Optimal. Leaf size=119 \[ \frac {8 x}{35 c^6 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 659, 191} \[ \frac {8 x}{35 c^6 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 659
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^6} \, dx &=\frac {\int \frac {1}{(c-a c x)^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4 \int \frac {1}{(c-a c x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{7 c^4}\\ &=\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {12 \int \frac {1}{(c-a c x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^5}\\ &=\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^6}\\ &=\frac {8 x}{35 c^6 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.51 \[ \frac {8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13}{35 a c^6 (a x-1)^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 144, normalized size = 1.21 \[ \frac {13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x - {\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {-a^{2} x^{2} + 1} + 13}{35 \, {\left (a^{6} c^{6} x^{5} - 3 \, a^{5} c^{6} x^{4} + 2 \, a^{4} c^{6} x^{3} + 2 \, a^{3} c^{6} x^{2} - 3 \, a^{2} c^{6} x + a c^{6}\right )}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{6} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 65, normalized size = 0.55 \[ \frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (8 x^{4} a^{4}-24 x^{3} a^{3}+20 a^{2} x^{2}+4 a x -13\right )}{35 \left (a x -1\right )^{5} c^{6} a \left (a x +1\right )^{2}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{6} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 347, normalized size = 2.92 \[ \frac {3\,a\,\sqrt {1-a^2\,x^2}}{40\,\left (a^4\,c^6\,x^2-2\,a^3\,c^6\,x+a^2\,c^6\right )}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^6\,x^2-2\,a^5\,c^6\,x+a^4\,c^6\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{14\,\left (a^6\,c^6\,x^4-4\,a^5\,c^6\,x^3+6\,a^4\,c^6\,x^2-4\,a^3\,c^6\,x+a^2\,c^6\right )}+\frac {\sqrt {1-a^2\,x^2}}{16\,\sqrt {-a^2}\,\left (c^6\,x\,\sqrt {-a^2}+\frac {c^6\,\sqrt {-a^2}}{a}\right )}+\frac {93\,\sqrt {1-a^2\,x^2}}{560\,\sqrt {-a^2}\,\left (c^6\,x\,\sqrt {-a^2}-\frac {c^6\,\sqrt {-a^2}}{a}\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{140\,\sqrt {-a^2}\,\left (3\,c^6\,x\,\sqrt {-a^2}-\frac {c^6\,\sqrt {-a^2}}{a}+a^2\,c^6\,x^3\,\sqrt {-a^2}-3\,a\,c^6\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\right )\, dx}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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