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