Optimal. Leaf size=40 \[ \frac {b \sqrt {1-c x}}{x \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{x} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6418, 97}
\begin {gather*} \frac {b \sqrt {1-c x}}{x \sqrt {\frac {1}{c x+1}}}-\frac {a+b \text {sech}^{-1}(c x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 97
Rule 6418
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^2} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{x}-\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=\frac {b \sqrt {1-c x}}{x \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{x}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 42, normalized size = 1.05 \begin {gather*} -\frac {a}{x}+b \left (c+\frac {1}{x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b \text {sech}^{-1}(c x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 58, normalized size = 1.45
method | result | size |
derivativedivides | \(c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\) | \(58\) |
default | \(c \left (-\frac {a}{c x}+b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 32, normalized size = 0.80 \begin {gather*} {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 66, normalized size = 1.65 \begin {gather*} \frac {b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - b \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - a}{x} \end {gather*}
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 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.48, size = 46, normalized size = 1.15 \begin {gather*} b\,c\,\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}-\frac {b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x}-\frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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