3.223 \(\int \frac{(a+b \tanh ^{-1}(c x^{3/2}))^2}{x^4} \, dx\)

Optimal. Leaf size=96 \[ \frac{1}{3} c^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2-\frac{2 b c \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )}{3 x^{3/2}}-\frac{\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{3 x^3}-\frac{1}{3} b^2 c^2 \log \left (1-c^2 x^3\right )+b^2 c^2 \log (x) \]

[Out]

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

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Rubi [F]  time = 0.0237288, 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 = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*x^(3/2)])^2/x^4,x]

[Out]

Defer[Int][(a + b*ArcTanh[c*x^(3/2)])^2/x^4, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{x^4} \, dx &=\int \frac{\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{x^4} \, dx\\ \end{align*}

Mathematica [A]  time = 0.139447, size = 123, normalized size = 1.28 \[ \frac{1}{3} \left (-\frac{a^2}{x^3}-b c^2 (a+b) \log \left (1-c x^{3/2}\right )+b c^2 (a-b) \log \left (c x^{3/2}+1\right )-\frac{2 a b c}{x^{3/2}}-\frac{2 b \tanh ^{-1}\left (c x^{3/2}\right ) \left (a+b c x^{3/2}\right )}{x^3}+\frac{b^2 \left (c^2 x^3-1\right ) \tanh ^{-1}\left (c x^{3/2}\right )^2}{x^3}+3 b^2 c^2 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^(3/2)])^2/x^4,x]

[Out]

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

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Maple [F]  time = 0.28, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b{\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(3/2)))^2/x^4,x)

[Out]

int((a+b*arctanh(c*x^(3/2)))^2/x^4,x)

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Maxima [B]  time = 0.998003, size = 236, normalized size = 2.46 \begin{align*} \frac{1}{3} \,{\left ({\left (c \log \left (c x^{\frac{3}{2}} + 1\right ) - c \log \left (c x^{\frac{3}{2}} - 1\right ) - \frac{2}{x^{\frac{3}{2}}}\right )} c - \frac{2 \, \operatorname{artanh}\left (c x^{\frac{3}{2}}\right )}{x^{3}}\right )} a b + \frac{1}{12} \,{\left ({\left (2 \,{\left (\log \left (c x^{\frac{3}{2}} - 1\right ) - 2\right )} \log \left (c x^{\frac{3}{2}} + 1\right ) - \log \left (c x^{\frac{3}{2}} + 1\right )^{2} - \log \left (c x^{\frac{3}{2}} - 1\right )^{2} - 4 \, \log \left (c x^{\frac{3}{2}} - 1\right ) + 12 \, \log \left (x\right )\right )} c^{2} + 4 \,{\left (c \log \left (c x^{\frac{3}{2}} + 1\right ) - c \log \left (c x^{\frac{3}{2}} - 1\right ) - \frac{2}{x^{\frac{3}{2}}}\right )} c \operatorname{artanh}\left (c x^{\frac{3}{2}}\right )\right )} b^{2} - \frac{b^{2} \operatorname{artanh}\left (c x^{\frac{3}{2}}\right )^{2}}{3 \, x^{3}} - \frac{a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(3/2)))^2/x^4,x, algorithm="maxima")

[Out]

1/3*((c*log(c*x^(3/2) + 1) - c*log(c*x^(3/2) - 1) - 2/x^(3/2))*c - 2*arctanh(c*x^(3/2))/x^3)*a*b + 1/12*((2*(l
og(c*x^(3/2) - 1) - 2)*log(c*x^(3/2) + 1) - log(c*x^(3/2) + 1)^2 - log(c*x^(3/2) - 1)^2 - 4*log(c*x^(3/2) - 1)
+ 12*log(x))*c^2 + 4*(c*log(c*x^(3/2) + 1) - c*log(c*x^(3/2) - 1) - 2/x^(3/2))*c*arctanh(c*x^(3/2)))*b^2 - 1/
3*b^2*arctanh(c*x^(3/2))^2/x^3 - 1/3*a^2/x^3

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Fricas [B]  time = 1.89078, size = 402, normalized size = 4.19 \begin{align*} \frac{24 \, b^{2} c^{2} x^{3} \log \left (\sqrt{x}\right ) + 4 \,{\left (a b - b^{2}\right )} c^{2} x^{3} \log \left (c x^{\frac{3}{2}} + 1\right ) - 4 \,{\left (a b + b^{2}\right )} c^{2} x^{3} \log \left (c x^{\frac{3}{2}} - 1\right ) - 8 \, a b c x^{\frac{3}{2}} +{\left (b^{2} c^{2} x^{3} - b^{2}\right )} \log \left (-\frac{c^{2} x^{3} + 2 \, c x^{\frac{3}{2}} + 1}{c^{2} x^{3} - 1}\right )^{2} - 4 \, a^{2} - 4 \,{\left (b^{2} c x^{\frac{3}{2}} + a b\right )} \log \left (-\frac{c^{2} x^{3} + 2 \, c x^{\frac{3}{2}} + 1}{c^{2} x^{3} - 1}\right )}{12 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(3/2)))^2/x^4,x, algorithm="fricas")

[Out]

1/12*(24*b^2*c^2*x^3*log(sqrt(x)) + 4*(a*b - b^2)*c^2*x^3*log(c*x^(3/2) + 1) - 4*(a*b + b^2)*c^2*x^3*log(c*x^(
3/2) - 1) - 8*a*b*c*x^(3/2) + (b^2*c^2*x^3 - b^2)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1))^2 - 4*a^2 -
4*(b^2*c*x^(3/2) + a*b)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1)))/x^3

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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**(3/2)))**2/x**4,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 x^{\frac{3}{2}}\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(3/2)))^2/x^4,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^(3/2)) + a)^2/x^4, x)