Integrand size = 31, antiderivative size = 67 \[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=\frac {F^{a f} \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
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Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2308, 2235} \[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=\frac {\sqrt {\pi } F^{a f} \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
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Rule 2235
Rule 2308
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^{a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g n} \\ & = \frac {F^{a f} \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=\frac {F^{a f} \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.22 (sec) , antiderivative size = 382, normalized size of antiderivative = 5.70
method | result | size |
risch | \(\frac {\sqrt {\pi }\, F^{f \left (-b \,\pi ^{2}+b \,\pi ^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )+b \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \pi ^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-b \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \pi ^{2} \operatorname {csgn}\left (i c \right )+i b \ln \left (c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \pi -i b \ln \left (c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \ln \left (c \right ) \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \ln \left (c \right ) \pi \,\operatorname {csgn}\left (i c \right )+b \ln \left (c \right )^{2}+a \right )} F^{-\frac {f b {\left (i \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \pi +i \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi \,\operatorname {csgn}\left (i c \right )-2 \ln \left (c \right )\right )}^{2}}{4}} \operatorname {erf}\left (\sqrt {-\ln \left (F \right ) b f}\, \ln \left (\left (e x +d \right )^{n}\right )-\frac {f b \left (2 \ln \left (c \right )-i \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \left (-\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )\right )\right ) \ln \left (F \right )}{2 \sqrt {-\ln \left (F \right ) b f}}\right )}{2 g e n \sqrt {-\ln \left (F \right ) b f}}\) | \(382\) |
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Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=-\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} F^{a f} \operatorname {erf}\left (\frac {\sqrt {-b f n^{2} \log \left (F\right )} {\left (n \log \left (e x + d\right ) + \log \left (c\right )\right )}}{n}\right )}{2 \, e g n} \]
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\[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=\frac {\int \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}}{d + e x}\, dx}{g} \]
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\[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=\int { \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}}{e g x + d g} \,d x } \]
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\[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=\int { \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}}{e g x + d g} \,d x } \]
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Time = 0.45 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx=\frac {F^{a\,f}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,f\,\ln \left (F\right )\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {b\,f\,\ln \left (F\right )}}\right )}{2\,e\,g\,n\,\sqrt {b\,f\,\ln \left (F\right )}} \]
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