Optimal. Leaf size=118 \[ -\frac {b^2 (1-c x) (1+c x)}{4 x^2}-\frac {1}{2} a b c^2 \text {sech}^{-1}(c x)-\frac {1}{4} b^2 c^2 \text {sech}^{-1}(c x)^2+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 118, 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, 5554,
3391} \begin {gather*} -\frac {1}{2} a b c^2 \text {sech}^{-1}(c x)+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{4} b^2 c^2 \text {sech}^{-1}(c x)^2-\frac {b^2 (1-c x) (c x+1)}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3391
Rule 5554
Rule 6420
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^3} \, dx &=-\left (c^2 \text {Subst}\left (\int (a+b x)^2 \cosh (x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}+\left (b c^2\right ) \text {Subst}\left (\int (a+b x) \sinh ^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {b^2 (1-c x) (1+c x)}{4 x^2}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{2} \left (b c^2\right ) \text {Subst}\left (\int (a+b x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {b^2 (1-c x) (1+c x)}{4 x^2}-\frac {1}{2} a b c^2 \text {sech}^{-1}(c x)-\frac {1}{4} b^2 c^2 \text {sech}^{-1}(c x)^2+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 183, normalized size = 1.55 \begin {gather*} \frac {-2 a^2-b^2+2 a b \sqrt {\frac {1-c x}{1+c x}}+2 a b c x \sqrt {\frac {1-c x}{1+c x}}+2 b \left (-2 a+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+b^2 \left (-2+c^2 x^2\right ) \text {sech}^{-1}(c x)^2-2 a b c^2 x^2 \log (x)+2 a b c^2 x^2 \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 192, normalized size = 1.63
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{2 c x}+\frac {\mathrm {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(192\) |
default | \(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{2 c x}+\frac {\mathrm {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+2 a b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 165, normalized size = 1.40 \begin {gather*} \frac {2 \, a b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, {\left (a b c^{2} x^{2} + b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, a b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, x^{2}} \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^{3}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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