Optimal. Leaf size=52 \[ \frac {2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \]
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Rubi [A]
time = 0.18, 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 = {6286, 1819,
272, 65, 214} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 1819
Rule 6286
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} \text {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 {\text {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 = 0.18, size = 66, normalized size = 1.27 \begin {gather*} \frac {2 \sqrt {c-a^2 c x^2}}{c-a c x}+\frac {\log (x)}{\sqrt {c}}-\frac {\log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 80, normalized size = 1.54
method | result | size |
default | \(-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {2 \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{a c \left (x -\frac {1}{a}\right )}\) | \(80\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 147, normalized size = 2.83 \begin {gather*} \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 ] \end {gather*}
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 \begin {gather*} - \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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