Optimal. Leaf size=75 \[ -\frac {1-a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.04, 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 = {6274, 653, 198,
197} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 653
Rule 6274
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 = 0.02, size = 59, normalized size = 0.79 \begin {gather*} -\frac {3-12 a x-12 a^2 x^2+8 a^3 x^3+8 a^4 x^4}{15 a c^3 (1-a x)^{3/2} (1+a x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.09, size = 524, normalized size = 6.99
method | result | size |
gosper | \(-\frac {8 a^{4} x^{4}+8 a^{3} x^{3}-12 a^{2} x^{2}-12 a x +3}{15 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a x +1\right ) a \,c^{3}}\) | \(58\) |
trager | \(-\frac {\left (8 a^{4} x^{4}+8 a^{3} x^{3}-12 a^{2} x^{2}-12 a x +3\right ) \sqrt {-a^{2} x^{2}+1}}{15 c^{3} \left (a x +1\right )^{3} \left (a x -1\right )^{2} a}\) | \(65\) |
default | \(-\frac {-\frac {5 \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{32 a}-\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}}{8 a^{4}}-\frac {3 \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}-a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )\right )}{16 a^{2}}+\frac {\frac {5 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{32}-\frac {5 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{32 \sqrt {a^{2}}}}{a}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{48 a^{4} \left (x -\frac {1}{a}\right )^{3}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{16 a^{4} \left (x +\frac {1}{a}\right )^{3}}-\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{8 a^{2}}}{c^{3}}\) | \(524\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (62) = 124\).
time = 0.34, size = 141, normalized size = 1.88 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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