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

Optimal. Leaf size=234 \[ 4 b^2 c^4 \log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b^2 c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x}+\frac {1}{2} c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3+2 b c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-\frac {3 b c^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 \sqrt {x}}-\frac {b c \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 x^{3/2}}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{2 x^2}-2 b^3 c^4 \text {Li}_2\left (\frac {2}{\sqrt {x} c+1}-1\right )+\frac {1}{2} b^3 c^4 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b^3 c^3}{2 \sqrt {x}} \]

[Out]

1/2*b^3*c^4*arctanh(c*x^(1/2))-1/2*b^2*c^2*(a+b*arctanh(c*x^(1/2)))/x+2*b*c^4*(a+b*arctanh(c*x^(1/2)))^2-1/2*b
*c*(a+b*arctanh(c*x^(1/2)))^2/x^(3/2)+1/2*c^4*(a+b*arctanh(c*x^(1/2)))^3-1/2*(a+b*arctanh(c*x^(1/2)))^3/x^2+4*
b^2*c^4*(a+b*arctanh(c*x^(1/2)))*ln(2-2/(1+c*x^(1/2)))-2*b^3*c^4*polylog(2,-1+2/(1+c*x^(1/2)))-1/2*b^3*c^3/x^(
1/2)-3/2*b*c^3*(a+b*arctanh(c*x^(1/2)))^2/x^(1/2)

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Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.72, size = 333, normalized size = 1.42 \[ -\frac {2 a^3+2 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (3 a^2+2 a b c \sqrt {x} \left (3 c^2 x+1\right )-8 b^2 c^4 x^2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+b^2 c^2 x \left (1-c^2 x\right )\right )+3 a^2 b c^4 x^2 \log \left (1-c \sqrt {x}\right )-3 a^2 b c^4 x^2 \log \left (c \sqrt {x}+1\right )+6 a^2 b c^3 x^{3/2}+2 a^2 b c \sqrt {x}-2 a b^2 c^4 x^2+2 a b^2 c^2 x-16 a b^2 c^4 x^2 \log \left (\frac {c \sqrt {x}}{\sqrt {1-c^2 x}}\right )-2 b^2 \tanh ^{-1}\left (c \sqrt {x}\right )^2 \left (3 a \left (c^4 x^2-1\right )+b c \sqrt {x} \left (4 c^3 x^{3/2}-3 c^2 x-1\right )\right )+8 b^3 c^4 x^2 \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )-2 b^3 \left (c^4 x^2-1\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3+2 b^3 c^3 x^{3/2}}{4 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(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*Log[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*ArcT
anh[c*Sqrt[x]])])/x^2

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fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c \sqrt {x}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c \sqrt {x}\right ) + a^{3}}{x^{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x^{3}}\,{d x} \]

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)

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maple [C]  time = 0.59, size = 1365, normalized size = 5.83 \[ \text {result too large to display} \]

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+(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+(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))^3*arctanh(c*x^(1/2))^2-3/4*I*c^4*b^3*Pi*csgn(I/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))^2*arctanh(c*x^(
1/2))^2-c*a*b^2*arctanh(c*x^(1/2))/x^(3/2)-3*c^3*a*b^2*arctanh(c*x^(1/2))/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))-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(-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))+3/4*I*c^4*b^3*Pi*arctanh(c*x^(1/2))^2-1/2*b^3/x^2*arctanh(c*x^(1/2))^3-4*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+4*c^4*b^3*dilog(1+(1+c*
x^(1/2))/(-c^2*x+1)^(1/2))+1/2*b^3*c^4*arctanh(c*x^(1/2))+3/8*I*c^4*b^3*Pi*csgn(I/(1+(1+c*x^(1/2))^2/(-c^2*x+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+(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))/(-c^2*x+1)^(1/2))^2*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*arct
anh(c*x^(1/2))^2-3/8*I*c^4*b^3*Pi*csgn(I/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(1+(
1+c*x^(1/2))^2/(-c^2*x+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))*csg
n(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))^2/(c^2*x-1))*csgn(
I*(1+c*x^(1/2))^2/(c^2*x-1)/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))^2*arctanh(c*x^(1/2))^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^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)-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^2*
a*b^2/x-3/2*c^4*b^3*arctanh(c*x^(1/2))^2*ln((1+c*x^(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)-2*c^4*a*b^2*ln(c*x^(1/2)-1)-2*c^4*a*b^2*ln(1+c*x^(1/2))-1/2*c*a^2*b/x^(3/2)-3/4*c^4*
a^2*b*ln(c*x^(1/2)-1)+3/4*c^4*b^3*arctanh(c*x^(1/2))^2*ln(1+c*x^(1/2))+3/4*c^4*a^2*b*ln(1+c*x^(1/2))+4*c^4*a*b
^2*ln(c*x^(1/2))-3/2*a^2*b/x^2*arctanh(c*x^(1/2))-3/2*a*b^2/x^2*arctanh(c*x^(1/2))^2-3/2*c^3*a^2*b/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))^2-3/4*c^4*b^3*arctanh(c*x^(1/2))^2*ln(c*x^(1/2)-1)-
1/2*a^3/x^2

