Optimal. Leaf size=112 \[ -\frac {(1-a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {2 \sqrt {1-a x} \sqrt {1+a x} \text {ArcSin}(a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \]
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Rubi [A]
time = 0.20, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6302, 6294,
6264, 79, 52, 41, 222} \begin {gather*} -\frac {2 \sqrt {a x+1} \sqrt {1-a x} \text {ArcSin}(a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {(1-a x)^2}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 (a x+1) (1-a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 52
Rule 79
Rule 222
Rule 6264
Rule 6294
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx\\ &=-\frac {\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=-\frac {\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {x \sqrt {1-a x}}{(1+a x)^{3/2}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=-\frac {(1-a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {\sqrt {1-a x}}{\sqrt {1+a x}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=-\frac {(1-a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=-\frac {(1-a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=-\frac {(1-a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {2 \sqrt {1-a x} \sqrt {1+a x} \sin ^{-1}(a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 68, normalized size = 0.61 \begin {gather*} \frac {-3+2 a x+a^2 x^2-2 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 179, normalized size = 1.60
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a \sqrt {a^{2} c}}+\frac {2 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-2 \left (x +\frac {1}{a}\right ) a c}}{a^{3} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {c \left (a^{2} x^{2}-1\right )}}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}\) | \(154\) |
default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \left (-\sqrt {c}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} x +2 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a c x -2 a \sqrt {\frac {c \left (a x +1\right ) \left (a x -1\right )}{a^{2}}}\, \sqrt {c}-\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a \sqrt {c}+2 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) c \right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,c^{\frac {3}{2}} a \left (a x +1\right )}\) | \(179\) |
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 = 0.36, size = 212, normalized size = 1.89 \begin {gather*} \left [\frac {{\left (a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}, \frac {2 \, {\left (a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{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*} \int \frac {a x - 1}{\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x-1}{\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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