Optimal. Leaf size=70 \[ \frac {a x+1}{c x \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{c x}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.12, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6148, 823, 807, 266, 63, 208} \[ \frac {a x+1}{c x \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{c x}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx &=\frac {\int \frac {1+a x}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {1+a x}{c x \sqrt {1-a^2 x^2}}+\frac {\int \frac {2 a^2+a^3 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {1+a x}{c x \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{c x}+\frac {a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {1+a x}{c x \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{c x}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac {1+a x}{c x \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{c x}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a c}\\ &=\frac {1+a x}{c x \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{c x}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 0.96 \[ \frac {2 a^2 x^2-a x \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+a x-1}{c x \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 78, normalized size = 1.11 \[ \frac {a^{2} x^{2} - a x + {\left (a^{2} x^{2} - a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )}}{a c x^{2} - c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.63, size = 159, normalized size = 2.27 \[ -\frac {a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c {\left | a \right |}} - \frac {{\left (a^{2} - \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, c x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 75, normalized size = 1.07 \[ -\frac {a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{x}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 104, normalized size = 1.49 \[ -\frac {\frac {a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c} - \frac {a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c} - \frac {2 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} c}}{2 \, a} + \frac {2 \, a^{2} x^{2} - 1}{\sqrt {a x + 1} \sqrt {-a x + 1} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 90, normalized size = 1.29 \[ \frac {a^2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{c\,x}-\frac {a\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{c} \]
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 \[ \frac {\int \frac {a}{- a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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