Optimal. Leaf size=97 \[ \frac {(1-a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \text {ArcSin}(a x)}{a c} \]
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Rubi [A]
time = 0.16, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6292, 6284,
1649, 21, 683, 655, 222} \begin {gather*} \frac {(1-a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \text {ArcSin}(a x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 222
Rule 655
Rule 683
Rule 1649
Rule 6284
Rule 6292
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {a^2 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \int \frac {x^2 (1-a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=\frac {(1-a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 \int \frac {\left (\frac {3}{a^2}-\frac {3 x}{a}\right ) (1-a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac {(1-a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {(1-a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {(1-a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {3 \int \frac {1-a x}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {(1-a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {(1-a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 78, normalized size = 0.80 \begin {gather*} \frac {-14-5 a x+16 a^2 x^2+3 a^3 x^3-9 (1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a c (1+a x) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(824\) vs.
\(2(89)=178\).
time = 0.76, size = 825, normalized size = 8.51
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {2 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a^{5} \left (x +\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a^{4} \left (x +\frac {1}{a}\right )}\right ) a^{2}}{c}\) | \(139\) |
default | \(\frac {a^{2} \left (-\frac {\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 )}{16 a^{3}}+\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}}}{4 a \left (x +\frac {1}{a}\right )^{3}}-\frac {3 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 )}{2}}{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 )^{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 )}{8 a^{4}}-\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{3 a \left (x +\frac {1}{a}\right )^{4}}-\frac {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 )^{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 )\right )}{3}}{2 a^{6}}+\frac {\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 )}{16 a^{3}}\right )}{c}\) | \(825\) |
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 [A]
time = 0.35, size = 101, normalized size = 1.04 \begin {gather*} -\frac {14 \, a^{2} x^{2} + 28 \, a x - 18 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} + 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{3} c x^{2} + 2 \, a^{2} c x + a c\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 {a^{2} \left (\int \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} + 3 a^{4} x^{4} + 2 a^{3} x^{3} - 2 a^{2} x^{2} - 3 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} + 3 a^{4} x^{4} + 2 a^{3} x^{3} - 2 a^{2} x^{2} - 3 a x - 1}\right )\, dx\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.85, size = 129, normalized size = 1.33 \begin {gather*} \frac {2\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^4\,x^2+2\,c\,a^3\,x+c\,a^2\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (\frac {c\,\sqrt {-a^2}}{a}+c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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