Optimal. Leaf size=49 \[ \frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e}-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2436, 2334,
2336, 2211, 2235} \begin {gather*} \frac {2 \sqrt {\pi } \text {Erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e}-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2334
Rule 2336
Rule 2436
Rubi steps
\begin {align*} \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\log ^{\frac {3}{2}}(c x)} \, dx,x,d+e x\right )}{e}\\ &=-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {\log (c x)}} \, dx,x,d+e x\right )}{e}\\ &=-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}}+\frac {2 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\log (c (d+e x))\right )}{c e}\\ &=-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}}+\frac {4 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\log (c (d+e x))}\right )}{c e}\\ &=\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e}-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 58, normalized size = 1.18 \begin {gather*} \frac {-2 c (d+e x)+2 \Gamma \left (\frac {1}{2},-\log (c (d+e x))\right ) \sqrt {-\log (c (d+e x))}}{c e \sqrt {\log (c (d+e x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\ln \left (c \left (e x +d \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 47, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {-\log \left (c x e + c d\right )} e^{\left (-1\right )} \Gamma \left (-\frac {1}{2}, -\log \left (c x e + c d\right )\right )}{c \sqrt {\log \left (c x e + c d\right )}} \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. 92 vs.
\(2 (44) = 88\).
time = 16.97, size = 92, normalized size = 1.88 \begin {gather*} \begin {cases} 0 & \text {for}\: c = 0 \\\frac {x}{\log {\left (c d \right )}^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {\left (- \log {\left (c d + c e x \right )}\right )^{\frac {3}{2}} \left (- 2 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )} + \frac {2 \left (c d + c e x\right )}{\sqrt {- \log {\left (c d + c e x \right )}}}\right )}{c e \log {\left (c d + c e x \right )}^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 67, normalized size = 1.37 \begin {gather*} -\frac {2\,\left (d+e\,x\right )}{e\,\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}}-\frac {2\,\sqrt {\pi }\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{c\,e\,{\ln \left (c\,\left (d+e\,x\right )\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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