Optimal. Leaf size=78 \[ -\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{x}+3 b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x}+6 b^3 c \sqrt{\frac{1}{c^2 x^2}+1} \]
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Rubi [A] time = 0.0989242, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6286, 3296, 2638} \[ -\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{x}+3 b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x}+6 b^3 c \sqrt{\frac{1}{c^2 x^2}+1} \]
Antiderivative was successfully verified.
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Rule 6286
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x^2} \, dx &=-\left (c \operatorname{Subst}\left (\int (a+b x)^3 \cosh (x) \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x}+(3 b c) \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=3 b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x}-\left (6 b^2 c\right ) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{x}+3 b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x}+\left (6 b^3 c\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=6 b^3 c \sqrt{1+\frac{1}{c^2 x^2}}-\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right )}{x}+3 b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x}\\ \end{align*}
Mathematica [A] time = 0.23174, size = 132, normalized size = 1.69 \[ -\frac{3 b \text{csch}^{-1}(c x) \left (a^2-2 a b c x \sqrt{\frac{1}{c^2 x^2}+1}+2 b^2\right )-3 a^2 b c x \sqrt{\frac{1}{c^2 x^2}+1}+a^3+3 b^2 \text{csch}^{-1}(c x)^2 \left (a-b c x \sqrt{\frac{1}{c^2 x^2}+1}\right )+6 a b^2-6 b^3 c x \sqrt{\frac{1}{c^2 x^2}+1}+b^3 \text{csch}^{-1}(c x)^3}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.186, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992252, size = 194, normalized size = 2.49 \begin{align*} -\frac{b^{3} \operatorname{arcsch}\left (c x\right )^{3}}{x} + 3 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} + 1} - \frac{\operatorname{arcsch}\left (c x\right )}{x}\right )} a^{2} b + 6 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} + 1} \operatorname{arcsch}\left (c x\right ) - \frac{1}{x}\right )} a b^{2} + 3 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} + 1} \operatorname{arcsch}\left (c x\right )^{2} + 2 \, c \sqrt{\frac{1}{c^{2} x^{2}} + 1} - \frac{2 \, \operatorname{arcsch}\left (c x\right )}{x}\right )} b^{3} - \frac{3 \, a b^{2} \operatorname{arcsch}\left (c x\right )^{2}}{x} - \frac{a^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.31187, size = 477, normalized size = 6.12 \begin{align*} -\frac{b^{3} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + a^{3} + 6 \, a b^{2} - 3 \,{\left (b^{3} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - a b^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \,{\left (2 \, a b^{2} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - a^{2} b - 2 \, b^{3}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \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{arcsch}\left (c x\right ) + a\right )}^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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