3.208 \(\int \frac{(a+b \tanh ^{-1}(c \sqrt{x}))^3}{x^3} \, dx\)
Optimal. Leaf size=234 \[ -2 b^3 c^4 \text{PolyLog}\left (2,\frac{2}{c \sqrt{x}+1}-1\right )-\frac{b^2 c^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 x}+4 b^2 c^4 \log \left (2-\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )+\frac{1}{2} c^4 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3+2 b c^4 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2-\frac{3 b c^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{2 \sqrt{x}}-\frac{b c \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{2 x^{3/2}}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{2 x^2}-\frac{b^3 c^3}{2 \sqrt{x}}+\frac{1}{2} b^3 c^4 \tanh ^{-1}\left (c \sqrt{x}\right ) \]
[Out]
-(b^3*c^3)/(2*Sqrt[x]) + (b^3*c^4*ArcTanh[c*Sqrt[x]])/2 - (b^2*c^2*(a + b*ArcTanh[c*Sqrt[x]]))/(2*x) + 2*b*c^4
*(a + b*ArcTanh[c*Sqrt[x]])^2 - (b*c*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*x^(3/2)) - (3*b*c^3*(a + b*ArcTanh[c*Sqr
t[x]])^2)/(2*Sqrt[x]) + (c^4*(a + b*ArcTanh[c*Sqrt[x]])^3)/2 - (a + b*ArcTanh[c*Sqrt[x]])^3/(2*x^2) + 4*b^2*c^
4*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - 2*b^3*c^4*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]
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Rubi [F] time = 0.0232819, 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 = {}
\[ \int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{x^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(a + b*ArcTanh[c*Sqrt[x]])^3/x^3,x]
[Out]
Defer[Int][(a + b*ArcTanh[c*Sqrt[x]])^3/x^3, x]
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{x^3} \, dx &=\int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{x^3} \, dx\\ \end{align*}
Mathematica [A] time = 0.692146, size = 333, normalized size = 1.42 \[ -\frac{8 b^3 c^4 x^2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+2 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (3 a^2+2 a b c \sqrt{x} \left (3 c^2 x+1\right )-8 b^2 c^4 x^2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+b^2 c^2 x \left (1-c^2 x\right )\right )+6 a^2 b c^3 x^{3/2}+3 a^2 b c^4 x^2 \log \left (1-c \sqrt{x}\right )-3 a^2 b c^4 x^2 \log \left (c \sqrt{x}+1\right )+2 a^2 b c \sqrt{x}+2 a^3-2 a b^2 c^4 x^2-16 a b^2 c^4 x^2 \log \left (\frac{c \sqrt{x}}{\sqrt{1-c^2 x}}\right )-2 b^2 \tanh ^{-1}\left (c \sqrt{x}\right )^2 \left (3 a \left (c^4 x^2-1\right )+b c \sqrt{x} \left (4 c^3 x^{3/2}-3 c^2 x-1\right )\right )+2 a b^2 c^2 x+2 b^3 c^3 x^{3/2}-2 b^3 \left (c^4 x^2-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^3}{4 x^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*ArcTanh[c*Sqrt[x]])^3/x^3,x]
[Out]
-(2*a^3 + 2*a^2*b*c*Sqrt[x] + 2*a*b^2*c^2*x + 6*a^2*b*c^3*x^(3/2) + 2*b^3*c^3*x^(3/2) - 2*a*b^2*c^4*x^2 - 2*b^
2*(b*c*Sqrt[x]*(-1 - 3*c^2*x + 4*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 + b^2*c^2*x*(1 - c^2*x) + 2*a*b*c*Sqrt[x]*(1 + 3*c^2*x)
- 8*b^2*c^4*x^2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + 3*a^2*b*c^4*x^2*Log[1 - c*Sqrt[x]] - 3*a^2*b*c^4*x^2*Lo
g[1 + c*Sqrt[x]] - 16*a*b^2*c^4*x^2*Log[(c*Sqrt[x])/Sqrt[1 - c^2*x]] + 8*b^3*c^4*x^2*PolyLog[2, E^(-2*ArcTanh[
c*Sqrt[x]])])/(4*x^2)
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Maple [C] time = 0.31, size = 1365, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arctanh(c*x^(1/2)))^3/x^3,x)
[Out]
-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))^2-3/4*
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+3/4*I*c^4*b^3*Pi*csgn(I/((1+c*x^(1/2))^2/(-c^2*x+1)+1))^3*arctanh(c*x^(
