Optimal. Leaf size=82 \[ 4 a \sqrt {c-\frac {c}{a x}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6268, 25, 528,
455, 52, 65, 214} \begin {gather*} \frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 a \sqrt {c-\frac {c}{a x}}-4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 52
Rule 65
Rule 214
Rule 455
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^2 (1+a x)} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x (1+a x)} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^2} \, dx}{c}\\ &=\frac {a \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+(2 a) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right )\\ &=4 a \sqrt {c-\frac {c}{a x}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+(4 a c) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=4 a \sqrt {c-\frac {c}{a x}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\left (8 a^2\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=4 a \sqrt {c-\frac {c}{a x}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 69, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {c-\frac {c}{a x}} (-1+7 a x)}{3 x}-4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \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. \(253\) vs.
\(2(67)=134\).
time = 0.42, size = 254, normalized size = 3.10
method | result | size |
risch | \(\frac {2 \left (7 a^{2} x^{2}-8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \left (a x -1\right )}+\frac {2 a \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {a c x \left (a x -1\right )}}{\sqrt {c}\, \left (a x -1\right )}\) | \(142\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (6 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, x^{3}-18 a^{\frac {5}{2}} \sqrt {a \,x^{2}-x}\, \sqrt {\frac {1}{a}}\, x^{3}-6 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{3}+12 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-9 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{3 x^{2} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(254\) |
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 = 0.40, size = 157, normalized size = 1.91 \begin {gather*} \left [\frac {2 \, {\left (3 \, \sqrt {2} a \sqrt {c} x \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}, \frac {2 \, {\left (6 \, \sqrt {2} a \sqrt {-c} x \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}\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 \left (- \frac {\sqrt {c - \frac {c}{a x}}}{a x^{3} + x^{2}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{3} + x^{2}}\, 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 [B]
time = 1.42, size = 67, normalized size = 0.82 \begin {gather*} 4\,a\,\sqrt {c-\frac {c}{a\,x}}+\frac {2\,a\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}-4\,\sqrt {2}\,a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-\frac {c}{a\,x}}}{2\,\sqrt {c}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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