Optimal. Leaf size=43 \[ \frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\text {ArcSin}(a x)}{a c} \]
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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6262, 677, 222}
\begin {gather*} \frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\text {ArcSin}(a x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 677
Rule 6262
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{c-a c x} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 1.07 \begin {gather*} \frac {2 \left (\frac {\sqrt {1+a x}}{\sqrt {1-a x}}+\text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 74, normalized size = 1.72
method | result | size |
default | \(-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{2} \left (x -\frac {1}{a}\right )}}{c}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 40, normalized size = 0.93 \begin {gather*} -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2} c x - a c} - \frac {\arcsin \left (a x\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 62, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (a x + {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} - 1\right )}}{a^{2} c x - a c} \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 {\int \frac {a x}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 53, normalized size = 1.23 \begin {gather*} -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c {\left | a \right |}} + \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 71, normalized size = 1.65 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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