Optimal. Leaf size=144 \[ \frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 x^3 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 144, 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 = {6160, 6150, 75} \[ \frac {a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}+\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 x^3 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2}}{x^3} \, dx}{\left (1-a^2 x^2\right )^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {(1-a x)^2 (1+a x)}{x^3} \, dx}{\left (1-a^2 x^2\right )^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (a^3+\frac {1}{x^3}-\frac {a}{x^2}-\frac {a^2}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{3/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}{2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{\left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^4}{\left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \log (x)}{\left (1-a^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.50 \[ -\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (2 a^3 x^3-3 a^2 x^2-2 a^2 x^2 \log (x)+2 a x-1\right )}{2 a^2 x \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.77, size = 377, normalized size = 2.62 \[ \left [\frac {{\left (a^{3} c x^{3} - a c x\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) + {\left (2 \, a^{3} c x^{3} - {\left (2 \, a^{3} + 2 \, a - 1\right )} c x^{2} + 2 \, a c x - c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} x^{3} - a^{2} x\right )}}, -\frac {2 \, {\left (a^{3} c x^{3} - a c x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) - {\left (2 \, a^{3} c x^{3} - {\left (2 \, a^{3} + 2 \, a - 1\right )} c x^{2} + 2 \, a c x - c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} x^{3} - a^{2} x\right )}}\right ] \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 0.49 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (-2 x^{3} a^{3}+2 a^{2} \ln \relax (x ) x^{2}-2 a x +1\right )}{2 \left (a^{2} x^{2}-1\right )^{2}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{a x + 1}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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