Optimal. Leaf size=45 \[ -\frac {1}{a c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6266, 6263,
272, 45} \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {1}{a c^3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 6263
Rule 6266
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\frac {a^3 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {a^3 \text {Subst}\left (\int \frac {x}{\left (1-a^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac {a^3 \text {Subst}\left (\int \left (\frac {1}{a^2 \left (1-a^2 x\right )^{3/2}}-\frac {1}{a^2 \sqrt {1-a^2 x}}\right ) \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac {1}{a c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 30, normalized size = 0.67 \begin {gather*} \frac {-2+a^2 x^2}{a c^3 \sqrt {1-a^2 x^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 = 0.75, size = 744, normalized size = 16.53
method | result | size |
trager | \(-\frac {\left (a^{2} x^{2}-2\right ) \sqrt {-a^{2} x^{2}+1}}{a \,c^{3} \left (a^{2} x^{2}-1\right )}\) | \(41\) |
gosper | \(\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a^{2} x^{2}-2\right )}{a \left (a x -1\right )^{2} c^{3} \left (a x +1\right )^{2}}\) | \(43\) |
risch | \(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {1}{a \,c^{3} \sqrt {-a^{2} x^{2}+1}}\) | \(50\) |
default | \(\frac {a^{3} \left (\frac {-\frac {3 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{16 a \left (x -\frac {1}{a}\right )^{2}}-\frac {9 a \left (\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{16}}{a^{5}}+\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {1}{a}\right )^{3}}+2 a \left (-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}-3 a \left (\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{8 a^{6}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{8 a^{6}}-\frac {3 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{16 a^{5}}\right )}{c^{3}}\) | \(744\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 45, normalized size = 1.00 \begin {gather*} -\frac {{\left (a^{2} x^{2} - 2\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{a^{3} c^{3} x^{2} - a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 53, normalized size = 1.18 \begin {gather*} -\frac {2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} - 2}{a^{3} c^{3} x^{2} - a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.53, size = 34, normalized size = 0.76 \begin {gather*} - \frac {2 \left (\frac {\sqrt {- a^{2} x^{2} + 1}}{2 c^{3}} + \frac {1}{2 c^{3} \sqrt {- a^{2} x^{2} + 1}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 38, normalized size = 0.84 \begin {gather*} -\frac {\frac {\sqrt {-a^{2} x^{2} + 1}}{c^{3}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} c^{3}}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.86, size = 116, normalized size = 2.58 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^3}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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