Integrand size = 12, antiderivative size = 25 \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2436, 2336, 2211, 2235} \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2336
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e} \]
[In]
[Out]
\[\int \frac {1}{\sqrt {\ln \left (c \left (e x +d \right )\right )}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.96 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=\begin {cases} 0 & \text {for}\: c = 0 \\\frac {x}{\sqrt {\log {\left (c d \right )}}} & \text {for}\: e = 0 \\\frac {\sqrt {\pi } \sqrt {- \log {\left (c d + c e x \right )}} \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )}}{c e \sqrt {\log {\left (c d + c e x \right )}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=-\frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c e x + c d\right )}\right )}{c e} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {\log \left (c e x + c d\right )}\right )}{c e} \]
[In]
[Out]
Time = 1.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\sqrt {\log (c (d+e x))}} \, dx=\frac {\sqrt {\pi }\,\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{c\,e\,\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}} \]
[In]
[Out]