Optimal. Leaf size=172 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {5 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{4 a c \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A] time = 0.14, antiderivative size = 172, 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 = {6197, 6193, 88} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {5 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{4 a c \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \, dx}{c \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^3 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^3}{(-1+a x) (1+a x)^2} \, dx}{c \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^3 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^3}+\frac {1}{4 a^3 (-1+a x)}+\frac {1}{2 a^3 (1+a x)^2}-\frac {5}{4 a^3 (1+a x)}\right ) \, dx}{c \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {5 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{4 a c \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 65, normalized size = 0.38 \[ \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (4 a x-\frac {2}{a x+1}+\log (1-a x)-5 \log (a x+1)\right )}{4 a \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 66, normalized size = 0.38 \[ \frac {{\left (4 \, a^{2} x^{2} + 4 \, a x - 5 \, {\left (a x + 1\right )} \log \left (a x + 1\right ) + {\left (a x + 1\right )} \log \left (a x - 1\right ) - 2\right )} \sqrt {a^{2} c}}{4 \, {\left (a^{3} c^{2} x + a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 100, normalized size = 0.58 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (4 a^{2} x^{2}+\ln \left (a x -1\right ) x a -5 a x \ln \left (a x +1\right )+4 a x +\ln \left (a x -1\right )-5 \ln \left (a x +1\right )-2\right ) \left (a x -1\right )}{4 a^{4} x^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}} \,d x \]
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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