Optimal. Leaf size=70 \[ -\frac {2 \left (a+\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
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Rubi [A]
time = 0.13, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6312, 866,
1819, 821, 272, 65, 214} \begin {gather*} -\frac {2 \left (a+\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 6312
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2 \left (c-\frac {c x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^2}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-c^2-\frac {2 c^2 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 63, normalized size = 0.90 \begin {gather*} \frac {a \sqrt {1-\frac {1}{a^2 x^2}} x (-3+a x)+2 (-1+a x) \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c (-1+a x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs.
\(2(64)=128\).
time = 0.10, size = 250, normalized size = 3.57
method | result | size |
risch | \(\frac {a x -1}{a c \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a \sqrt {a^{2}}}-\frac {2 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{a^{3} \left (x -\frac {1}{a}\right )}\right ) a \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(143\) |
default | \(\frac {2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+2 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -\left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-4 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )+2 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 116, normalized size = 1.66 \begin {gather*} -2 \, a {\left (\frac {\frac {2 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 94, normalized size = 1.34 \begin {gather*} \frac {2 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 2 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} x^{2} - 2 \, a x - 3\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \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 \int \frac {x}{a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c} \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 [B]
time = 1.22, size = 62, normalized size = 0.89 \begin {gather*} \frac {2\,a\,x+8\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}-6}{2\,a\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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