Optimal. Leaf size=35 \[ c \sqrt {1-a^2 x^2}-c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6128, 266, 50, 63, 208} \[ c \sqrt {1-a^2 x^2}-c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)}{x} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x} \, dx\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x} \, dx,x,x^2\right )\\ &=c \sqrt {1-a^2 x^2}+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=c \sqrt {1-a^2 x^2}-\frac {c \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2}\\ &=c \sqrt {1-a^2 x^2}-c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [B] time = 0.08, size = 79, normalized size = 2.26 \[ c \left (-\frac {a^2 x^2}{\sqrt {1-a^2 x^2}}+\frac {1}{\sqrt {1-a^2 x^2}}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\sin ^{-1}(a x)+2 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 36, normalized size = 1.03 \[ c \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} c \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 53, normalized size = 1.51 \[ -\frac {1}{2} \, c \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) + \frac {1}{2} \, c \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) + \sqrt {-a^{2} x^{2} + 1} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 32, normalized size = 0.91 \[ -c \left (-\sqrt {-a^{2} x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 44, normalized size = 1.26 \[ -c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \sqrt {-a^{2} x^{2} + 1} c \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 31, normalized size = 0.89 \[ -c\,\left (\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\sqrt {1-a^2\,x^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.27, size = 66, normalized size = 1.89 \[ \frac {a^{2} c \left (\begin {cases} - x^{2} & \text {for}\: a^{2} = 0 \\\frac {2 \sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right )}{2} - \frac {c \left (- \log {\left (-1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )} + \log {\left (1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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