Optimal. Leaf size=45 \[ \frac {2 (a x+1)}{c \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.17, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6128, 852, 1805, 12, 266, 63, 208} \[ \frac {2 (a x+1)}{c \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 852
Rule 1805
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x (c-a c x)} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x (c-a c x)^2} \, dx\\ &=\frac {\int \frac {(c+a c x)^2}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {2 (1+a x)}{c \sqrt {1-a^2 x^2}}+\frac {\int \frac {c^2}{x \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {2 (1+a x)}{c \sqrt {1-a^2 x^2}}+\frac {\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 (1+a x)}{c \sqrt {1-a^2 x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac {2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\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^2 c}\\ &=\frac {2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 1.22 \[ \frac {-\sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+2 a x+2}{c \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 56, normalized size = 1.24 \[ \frac {2 \, a x + {\left (a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} - 2}{a c x - c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 80, normalized size = 1.78 \[ -\frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c {\left | a \right |}} + \frac {4 \, a}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 61, normalized size = 1.36 \[ -\frac {\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}}{c} \]
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 \[ -\int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 68, normalized size = 1.51 \[ \frac {2\,a\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{c} \]
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 \[ - \frac {\int \frac {a x}{a x^{2} \sqrt {- a^{2} x^{2} + 1} - x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a x^{2} \sqrt {- a^{2} x^{2} + 1} - x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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