Optimal. Leaf size=90 \[ \frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1+a x) \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \tanh ^{-1}(a x)}{2 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6327, 6328, 46,
213} \begin {gather*} \frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (a x+1) \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \tanh ^{-1}(a x)}{2 \left (c-a^2 c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 6327
Rule 6328
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{(-1+a x) (1+a x)^2} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (-\frac {1}{2 (1+a x)^2}+\frac {1}{2 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1+a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{-1+a^2 x^2} \, dx}{2 \left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1+a x) \left (c-a^2 c x^2\right )^{3/2}}-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \tanh ^{-1}(a x)}{2 \left (c-a^2 c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 54, normalized size = 0.60 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-1+(1+a x) \tanh ^{-1}(a x)\right )}{2 (c+a c x) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 84, normalized size = 0.93
method | result | size |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x +1\right ) a x -x \ln \left (a x -1\right ) a +\ln \left (a x +1\right )-\ln \left (a x -1\right )-2\right )}{4 \left (a^{2} x^{2}-1\right ) c^{2} a}\) | \(84\) |
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 = 0.35, size = 83, normalized size = 0.92 \begin {gather*} -\frac {{\left (a^{2} x + a\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c} \sqrt {-c} x + c}{a^{2} x^{2} - 1}\right ) - 2 \, \sqrt {-a^{2} c}}{4 \, {\left (a^{3} c^{2} x + a^{2} c^{2}\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 {\frac {a x - 1}{a x + 1}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, 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 {\frac {a\,x-1}{a\,x+1}}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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