Optimal. Leaf size=45 \[ -\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {1}{a c^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6131, 6128, 266, 43} \[ -\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {1}{a c^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\frac {a^3 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {a^3 \operatorname {Subst}\left (\int \frac {x}{\left (1-a^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 \left (1-a^2 x\right )^{3/2}}-\frac {1}{a^2 \sqrt {1-a^2 x}}\right ) \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac {1}{a c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 30, normalized size = 0.67 \[ \frac {a^2 x^2-2}{a c^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 53, normalized size = 1.18 \[ -\frac {2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} - 2}{a^{3} c^{3} x^{2} - a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 38, normalized size = 0.84 \[ -\frac {\frac {\sqrt {-a^{2} x^{2} + 1}}{c^{3}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} c^{3}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 0.96 \[ \frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a^{2} x^{2}-2\right )}{a \left (a x -1\right )^{2} c^{3} \left (a x +1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 45, normalized size = 1.00 \[ -\frac {{\left (a^{2} x^{2} - 2\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{a^{3} c^{3} x^{2} - a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.86, size = 116, normalized size = 2.58 \[ \frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^3}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.32, size = 34, normalized size = 0.76 \[ - \frac {2 \left (\frac {\sqrt {- a^{2} x^{2} + 1}}{2 c^{3}} + \frac {1}{2 c^{3} \sqrt {- a^{2} x^{2} + 1}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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