Integrand size = 12, antiderivative size = 49 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e}-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}} \]
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Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2436, 2334, 2336, 2211, 2235} \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e}-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}} \]
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Rule 2211
Rule 2235
Rule 2334
Rule 2336
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\frac {-2 c (d+e x)+2 \Gamma \left (\frac {1}{2},-\log (c (d+e x))\right ) \sqrt {-\log (c (d+e x))}}{c e \sqrt {\log (c (d+e x))}} \]
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\[\int \frac {1}{\ln \left (c \left (e x +d \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\text {Exception raised: TypeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (44) = 88\).
Time = 13.49 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\begin {cases} 0 & \text {for}\: c = 0 \\\frac {x}{\log {\left (c d \right )}^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {\left (- \log {\left (c d + c e x \right )}\right )^{\frac {3}{2}} \left (- 2 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )} + \frac {2 \left (c d + c e x\right )}{\sqrt {- \log {\left (c d + c e x \right )}}}\right )}{c e \log {\left (c d + c e x \right )}^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=-\frac {\sqrt {-\log \left (c e x + c d\right )} \Gamma \left (-\frac {1}{2}, -\log \left (c e x + c d\right )\right )}{c e \sqrt {\log \left (c e x + c d\right )}} \]
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\[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\int { \frac {1}{\log \left ({\left (e x + d\right )} c\right )^{\frac {3}{2}}} \,d x } \]
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Time = 1.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=-\frac {2\,\left (d+e\,x\right )}{e\,\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}}-\frac {2\,\sqrt {\pi }\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{c\,e\,{\ln \left (c\,\left (d+e\,x\right )\right )}^{3/2}} \]
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