Optimal. Leaf size=37 \[ -\frac {\sqrt {c x-1}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 50, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5713, 5676} \[ -\frac {\sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5676
Rule 5713
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 1.35 \[ -\frac {\sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 75, normalized size = 2.03 \[ \frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1}}{a b c^{3} x^{2} - a b c + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )} \]
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 \[ \int \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 57, normalized size = 1.54 \[ \frac {\sqrt {-\left (c x -1\right ) \left (c x +1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}}{c \left (c^{2} x^{2}-1\right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b} \]
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 \[ -\frac {c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x + {\left (a b c^{3} x^{2} - a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} + \int -\frac {c^{2} x^{2} - {\left (c x + 1\right )} {\left (c x - 1\right )} - 1}{{\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} b^{2} c^{2} x^{2} + 2 \, {\left (b^{2} c^{3} x^{3} - b^{2} c x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} a b c^{2} x^{2} + 2 \, {\left (a b c^{3} x^{3} - a b c x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 59, normalized size = 1.59 \[ -\frac {b\,\sqrt {1-c^2\,x^2}\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{c\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (b^2-b^2\,c^2\,x^2\right )} \]
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 \[ \int \frac {1}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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