Optimal. Leaf size=32 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2282, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b e^{c+d x}}\right )}{b d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 83, normalized size = 2.59 \[ \left [\frac {\log \left ({\left (b e^{\left (d x + c\right )} - 2 \, \sqrt {b e^{\left (d x + c\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-d x - c\right )}\right )}{\sqrt {a} d}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a} \sqrt {-a}}{a}\right )}{a d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 29, normalized size = 0.91 \[ \frac {2 \, \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.81 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b \,{\mathrm e}^{d x +c}+a}}{\sqrt {a}}\right )}{\sqrt {a}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.69, size = 45, normalized size = 1.41 \[ \frac {\log \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a} - \sqrt {a}}{\sqrt {b e^{\left (d x + c\right )} + a} + \sqrt {a}}\right )}{\sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.62, size = 25, normalized size = 0.78 \[ -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}}{\sqrt {a}}\right )}{\sqrt {a}\,d} \]
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 \[ \int \frac {1}{\sqrt {a + b e^{c + d x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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