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

Optimal. Leaf size=142 \[ -3 b^3 c^2 \text{PolyLog}\left (2,\frac{2}{c \sqrt{x}+1}-1\right )+6 b^2 c^2 \log \left (2-\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )+3 b c^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2+c^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3-\frac{3 b c \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{\sqrt{x}}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{x} \]

[Out]

3*b*c^2*(a + b*ArcTanh[c*Sqrt[x]])^2 - (3*b*c*(a + b*ArcTanh[c*Sqrt[x]])^2)/Sqrt[x] + c^2*(a + b*ArcTanh[c*Sqr
t[x]])^3 - (a + b*ArcTanh[c*Sqrt[x]])^3/x + 6*b^2*c^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] -
3*b^3*c^2*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]

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Rubi [F]  time = 0.0222653, 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 = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.307345, size = 230, normalized size = 1.62 \[ \frac{-6 b^3 c^2 x \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+a \left (-2 a^2-3 a b c^2 x \log \left (1-c \sqrt{x}\right )+3 a b c^2 x \log \left (c \sqrt{x}+1\right )-6 a b c \sqrt{x}+12 b^2 c^2 x \log \left (\frac{c \sqrt{x}}{\sqrt{1-c^2 x}}\right )\right )-6 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (a^2+2 a b c \sqrt{x}-2 b^2 c^2 x \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )\right )+6 b^2 \left (c \sqrt{x}-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2 \left (a c \sqrt{x}+a+b c \sqrt{x}\right )+2 b^3 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^3}{2 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*b^2*(-1 + c*Sqrt[x])*(a + a*c*Sqrt[x] + b*c*Sqrt[x])*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^2*x)*ArcTanh[c*Sq
rt[x]]^3 - 6*b*ArcTanh[c*Sqrt[x]]*(a^2 + 2*a*b*c*Sqrt[x] - 2*b^2*c^2*x*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + a
*(-2*a^2 - 6*a*b*c*Sqrt[x] - 3*a*b*c^2*x*Log[1 - c*Sqrt[x]] + 3*a*b*c^2*x*Log[1 + c*Sqrt[x]] + 12*b^2*c^2*x*Lo
g[(c*Sqrt[x])/Sqrt[1 - c^2*x]]) - 6*b^3*c^2*x*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])])/(2*x)

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Maple [C]  time = 0.412, size = 5199, normalized size = 36.6 \begin{align*} \text{output 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^2,x)

[Out]

result too large to display

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Maxima [B]  time = 5.86528, size = 713, normalized size = 5.02 \begin{align*} -3 \,{\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^{2} - 3 \,{\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^{2} + 3 \,{\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^{2} - 3 \, a b^{2} c^{2} \log \left (c \sqrt{x} - 1\right ) - \frac{3}{4} \,{\left ({\left (2 \, c \log \left (c \sqrt{x} - 1\right ) - c \log \left (x\right ) + \frac{2}{\sqrt{x}}\right )} c - \frac{2 \, \log \left (-c \sqrt{x} + 1\right )}{x}\right )} a^{2} b - \frac{a^{3}}{x} + \frac{3}{2} \,{\left (a^{2} b c^{2} - 2 \, a b^{2} c^{2}\right )} \log \left (c \sqrt{x} + 1\right ) - \frac{3}{4} \,{\left (a^{2} b c^{2} - 4 \, a b^{2} c^{2}\right )} \log \left (x\right ) - \frac{12 \, a^{2} b c \sqrt{x} -{\left (b^{3} c^{2} x - b^{3}\right )} \log \left (c \sqrt{x} + 1\right )^{3} +{\left (b^{3} c^{2} x - b^{3}\right )} \log \left (-c \sqrt{x} + 1\right )^{3} + 6 \,{\left (b^{3} c \sqrt{x} + a b^{2} -{\left (a b^{2} c^{2} - b^{3} c^{2}\right )} x\right )} \log \left (c \sqrt{x} + 1\right )^{2} + 3 \,{\left (2 \, b^{3} c \sqrt{x} + 2 \, a b^{2} - 2 \,{\left (a b^{2} c^{2} + b^{3} c^{2}\right )} x -{\left (b^{3} c^{2} x - b^{3}\right )} \log \left (c \sqrt{x} + 1\right )\right )} \log \left (-c \sqrt{x} + 1\right )^{2} + 12 \,{\left (2 \, a b^{2} c \sqrt{x} + a^{2} b\right )} \log \left (c \sqrt{x} + 1\right ) - 3 \,{\left (8 \, a b^{2} c \sqrt{x} -{\left (b^{3} c^{2} x - b^{3}\right )} \log \left (c \sqrt{x} + 1\right )^{2} + 4 \,{\left (b^{3} c \sqrt{x} + a b^{2} -{\left (a b^{2} c^{2} - b^{3} c^{2}\right )} x\right )} \log \left (c \sqrt{x} + 1\right )\right )} \log \left (-c \sqrt{x} + 1\right )}{8 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b^3*c^2 - 3*(log(c*sqrt(x))*log
(-c*sqrt(x) + 1) + dilog(-c*sqrt(x) + 1))*b^3*c^2 + 3*(log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog(c*sqrt(x) +
1))*b^3*c^2 - 3*a*b^2*c^2*log(c*sqrt(x) - 1) - 3/4*((2*c*log(c*sqrt(x) - 1) - c*log(x) + 2/sqrt(x))*c - 2*log(
-c*sqrt(x) + 1)/x)*a^2*b - a^3/x + 3/2*(a^2*b*c^2 - 2*a*b^2*c^2)*log(c*sqrt(x) + 1) - 3/4*(a^2*b*c^2 - 4*a*b^2
*c^2)*log(x) - 1/8*(12*a^2*b*c*sqrt(x) - (b^3*c^2*x - b^3)*log(c*sqrt(x) + 1)^3 + (b^3*c^2*x - b^3)*log(-c*sqr
t(x) + 1)^3 + 6*(b^3*c*sqrt(x) + a*b^2 - (a*b^2*c^2 - b^3*c^2)*x)*log(c*sqrt(x) + 1)^2 + 3*(2*b^3*c*sqrt(x) +
2*a*b^2 - 2*(a*b^2*c^2 + b^3*c^2)*x - (b^3*c^2*x - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 + 12*(2*a*b^
2*c*sqrt(x) + a^2*b)*log(c*sqrt(x) + 1) - 3*(8*a*b^2*c*sqrt(x) - (b^3*c^2*x - b^3)*log(c*sqrt(x) + 1)^2 + 4*(b
^3*c*sqrt(x) + a*b^2 - (a*b^2*c^2 - b^3*c^2)*x)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1))/x

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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^{2}}, 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^2,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^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{2}}\,{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^2,x, algorithm="giac")

[Out]

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