1/2))^2-1/2*a^3/x^2+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))-1/2*c^2*b^3*arc
tanh(c*x^(1/2))/x-1/2*c*b^3*arctanh(c*x^(1/2))^2/x^(3/2)-3/2*c^3*b^3*arctanh(c*x^(1/2))^2/x^(1/2)-3/2*c^3*a^2*
b/x^(1/2)-1/2*c*a^2*b/x^(3/2)+4*c^4*a*b^2*ln(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)-2*c^4*a*b^2*ln(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))+4*c^4*b^3*arctanh(c*x^(1/2))*ln(1+(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-1/2*c^4*b^3/(c*x^(1/2)+1-(-c
^2*x+1)^(1/2))*(-c^2*x+1)^(1/2)+1/2*c^4*b^3/((-c^2*x+1)^(1/2)+c*x^(1/2)+1)*(-c^2*x+1)^(1/2)-3/4*c^4*b^3*arctan
h(c*x^(1/2))^2*ln(c*x^(1/2)-1)+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))*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/x^2*a
rctanh(c*x^(1/2))^3+4*c^4*b^3*dilog(1+(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-4*c^4*b^3*dilog((1+c*x^(1/2))/(-c^2*x+1)
^(1/2))-2*c^4*b^3*arctanh(c*x^(1/2))^2+1/2*c^4*b^3*arctanh(c*x^(1/2))^3+1/2*b^3*c^4*arctanh(c*x^(1/2))-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))^2-3/8*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))*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+3/4*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))-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*I*c^4*b^3*Pi*arctanh(c*x^(1/2))^2-1/2*
c^2*a*b^2/x
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Maxima [B] time = 6.64872, size = 949, normalized size = 4.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x^(1/2)))^3/x^3,x, algorithm="maxima")
[Out]
-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 - 2*(log(c*sqrt(x))*log
(-c*sqrt(x) + 1) + dilog(-c*sqrt(x) + 1))*b^3*c^4 + 2*(log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog(c*sqrt(x) +
1))*b^3*c^4 - 1/8*((6*c^3*log(c*sqrt(x) - 1) - 3*c^3*log(x) + (6*c^2*x + 3*c*sqrt(x) + 2)/x^(3/2))*c - 6*log(-
c*sqrt(x) + 1)/x^2)*a^2*b + 1/4*(3*a^2*b*c^4 - 8*a*b^2*c^4 + b^3*c^4)*log(c*sqrt(x) + 1) - 1/4*(8*a*b^2*c^4 +
b^3*c^4)*log(c*sqrt(x) - 1) - 1/8*(3*a^2*b*c^4 - 16*a*b^2*c^4)*log(x) - 1/2*a^3/x^2 - 1/16*(4*a^2*b*c*sqrt(x)
- (b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^3 + (b^3*c^4*x^2 - b^3)*log(-c*sqrt(x) + 1)^3 + 2*(3*b^3*c^3*x^(3/2)
+ b^3*c*sqrt(x) + 3*a*b^2 - (3*a*b^2*c^4 - 4*b^3*c^4)*x^2)*log(c*sqrt(x) + 1)^2 + (6*b^3*c^3*x^(3/2) + 2*b^3*c
*sqrt(x) + 6*a*b^2 - 2*(3*a*b^2*c^4 + 4*b^3*c^4)*x^2 - 3*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x
) + 1)^2 + 4*(3*a^2*b*c^3 + 2*b^3*c^3)*x^(3/2) - 2*(3*a^2*b*c^2 - 4*a*b^2*c^2)*x + 4*(6*a*b^2*c^3*x^(3/2) + b^
3*c^2*x + 2*a*b^2*c*sqrt(x) + 3*a^2*b)*log(c*sqrt(x) + 1) - (24*a*b^2*c^3*x^(3/2) + 4*b^3*c^2*x + 8*a*b^2*c*sq
rt(x) - 3*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^2 + 4*(3*b^3*c^3*x^(3/2) + b^3*c*sqrt(x) + 3*a*b^2 - (3*a*b^2
*c^4 - 4*b^3*c^4)*x^2)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1))/x^2
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{artanh}\left (c \sqrt{x}\right )^{3} + 3 \, a b^{2} \operatorname{artanh}\left (c \sqrt{x}\right )^{2} + 3 \, a^{2} b \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((a+b*arctanh(c*x^(1/2)))^3/x^3,x, algorithm="fricas")
[Out]
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, 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((a+b*atanh(c*x**(1/2)))**3/x**3,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((a+b*arctanh(c*x^(1/2)))^3/x^3,x, algorithm="giac")
[Out]
integrate((b*arctanh(c*sqrt(x)) + a)^3/x^3, x)