Optimal. Leaf size=62 \[ \frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}{a}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {c^2 \csc ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.10, antiderivative size = 62, 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 = {6177, 813, 844, 216, 266, 63, 208} \[ \frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}{a}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {c^2 \csc ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 813
Rule 844
Rule 6177
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {\frac {2 c}{a}+\frac {2 c x}{a^2}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}-\left (a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [B] time = 0.17, size = 158, normalized size = 2.55 \[ \frac {c^2 \left (2 a^3 x^3+2 a^2 x^2-2 a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {1}{a x}\right )-2 a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-2 a x-2\right )}{2 a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.42, size = 119, normalized size = 1.92 \[ -\frac {2 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c^{2} x^{2} + 2 \, a c^{2} x + c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 113, normalized size = 1.82 \[ -a c^{2} {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {\log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac {4 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 168, normalized size = 2.71 \[ -\frac {\left (a x -1\right ) c^{2} \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 125, normalized size = 2.02 \[ -{\left (\frac {4 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 90, normalized size = 1.45 \[ \frac {4\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c^{2} \left (\int \frac {a^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {2 a}{x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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