Optimal. Leaf size=116 \[ \sqrt {c-\frac {c}{a^2 x^2}} x-\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x \text {ArcSin}(a x)}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}} \]
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Rubi [A]
time = 0.24, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6302, 6294,
6264, 104, 163, 41, 222, 94, 214} \begin {gather*} -\frac {2 x \text {ArcSin}(a x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a x} \sqrt {a x+1}}+x \sqrt {c-\frac {c}{a^2 x^2}}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 94
Rule 104
Rule 163
Rule 214
Rule 222
Rule 6264
Rule 6294
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^{3/2}}{x \sqrt {1-a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}} x+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {-a-2 a^2 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{a \sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}} x-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {\left (2 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}} x+\frac {\left (a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {\left (2 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}} x-\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 80, normalized size = 0.69 \begin {gather*} \frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2}-\text {ArcTan}\left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+2 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{\sqrt {-1+a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 197, normalized size = 1.70
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (2 \sqrt {\frac {c \left (a x +1\right ) \left (a x -1\right )}{a^{2}}}\, a^{2} \sqrt {-\frac {c}{a^{2}}}+2 \sqrt {c}\, \ln \left (\frac {\sqrt {\frac {c \left (a x +1\right ) \left (a x -1\right )}{a^{2}}}\, \sqrt {c}+c x}{\sqrt {c}}\right ) a \sqrt {-\frac {c}{a^{2}}}-\sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-c \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right )\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} \sqrt {-\frac {c}{a^{2}}}}\) | \(197\) |
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 = 0.37, size = 267, normalized size = 2.30 \begin {gather*} \left [\frac {2 \, a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 4 \, \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 ) + \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right )}{2 \, a}, \frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - \sqrt {c} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\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 )}{a}\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 {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{a\,x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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