Optimal. Leaf size=263 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A] time = 0.17, antiderivative size = 263, 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} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]
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 )^{5/2}} \, 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 )^{5/2}} \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^5 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^5}{(-1+a x)^2 (1+a x)^3} \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^5 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^5}+\frac {1}{8 a^5 (-1+a x)^2}+\frac {7}{16 a^5 (-1+a x)}-\frac {1}{4 a^5 (1+a x)^3}+\frac {1}{a^5 (1+a x)^2}-\frac {23}{16 a^5 (1+a x)}\right ) \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 85, normalized size = 0.32 \[ \frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (2 \left (8 a x+\frac {1}{1-a x}-\frac {8}{a x+1}+\frac {1}{(a x+1)^2}\right )+7 \log (1-a x)-23 \log (a x+1)\right )}{16 a \left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 135, normalized size = 0.51 \[ \frac {{\left (16 \, a^{4} x^{4} + 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} - 18 \, a x - 23 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \log \left (a x + 1\right ) + 7 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \log \left (a x - 1\right ) + 12\right )} \sqrt {a^{2} c}}{16 \, {\left (a^{5} c^{3} x^{3} + a^{4} c^{3} x^{2} - a^{3} c^{3} x - a^{2} c^{3}\right )}} \]
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 = 175, normalized size = 0.67 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (a x -1\right ) \left (16 x^{4} a^{4}+7 \ln \left (a x -1\right ) x^{3} a^{3}-23 a^{3} x^{3} \ln \left (a x +1\right )+16 x^{3} a^{3}+7 \ln \left (a x -1\right ) x^{2} a^{2}-23 \ln \left (a x +1\right ) x^{2} a^{2}-34 a^{2} x^{2}-7 \ln \left (a x -1\right ) x a +23 a x \ln \left (a x +1\right )-18 a x -7 \ln \left (a x -1\right )+23 \ln \left (a x +1\right )+12\right )}{16 a^{6} x^{5} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}} \,d x \]
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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