2))^2*x^(1/2)+1/2/c*b^3*arctanh(c*x^(1/2))^2*x^(3/2)+3/2/c^3*a^2*b*x^(1/2)+1/2/c*a^2*b*x^(3/2)-3/8*I/c^
4*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+3/4*I/c^4*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+3/8*I/c^4*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+3/8*I/c^4*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*x^2*a^3-1/2/c^4*b^3
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Maxima [B] time = 3.39122, size = 1598, normalized size = 6.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arctanh(c*x^(1/2)))^3,x, algorithm="maxima")
[Out]
1/2*a^3*x^2 - 1/32*a*b^2*c*((3*c^3*x^2 + 10*c*x - 2*(3*c^3*x^2 + 4*c^2*x^(3/2) + 6*c*x + 12*sqrt(x))*log(c*sqr
t(x) + 1))/c^4 - 14*log(c*sqrt(x) + 1)/c^5 - 14*log(c*sqrt(x) - 1)/c^5) - 1/16*(12*x^2*log(c*sqrt(x) + 1) - c*
((3*c^3*x^2 - 4*c^2*x^(3/2) + 6*c*x - 12*sqrt(x))/c^4 + 12*log(c*sqrt(x) + 1)/c^5))*a*b^2*log(-c*sqrt(x) + 1)
+ 1/16*(12*x^2*log(c*sqrt(x) + 1) - c*((3*c^3*x^2 - 4*c^2*x^(3/2) + 6*c*x - 12*sqrt(x))/c^4 + 12*log(c*sqrt(x)
+ 1)/c^5))*a^2*b - 1/16*(12*x^2*log(-c*sqrt(x) + 1) - c*((3*c^3*x^2 + 4*c^2*x^(3/2) + 6*c*x + 12*sqrt(x))/c^4
+ 12*log(c*sqrt(x) - 1)/c^5))*a^2*b + 1/192*(9*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(
x) - 1)^4 + 32*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 216*(2*log(-c*sqrt(x)
+ 1)^2 - 2*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^2 + 288*(log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt(x) + 1) +
2)*(c*sqrt(x) - 1))*a*b^2/c^4 - 1/4608*(9*(32*log(-c*sqrt(x) + 1)^3 - 24*log(-c*sqrt(x) + 1)^2 + 12*log(-c*sqr
t(x) + 1) - 3)*(c*sqrt(x) - 1)^4 + 128*(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 + 432*(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 + 1152*(log(-c*sqrt(x) + 1)^3 - 3*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 6)*(c*s
qrt(x) - 1))*b^3/c^4 + 2*(log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b^3/c^4 -
319/384*b^3*log(c*sqrt(x) - 1)/c^4 + 1/16*(25*a*b^2 - 4*b^3)*log(c*sqrt(x) + 1)/c^4 + 1/4608*(288*(b^3*c^4*x^
2 - b^3)*log(c*sqrt(x) + 1)^3 + 27*(8*a*b^2*c^4 - b^3*c^4)*x^2 + 576*(3*a*b^2*c^4*x^2 + b^3*c^3*x^(3/2) + 3*b^
3*c*sqrt(x) - 3*a*b^2 + 4*b^3)*log(c*sqrt(x) + 1)^2 - 72*(3*b^3*c^4*x^2 - 4*b^3*c^3*x^(3/2) + 6*b^3*c^2*x - 12
*b^3*c*sqrt(x) + 7*b^3 - 12*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 - 4*(168*a*b^2*c^3 +
37*b^3*c^3)*x^(3/2) + 6*(312*a*b^2*c^2 - 115*b^3*c^2)*x - 288*(3*a*b^2*c^4*x^2 - 4*a*b^2*c^3*x^(3/2) - 12*a*b
^2*c*sqrt(x) + 2*(3*a*b^2*c^2 - 2*b^3*c^2)*x)*log(c*sqrt(x) + 1) + 12*(9*b^3*c^4*x^2 + 28*b^3*c^3*x^(3/2) - 18
*b^3*c^2*x + 300*b^3*c*sqrt(x) - 72*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^2 - 96*(b^3*c^3*x^(3/2) + 3*b^3*c*s
qrt(x) + 4*b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1) - 12*(600*a*b^2*c + 223*b^3*c)*sqrt(x))/c^4
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x \operatorname{artanh}\left (c \sqrt{x}\right )^{3} + 3 \, a b^{2} x \operatorname{artanh}\left (c \sqrt{x}\right )^{2} + 3 \, a^{2} b x \operatorname{artanh}\left (c \sqrt{x}\right ) + a^{3} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arctanh(c*x^(1/2)))^3,x, algorithm="fricas")
[Out]
integral(b^3*x*arctanh(c*sqrt(x))^3 + 3*a*b^2*x*arctanh(c*sqrt(x))^2 + 3*a^2*b*x*arctanh(c*sqrt(x)) + a^3*x, x
)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*atanh(c*x**(1/2)))**3,x)
[Out]
Integral(x*(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\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arctanh(c*x^(1/2)))^3,x, algorithm="giac")
[Out]
integrate((b*arctanh(c*sqrt(x)) + a)^3*x, x)