Optimal. Leaf size=91 \[ \frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \tanh ^{-1}(a x)}{2 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6192, 6193, 44, 207} \[ \frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \tanh ^{-1}(a x)}{2 \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6192
Rule 6193
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)^2 (1+a x)} \, 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 = 56, normalized size = 0.62 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left ((a x-1) \tanh ^{-1}(a x)-1\right )}{2 c (a x-1) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 86, normalized size = 0.95 \[ -\frac {{\left (a^{2} x - a\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c} \sqrt {-c} x + c}{a^{2} x^{2} - 1}\right ) + 2 \, \sqrt {-a^{2} c}}{4 \, {\left (a^{3} c^{2} x - a^{2} c^{2}\right )}} \]
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 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 84, normalized size = 0.92 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x -1\right ) x a -a x \ln \left (a x +1\right )-\ln \left (a x -1\right )+\ln \left (a x +1\right )+2\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-1\right ) c^{2} a} \]
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 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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