Optimal. Leaf size=122 \[ -\frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 x (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {3 \sqrt {1-a^2 x^2} \log (a x+1)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A] time = 0.14, antiderivative size = 122, 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, 77} \[ -\frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 x (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {3 \sqrt {1-a^2 x^2} \log (a x+1)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-3 \tanh ^{-1}(a x)} x}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x (1-a x)}{(1+a x)^2} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (-\frac {1}{a}-\frac {2}{a (1+a x)^2}+\frac {3}{a (1+a x)}\right ) \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)}+\frac {3 \sqrt {1-a^2 x^2} \log (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.48 \[ \frac {\sqrt {1-a^2 x^2} \left (-a x+\frac {2}{a x+1}+3 \log (a x+1)\right )}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.73, size = 439, normalized size = 3.60 \[ \left [-\frac {3 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) + 2 \, {\left (a^{3} x^{3} + 3 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c\right )}}, \frac {3 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right ) - {\left (a^{3} x^{3} + 3 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{4} c x^{3} + a^{3} c x^{2} - a^{2} c x - a c}\right ] \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 0.64 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} x^{2}+3 a x \ln \left (a x +1\right )-a x +3 \ln \left (a x +1\right )+2\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2} \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.34, size = 53, normalized size = 0.43 \[ -\frac {i \, a^{2} \sqrt {c} x^{2} + i \, a \sqrt {c} x - 2 i \, \sqrt {c}}{a^{2} c x + a c} + \frac {3 i \, \log \left (a x + 1\right )}{a \sqrt {c}} \]
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 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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