Optimal. Leaf size=46 \[ \frac {1}{2} a^2 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {c \sqrt {1-a^2 x^2}}{2 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 46, 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, 47, 63, 208} \[ \frac {1}{2} a^2 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {c \sqrt {1-a^2 x^2}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)}{x^3} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {1}{4} \left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}+\frac {1}{2} 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 )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2}+\frac {1}{2} a^2 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 1.46 \[ \frac {c \left (a^2 x^2+a^2 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-1\right )}{2 x^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 47, normalized size = 1.02 \[ -\frac {a^{2} c x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 70, normalized size = 1.52 \[ \frac {a^{4} c \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a^{4} c \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c}{x^{2}}}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 0.87 \[ -c \left (-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 51, normalized size = 1.11 \[ \frac {1}{2} \, a^{2} c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 38, normalized size = 0.83 \[ \frac {a^2\,c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{2}-\frac {c\,\sqrt {1-a^2\,x^2}}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.45, size = 73, normalized size = 1.59 \[ - a^{2} c \left (\frac {\log {\left (\sqrt {- a^{2} x^{2} + 1} - 1 \right )}}{4} - \frac {\log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{4} - \frac {1}{4 \left (\sqrt {- a^{2} x^{2} + 1} + 1\right )} - \frac {1}{4 \left (\sqrt {- a^{2} x^{2} + 1} - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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