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 = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {2436, 2336,
2211, 2235} \begin {gather*} \frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2336
Rule 2436
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\log (c x)}} \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\log (c (d+e x))\right )}{c e}\\ &=\frac {2 \text {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*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {\ln \left (c \left (e x +d \right )\right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.26, size = 25, normalized size = 1.00 \begin {gather*} -\frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (20) = 40\).
time = 1.12, size = 63, normalized size = 2.52 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 2.89, size = 25, normalized size = 1.00 \begin {gather*} \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 45, normalized size = 1.80 \begin {gather*} \frac {\sqrt {\pi }\,\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{c\,e\,\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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