Optimal. Leaf size=175 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {5 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{4 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {\left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{4 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6160, 6150, 88} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {5 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{4 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {\left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{4 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {x^3}{(1-a x)^2 (1+a x)} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2} \int \left (\frac {1}{a^3}+\frac {1}{2 a^3 (-1+a x)^2}+\frac {5}{4 a^3 (-1+a x)}-\frac {1}{4 a^3 (1+a x)}\right ) \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)}+\frac {5 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{4 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {\left (1-a^2 x^2\right )^{3/2} \log (1+a x)}{4 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 0.52 \[ -\frac {\sqrt {1-a^2 x^2} \left (a^2 x^2-1\right ) \left (\frac {1}{2 a^4 (1-a x)}+\frac {5 \log (1-a x)}{4 a^4}-\frac {\log (a x+1)}{4 a^4}+\frac {x}{a^3}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} a^{4} x^{4} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right ) \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 93, normalized size = 0.53 \[ -\frac {\left (4 a^{2} x^{2}+5 \ln \left (a x -1\right ) x a -a x \ln \left (a x +1\right )-4 a x -5 \ln \left (a x -1\right )+\ln \left (a x +1\right )-2\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}{4 a^{4} x^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {a\,x+1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {a x + 1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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