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maxima [B]  time = 1.96, size = 703, normalized size = 3.00 \[ -2 \, {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b^{3} c^{4} - 2 \, {\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b^{3} c^{4} + 2 \, {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b^{3} c^{4} - \frac {1}{8} \, {\left ({\left (6 \, c^{3} \log \left (c \sqrt {x} - 1\right ) - 3 \, c^{3} \log \relax (x) + \frac {6 \, c^{2} x + 3 \, c \sqrt {x} + 2}{x^{\frac {3}{2}}}\right )} c - \frac {6 \, \log \left (-c \sqrt {x} + 1\right )}{x^{2}}\right )} a^{2} b + \frac {1}{4} \, {\left (3 \, a^{2} b c^{4} - 8 \, a b^{2} c^{4} + b^{3} c^{4}\right )} \log \left (c \sqrt {x} + 1\right ) - \frac {1}{4} \, {\left (8 \, a b^{2} c^{4} + b^{3} c^{4}\right )} \log \left (c \sqrt {x} - 1\right ) - \frac {1}{8} \, {\left (3 \, a^{2} b c^{4} - 16 \, a b^{2} c^{4}\right )} \log \relax (x) - \frac {a^{3}}{2 \, x^{2}} - \frac {4 \, a^{2} b c \sqrt {x} - {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{3} + {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (-c \sqrt {x} + 1\right )^{3} + 2 \, {\left (3 \, b^{3} c^{3} x^{\frac {3}{2}} + b^{3} c \sqrt {x} + 3 \, a b^{2} - {\left (3 \, a b^{2} c^{4} - 4 \, b^{3} c^{4}\right )} x^{2}\right )} \log \left (c \sqrt {x} + 1\right )^{2} + {\left (6 \, b^{3} c^{3} x^{\frac {3}{2}} + 2 \, b^{3} c \sqrt {x} + 6 \, a b^{2} - 2 \, {\left (3 \, a b^{2} c^{4} + 4 \, b^{3} c^{4}\right )} x^{2} - 3 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )^{2} + 4 \, {\left (3 \, a^{2} b c^{3} + 2 \, b^{3} c^{3}\right )} x^{\frac {3}{2}} - 2 \, {\left (3 \, a^{2} b c^{2} - 4 \, a b^{2} c^{2}\right )} x + 4 \, {\left (6 \, a b^{2} c^{3} x^{\frac {3}{2}} + b^{3} c^{2} x + 2 \, a b^{2} c \sqrt {x} + 3 \, a^{2} b\right )} \log \left (c \sqrt {x} + 1\right ) - {\left (24 \, a b^{2} c^{3} x^{\frac {3}{2}} + 4 \, b^{3} c^{2} x + 8 \, a b^{2} c \sqrt {x} - 3 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} + 4 \, {\left (3 \, b^{3} c^{3} x^{\frac {3}{2}} + b^{3} c \sqrt {x} + 3 \, a b^{2} - {\left (3 \, a b^{2} c^{4} - 4 \, b^{3} c^{4}\right )} x^{2}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{16 \, x^{2}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^(1/2)))^3/x^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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