Optimal. Leaf size=69 \[ \frac {2 a (a x+1)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c x}-\frac {2 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.19, antiderivative size = 69, 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, 807, 266, 63, 208} \[ \frac {2 a (a x+1)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c x}-\frac {2 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^2 (c-a c x)^2} \, dx\\ &=\frac {\int \frac {(c+a c x)^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {2 a (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\int \frac {-c^2-2 a c^2 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {2 a (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c x}+\frac {(2 a) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 a (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\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 )}{c}\\ &=\frac {2 a (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c x}-\frac {2 \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 {2 a (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c x}-\frac {2 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 68, normalized size = 0.99 \[ \frac {3 a^2 x^2-2 a x \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+2 a x-1}{c x \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 80, normalized size = 1.16 \[ \frac {2 \, a^{2} x^{2} - 2 \, a x + 2 \, {\left (a^{2} x^{2} - a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (3 \, a x - 1\right )}}{a c x^{2} - c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 159, normalized size = 2.30 \[ -\frac {2 \, a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c {\left | a \right |}} - \frac {{\left (a^{2} - \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, c x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 1.12 \[ -\frac {2 a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{x}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 91, normalized size = 1.32 \[ \frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{c\,x}-\frac {2\,a\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{c} \]
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 \[ - \frac {\int \frac {a x}{a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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