Optimal. Leaf size=61 \[ -\frac {2 b^2}{x}+\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x} \]
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Rubi [A]
time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6420, 3377,
2718} \begin {gather*} \frac {2 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {2 b^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 6420
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx &=-\left (c \text {Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x}+(2 b c) \text {Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\left (2 b^2 c\right ) \text {Subst}\left (\int \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{x}+\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 87, normalized size = 1.43 \begin {gather*} -\frac {a^2+2 b^2-2 a b \sqrt {\frac {1-c x}{1+c x}} (1+c x)-2 b \left (-a+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+b^2 \text {sech}^{-1}(c x)^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs.
\(2(59)=118\).
time = 0.25, size = 124, normalized size = 2.03
method | result | size |
derivativedivides | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{c x}+2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+2 a b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\) | \(124\) |
default | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{c x}+2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+2 a b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 78, normalized size = 1.28 \begin {gather*} 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} a b + 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right ) - \frac {1}{x}\right )} b^{2} - \frac {b^{2} \operatorname {arsech}\left (c x\right )^{2}}{x} - \frac {a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (59) = 118\).
time = 0.35, size = 143, normalized size = 2.34 \begin {gather*} \frac {2 \, a b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - b^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - a^{2} - 2 \, b^{2} + 2 \, {\left (b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{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 {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}{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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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