Optimal. Leaf size=77 \[ \frac {2 a (a x+1)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \]
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Rubi [A] time = 0.25, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6151, 1805, 807, 266, 63, 208} \[ \frac {2 a (a x+1)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1805
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx &=c \int \frac {(1+a x)^2}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 a (1+a x)}{\sqrt {c-a^2 c x^2}}-\int \frac {-1-2 a x}{x^2 \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {2 a (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}+(2 a) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {2 a (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}+a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac {2 a (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}-\frac {2 \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 c}\\ &=\frac {2 a (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}-\frac {2 a \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 = 0.14, size = 78, normalized size = 1.01 \[ \frac {(1-3 a x) \sqrt {c-a^2 c x^2}}{c x (a x-1)}-\frac {2 a \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )}{\sqrt {c}}+\frac {2 a \log (x)}{\sqrt {c}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.70, size = 178, normalized size = 2.31 \[ \left [\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a x - 1\right )}}{a c x^{2} - c x}, -\frac {2 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a x - 1\right )}}{a c x^{2} - c x}\right ] \]
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 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 99, normalized size = 1.29 \[ -\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {\sqrt {-a^{2} c \,x^{2}+c}}{c x}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{c \left (x -\frac {1}{a}\right )} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {{\left (a\,x+1\right )}^2}{x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {a x}{a x^{3} \sqrt {- a^{2} c x^{2} + c} - x^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a x^{3} \sqrt {- a^{2} c x^{2} + c} - x^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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