Optimal. Leaf size=77 \[ \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{(1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1+a x)}{\sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.11, 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 = {6327, 6328, 45}
\begin {gather*} \frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{(a x+1) \sqrt {c-a^2 c x^2}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{\sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6327
Rule 6328
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\left (\sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}} x} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {-1+a x}{(1+a x)^2} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \left (-\frac {2}{(1+a x)^2}+\frac {1}{1+a x}\right ) \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{(1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1+a x)}{\sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 52, normalized size = 0.68 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} x (2+(1+a x) \log (1+a x))}{(1+a x) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 62, normalized size = 0.81
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x +1\right ) a x +\ln \left (a x +1\right )+2\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a c \left (a x -1\right )^{2}}\) | \(62\) |
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 = 0.34, size = 38, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {-a^{2} c} {\left ({\left (a x + 1\right )} \log \left (a x + 1\right ) + 2\right )}}{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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\sqrt {- c \left (a x - 1\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 {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{\sqrt {c-a^2\,c\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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