Optimal. Leaf size=75 \[ \frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6139, 639, 192, 191} \[ \frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 639
Rule 6139
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^3}\\ &=-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3}\\ &=-\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.79 \[ -\frac {8 a^4 x^4+8 a^3 x^3-12 a^2 x^2-12 a x+3}{15 a c^3 (1-a x)^{3/2} (a x+1)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 141, normalized size = 1.88 \[ -\frac {3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 3 \, a x + {\left (8 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 12 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} + 3}{15 \, {\left (a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\right )}} \]
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 \[ \int -\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{3} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 58, normalized size = 0.77 \[ -\frac {8 x^{4} a^{4}+8 x^{3} a^{3}-12 a^{2} x^{2}-12 a x +3}{15 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a x +1\right ) c^{3} a} \]
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 \[ -\int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{3} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 64, normalized size = 0.85 \[ -\frac {\sqrt {1-a^2\,x^2}\,\left (8\,a^4\,x^4+8\,a^3\,x^3-12\,a^2\,x^2-12\,a\,x+3\right )}{15\,a\,c^3\,{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^3} \]
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 \[ \frac {\int \frac {1}{a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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