Optimal. Leaf size=52 \[ \frac {2 (a x+1)}{\sqrt {c-a^2 c x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \]
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Rubi [A] time = 0.23, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6151, 1805, 266, 63, 208} \[ \frac {2 (a x+1)}{\sqrt {c-a^2 c x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 1805
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x \sqrt {c-a^2 c x^2}} \, dx &=c \int \frac {(1+a x)^2}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 (1+a x)}{\sqrt {c-a^2 c x^2}}+\int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {2 (1+a x)}{\sqrt {c-a^2 c x^2}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac {2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )}{a^2 c}\\ &=\frac {2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 66, normalized size = 1.27 \[ \frac {2 \sqrt {c-a^2 c x^2}}{c-a c x}-\frac {\log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )}{\sqrt {c}}+\frac {\log (x)}{\sqrt {c}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.92, size = 147, normalized size = 2.83 \[ \left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 4 \, \sqrt {-a^{2} c x^{2} + c}}{2 \, {\left (a c x - c\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + 2 \, \sqrt {-a^{2} c x^{2} + c}}{a c x - c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 80, normalized size = 1.54 \[ -\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{a c \left (x -\frac {1}{a}\right )} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {{\left (a\,x+1\right )}^2}{x\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {a x}{a x^{2} \sqrt {- a^{2} c x^{2} + c} - x \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a x^{2} \sqrt {- a^{2} c x^{2} + c} - x \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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