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 = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 215
Rule 321
Rule 335
Rule 6284
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.98, size = 76, normalized size = 1.52 \[ \frac {b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - {\left (b c^{2} x^{2} + 2 \, b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, a}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 100, normalized size = 2.00 \[ c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\mathrm {arccsch}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\right )}{4 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 105, normalized size = 2.10 \[ \frac {1}{8} \, b {\left (\frac {\frac {2 \, c^{4} x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - 1} - c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c} - \frac {4 \, \operatorname {arcsch}\left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.27, size = 51, normalized size = 1.02 \[ \frac {b\,c\,\sqrt {\frac {1}{c^2\,x^2}+1}}{4\,x}-\frac {b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\,\left (\frac {c^2\,x}{4}+\frac {1}{2\,x}\right )}{x}-\frac {a}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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