Optimal. Leaf size=79 \[ \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
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Rubi [A]
time = 0.11, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6312, 893, 889,
214} \begin {gather*} \frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 889
Rule 893
Rule 6312
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a^3}\\ &=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 65, normalized size = 0.82 \begin {gather*} \frac {\sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} x-\frac {3 \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a}\right )}{\sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 101, normalized size = 1.28
method | result | size |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}-3 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right ) \sqrt {x \left (a x +1\right )}\, \sqrt {a}}\) | \(101\) |
risch | \(\frac {x \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}-\frac {3 \ln \left (\frac {\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c a x \left (a x +1\right )}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(139\) |
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. 142 vs.
\(2 (67) = 134\).
time = 0.38, size = 297, normalized size = 3.76 \begin {gather*} \left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\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 \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (-1 + \frac {1}{a x}\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 \sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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