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

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

[Out]

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

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Rubi [F]  time = 0.0063135, 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 = {} \[ \int \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*Sqrt[x]])^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.276754, size = 201, normalized size = 1.42 \[ \frac{6 b^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+a \left (2 a^2 c^2 x+6 a b c \sqrt{x}+3 a b \log \left (1-c \sqrt{x}\right )-3 a b \log \left (c \sqrt{x}+1\right )+6 b^2 \log \left (1-c^2 x\right )\right )+6 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (a^2 c^2 x+2 a b c \sqrt{x}-2 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )\right )+6 b^2 \left (c \sqrt{x}-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2 \left (a c \sqrt{x}+a+b\right )+2 b^3 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^3}{2 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*b^2*(-1 + c*Sqrt[x])*(a + b + a*c*Sqrt[x])*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^3 +
6*b*ArcTanh[c*Sqrt[x]]*(2*a*b*c*Sqrt[x] + a^2*c^2*x - 2*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + a*(6*a*b*c*
Sqrt[x] + 2*a^2*c^2*x + 3*a*b*Log[1 - c*Sqrt[x]] - 3*a*b*Log[1 + c*Sqrt[x]] + 6*b^2*Log[1 - c^2*x]) + 6*b^3*Po
lyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(2*c^2)

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Maple [C]  time = 0.388, size = 6235, normalized size = 43.9 \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)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \,{\left (c{\left (\frac{2 \, \sqrt{x}}{c^{2}} - \frac{\log \left (c \sqrt{x} + 1\right )}{c^{3}} + \frac{\log \left (c \sqrt{x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname{artanh}\left (c \sqrt{x}\right )\right )} a^{2} b + \frac{3}{4} \,{\left (4 \, c{\left (\frac{2 \, \sqrt{x}}{c^{2}} - \frac{\log \left (c \sqrt{x} + 1\right )}{c^{3}} + \frac{\log \left (c \sqrt{x} - 1\right )}{c^{3}}\right )} \operatorname{artanh}\left (c \sqrt{x}\right ) + 4 \, x \operatorname{artanh}\left (c \sqrt{x}\right )^{2} - \frac{2 \,{\left (\log \left (c \sqrt{x} - 1\right ) - 2\right )} \log \left (c \sqrt{x} + 1\right ) - \log \left (c \sqrt{x} + 1\right )^{2} - \log \left (c \sqrt{x} - 1\right )^{2} - 4 \, \log \left (c \sqrt{x} - 1\right )}{c^{2}}\right )} a b^{2} + a^{3} x - \frac{1}{32} \, b^{3}{\left (\frac{{\left (4 \, \log \left (-c \sqrt{x} + 1\right )^{3} - 6 \, \log \left (-c \sqrt{x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt{x} + 1\right ) - 3\right )}{\left (c \sqrt{x} - 1\right )}^{2} + 8 \,{\left (\log \left (-c \sqrt{x} + 1\right )^{3} - 3 \, \log \left (-c \sqrt{x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt{x} + 1\right ) - 6\right )}{\left (c \sqrt{x} - 1\right )}}{c^{2}} - 4 \, \int \log \left (c \sqrt{x} + 1\right )^{3} - 3 \, \log \left (c \sqrt{x} + 1\right )^{2} \log \left (-c \sqrt{x} + 1\right ) + 3 \, \log \left (c \sqrt{x} + 1\right ) \log \left (-c \sqrt{x} + 1\right )^{2}\,{d 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, algorithm="maxima")

[Out]

3/2*(c*(2*sqrt(x)/c^2 - log(c*sqrt(x) + 1)/c^3 + log(c*sqrt(x) - 1)/c^3) + 2*x*arctanh(c*sqrt(x)))*a^2*b + 3/4
*(4*c*(2*sqrt(x)/c^2 - log(c*sqrt(x) + 1)/c^3 + log(c*sqrt(x) - 1)/c^3)*arctanh(c*sqrt(x)) + 4*x*arctanh(c*sqr
t(x))^2 - (2*(log(c*sqrt(x) - 1) - 2)*log(c*sqrt(x) + 1) - log(c*sqrt(x) + 1)^2 - log(c*sqrt(x) - 1)^2 - 4*log
(c*sqrt(x) - 1))/c^2)*a*b^2 + a^3*x - 1/32*b^3*(((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 + 8*(log(-c*sqrt(x) + 1)^3 - 3*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) +
1) - 6)*(c*sqrt(x) - 1))/c^2 - 4*integrate(log(c*sqrt(x) + 1)^3 - 3*log(c*sqrt(x) + 1)^2*log(-c*sqrt(x) + 1)
+ 3*log(c*sqrt(x) + 1)*log(-c*sqrt(x) + 1)^2, x))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (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\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))^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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((a+b*atanh(c*x**(1/2)))**3,x)

[Out]

Integral((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}\,{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, algorithm="giac")

[Out]

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