Optimal. Leaf size=50 \[ -\frac{a+b \text{csch}^{-1}(c x)}{2 x^2}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 \text{csch}^{-1}(c x) \]
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Rubi [A] time = 0.0372564, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6284, 335, 321, 215} \[ -\frac{a+b \text{csch}^{-1}(c x)}{2 x^2}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 6284
Rule 335
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{x^3} \, dx &=-\frac{a+b \text{csch}^{-1}(c x)}{2 x^2}-\frac{b \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^4} \, dx}{2 c}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{2 x^2}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{4 x}-\frac{a+b \text{csch}^{-1}(c x)}{2 x^2}-\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 \text{csch}^{-1}(c x)-\frac{a+b \text{csch}^{-1}(c x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0331019, size = 66, normalized size = 1.32 \[ -\frac{a}{2 x^2}+\frac{b c \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 \sinh ^{-1}\left (\frac{1}{c x}\right )-\frac{b \text{csch}^{-1}(c x)}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.182, size = 100, normalized size = 2. \begin{align*}{c}^{2} \left ( -{\frac{a}{2\,{c}^{2}{x}^{2}}}+b \left ( -{\frac{{\rm arccsch} \left (cx\right )}{2\,{c}^{2}{x}^{2}}}-{\frac{1}{4\,{c}^{3}{x}^{3}}\sqrt{{c}^{2}{x}^{2}+1} \left ({\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ){c}^{2}{x}^{2}-\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.997988, size = 142, normalized size = 2.84 \begin{align*} \frac{1}{8} \, b{\left (\frac{\frac{2 \, c^{4} x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - 1} - c^{3} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) + c^{3} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c} - \frac{4 \, \operatorname{arcsch}\left (c x\right )}{x^{2}}\right )} - \frac{a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2034, size = 167, normalized size = 3.34 \begin{align*} \frac{b c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} -{\left (b c^{2} x^{2} + 2 \, b\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, a}{4 \, x^{2}} \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{a + b \operatorname{acsch}{\left (c x \right )}}{x^{3}}\, 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{b \operatorname{arcsch}\left (c x\right ) + a}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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