Optimal. Leaf size=113 \[ \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A]
time = 0.09, antiderivative size = 113, 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 = {6332, 6328, 78}
\begin {gather*} \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6328
Rule 6332
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x (-1+a x)}{(1+a x)^2} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a}+\frac {2}{a (1+a x)^2}-\frac {3}{a (1+a x)}\right ) \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 54, normalized size = 0.48 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (a x-\frac {2}{1+a x}-3 \log (1+a x)\right )}{a \sqrt {c-\frac {c}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 87, normalized size = 0.77
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \left (-a^{2} x^{2}+3 \ln \left (a x +1\right ) a x -a x +3 \ln \left (a x +1\right )+2\right )}{\left (a x -1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2}}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 47, normalized size = 0.42 \begin {gather*} \frac {{\left (a^{2} x^{2} + a x - 3 \, {\left (a x + 1\right )} \log \left (a x + 1\right ) - 2\right )} \sqrt {a^{2} c}}{a^{3} c x + a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{\sqrt {c-\frac {c}{a^2\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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