Optimal. Leaf size=25 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e} \]
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Rubi [A] time = 0.0219619, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2389, 2299, 2180, 2204} \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2299
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\log (c (d+e x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\log (c x)}} \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\log (c (d+e x))\right )}{c e}\\ &=\frac{2 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\log (c (d+e x))}\right )}{c e}\\ &=\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e}\\ \end{align*}
Mathematica [A] time = 0.0019845, size = 25, normalized size = 1. \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.229, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{\ln \left ( c \left ( ex+d \right ) \right ) }}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.22635, size = 34, normalized size = 1.36 \begin{align*} -\frac{i \, \sqrt{\pi } \operatorname{erf}\left (i \, \sqrt{\log \left (c e x + c d\right )}\right )}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.08629, size = 63, normalized size = 2.52 \begin{align*} \begin{cases} 0 & \text{for}\: c = 0 \\\frac{x}{\sqrt{\log{\left (c d \right )}}} & \text{for}\: e = 0 \\\frac{\sqrt{\pi } \sqrt{- \log{\left (c d + c e x \right )}} \operatorname{erfc}{\left (\sqrt{- \log{\left (c d + c e x \right )}} \right )}}{c e \sqrt{\log{\left (c d + c e x \right )}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31681, size = 35, normalized size = 1.4 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (-i \sqrt{\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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