Optimal. Leaf size=50 \[ \frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}+\frac {2 c \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.15, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6131, 6128, 1809, 844, 216, 266, 63, 208} \[ \frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}+\frac {2 c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1809
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\frac {c \int \frac {e^{-\tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac {c \int \frac {(1-a x)^2}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \int \frac {-a^2+2 a^3 x}{x \sqrt {1-a^2 x^2}} \, dx}{a^3}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+(2 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {c \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {2 c \sin ^{-1}(a x)}{a}-\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {2 c \sin ^{-1}(a x)}{a}+\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^3}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {2 c \sin ^{-1}(a x)}{a}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 47, normalized size = 0.94 \[ \frac {c \left (\sqrt {1-a^2 x^2}+\log \left (\sqrt {1-a^2 x^2}+1\right )+2 \sin ^{-1}(a x)-\log (x)\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 67, normalized size = 1.34 \[ -\frac {4 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt {-a^{2} x^{2} + 1} c}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 68, normalized size = 1.36 \[ \frac {2 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 106, normalized size = 2.12 \[ -\frac {c \sqrt {-a^{2} x^{2}+1}}{a}+\frac {c \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}+\frac {2 c \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a}+\frac {2 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 70, normalized size = 1.40 \[ a c {\left (\frac {\arcsin \left (a x\right )}{a^{2}} + \frac {\log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a^{2}}\right )} + c {\left (\frac {\arcsin \left (a x\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 56, normalized size = 1.12 \[ \frac {c\,\sqrt {1-a^2\,x^2}}{a}+\frac {c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{a}+\frac {2\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}} \]
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 \[ \frac {c \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a x^{2} + x}\right )\, dx + \int \frac {a x \sqrt {- a^{2} x^{2} + 1}}{a x^{2} + x}\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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