Optimal. Leaf size=80 \[ -\frac {2 x}{15 a c^3 \sqrt {1-a^2 x^2}}-\frac {x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a x+1}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6148, 778, 192, 191} \[ -\frac {2 x}{15 a c^3 \sqrt {1-a^2 x^2}}-\frac {x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a x+1}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 778
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {1+a x}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a c^3}\\ &=\frac {1+a x}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a c^3}\\ &=\frac {1+a x}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x}{15 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 x}{15 a c^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 68, normalized size = 0.85 \[ \frac {-2 a^4 x^4+2 a^3 x^3+3 a^2 x^2-3 a x+3}{15 a^2 c^3 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 145, normalized size = 1.81 \[ \frac {3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 3 \, a x + {\left (2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 3\right )} \sqrt {-a^{2} x^{2} + 1} - 3}{15 \, {\left (a^{7} c^{3} x^{5} - a^{6} c^{3} x^{4} - 2 \, a^{5} c^{3} x^{3} + 2 \, a^{4} c^{3} x^{2} + a^{3} c^{3} x - a^{2} c^{3}\right )}} \]
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 \[ \int -\frac {{\left (a x + 1\right )} x}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 58, normalized size = 0.72 \[ \frac {2 x^{4} a^{4}-2 x^{3} a^{3}-3 a^{2} x^{2}+3 a x -3}{15 \left (a x -1\right ) c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a^{2}} \]
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 \[ -a \int \frac {x^{2}}{{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} + \frac {1}{5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 271, normalized size = 3.39 \[ \frac {\sqrt {1-a^2\,x^2}}{30\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {5\,\sqrt {1-a^2\,x^2}}{48\,\left (c^3\,\sqrt {-a^2}+a\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {7\,\sqrt {1-a^2\,x^2}}{240\,\left (c^3\,\sqrt {-a^2}-a\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (c^3\,\sqrt {-a^2}+3\,a^2\,c^3\,x^2\,\sqrt {-a^2}-a^3\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x\,\sqrt {-a^2}\right )} \]
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 \[ \frac {\int \frac {x}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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