Optimal. Leaf size=77 \[ \frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \]
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Rubi [A]
time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6295, 6285, 45}
\begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-a^2 x^2} \log (a x+1)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rule 6295
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-\tanh ^{-1}(a x)} x}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x}{1+a x} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{a}-\frac {1}{a (1+a x)}\right ) \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 50, normalized size = 0.65 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (\frac {x}{a}-\frac {\log (1+a x)}{a^2}\right )}{\sqrt {c-\frac {c}{a^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 51, normalized size = 0.66
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-a x +\ln \left (a x +1\right )\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.28, size = 21, normalized size = 0.27 \begin {gather*} \frac {i \, x}{\sqrt {c}} - \frac {i \, \log \left (a x + 1\right )}{a \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs.
\(2 (69) = 138\).
time = 0.38, size = 367, normalized size = 4.77 \begin {gather*} \left [\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right )}{2 \, {\left (a^{3} c x^{2} - a c\right )}}, \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right )}{a^{3} c x^{2} - a c}\right ] \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*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-a^2\,x^2}}{\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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