Optimal. Leaf size=105 \[ -\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \]
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Rubi [A] time = 0.29, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 6177
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2 \left (c-\frac {c x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^3}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^5}\\ &=-\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-3 c^3-\frac {9 c^3 x}{a}-\frac {8 c^3 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^5}\\ &=-\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\operatorname {Subst}\left (\int \frac {3 c^3+\frac {9 c^3 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 c^5}\\ &=-\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^2}\\ &=-\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c^2}\\ &=-\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^2}\\ &=-\frac {4 \left (a+\frac {1}{x}\right )}{3 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {9 a+\frac {11}{x}}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 94, normalized size = 0.90 \[ \frac {3 a^3 x^3-16 a^2 x^2+9 a x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-5 a x+14}{3 a^2 c^2 x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 134, normalized size = 1.28 \[ \frac {9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 148, normalized size = 1.41 \[ \frac {1}{3} \, a {\left (\frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {9 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac {{\left (a x + 1\right )} {\left (\frac {12 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 339, normalized size = 3.23 \[ \frac {9 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-6 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+5 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -9 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-9 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{3 a \left (a x -1\right )^{2} \sqrt {a^{2}}\, c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 137, normalized size = 1.30 \[ \frac {1}{3} \, a {\left (\frac {\frac {11 \, {\left (a x - 1\right )}}{a x + 1} - \frac {18 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 104, normalized size = 0.99 \[ \frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2}-\frac {\frac {11\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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 \[ \frac {a^{2} \int \frac {x^{2}}{a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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