Optimal. Leaf size=27 \[ c x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {c \csc ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6177, 277, 216} \[ c x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {c \csc ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 216
Rule 277
Rule 6177
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {c \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=c \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {c \csc ^{-1}(a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 31, normalized size = 1.15 \[ \frac {c \left (a x \sqrt {1-\frac {1}{a^2 x^2}}+\sin ^{-1}\left (\frac {1}{a x}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 48, normalized size = 1.78 \[ -\frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 86, normalized size = 3.19 \[ -\frac {1}{2} \, a c {\left (\frac {\pi + 2 \, \arctan \left (\frac {\frac {a x - 1}{a x + 1} - 1}{2 \, \sqrt {\frac {a x - 1}{a x + 1}}}\right )}{a^{2}} + \frac {4}{a^{2} {\left (\sqrt {\frac {a x - 1}{a x + 1}} - \frac {1}{\sqrt {\frac {a x - 1}{a x + 1}}}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 63, normalized size = 2.33 \[ \frac {\left (a x -1\right ) c \left (\sqrt {a^{2} x^{2}-1}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 66, normalized size = 2.44 \[ -2 \, a {\left (\frac {c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac {c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 60, normalized size = 2.22 \[ \frac {2\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}}-\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c \left (\int \frac {a}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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