Optimal. Leaf size=50 \[ \frac {c \sqrt {1-a^2 x^2}}{a}+\frac {2 c \text {ArcSin}(a x)}{a}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a} \]
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Rubi [A]
time = 0.10, 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 = {6266, 6263,
1823, 858, 222, 272, 65, 214} \begin {gather*} \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 \text {ArcSin}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1823
Rule 6263
Rule 6266
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 \text {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 \text {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 \begin {gather*} \frac {c \left (\sqrt {1-a^2 x^2}+2 \text {ArcSin}(a x)-\log (x)+\log \left (1+\sqrt {1-a^2 x^2}\right )\right )}{a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs.
\(2(46)=92\).
time = 0.75, size = 98, normalized size = 1.96
method | result | size |
default | \(\frac {c \left (2 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {2 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}-\sqrt {-a^{2} x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 70, normalized size = 1.40 \begin {gather*} 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 67, normalized size = 1.34 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 68, normalized size = 1.36 \begin {gather*} \frac {2 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\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} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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