Optimal. Leaf size=87 \[ \frac {2 x}{5 c^5 \sqrt {1-a^2 x^2}}+\frac {1}{5 a c^5 (1-a x) \sqrt {1-a^2 x^2}}+\frac {1}{5 a c^5 (1-a x)^2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 659, 191} \[ \frac {2 x}{5 c^5 \sqrt {1-a^2 x^2}}+\frac {1}{5 a c^5 (1-a x) \sqrt {1-a^2 x^2}}+\frac {1}{5 a c^5 (1-a x)^2 \sqrt {1-a^2 x^2}} \]
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)^5} \, dx &=\frac {\int \frac {1}{(c-a c x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{5 a c^5 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {3 \int \frac {1}{(c-a c x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{5 c^4}\\ &=\frac {1}{5 a c^5 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{5 a c^5 (1-a x) \sqrt {1-a^2 x^2}}+\frac {2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{5 c^5}\\ &=\frac {2 x}{5 c^5 \sqrt {1-a^2 x^2}}+\frac {1}{5 a c^5 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{5 a c^5 (1-a x) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.60 \[ \frac {2 a^3 x^3-4 a^2 x^2+a x+2}{5 a c^5 (a x-1)^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 98, normalized size = 1.13 \[ \frac {2 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 4 \, a x - {\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} - 2}{5 \, {\left (a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.34, size = 174, normalized size = 2.00 \[ \frac {1}{40} \, {\left (a {\left (\frac {5}{a^{3} c^{7} \sqrt {-\frac {2 \, c}{a c x - c} - 1}} - \frac {a^{12} c^{28} {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 5 \, a^{12} c^{28} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 15 \, a^{12} c^{28} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{a^{15} c^{35}}\right )} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c) + \frac {16 i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}{a^{2} c^{7}}\right )} c^{2} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 56, normalized size = 0.64 \[ \frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (2 x^{3} a^{3}-4 a^{2} x^{2}+a x +2\right )}{5 \left (a x -1\right )^{4} c^{5} a \left (a x +1\right )^{2}} \]
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 \[ -\int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{5} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 233, normalized size = 2.68 \[ \frac {3\,a\,\sqrt {1-a^2\,x^2}}{20\,\left (a^4\,c^5\,x^2-2\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {\sqrt {1-a^2\,x^2}}{8\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}\right )}+\frac {11\,\sqrt {1-a^2\,x^2}}{40\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{10\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}-3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ - \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\right )\, dx}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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