Integrand size = 19, antiderivative size = 280 \[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) e^{\frac {i (1+m) \left (1+m+2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) e^{-\frac {i (1+m) \left (1+m-2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m-i a b d^2 n \pi -i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}+\frac {(e x)^{1+m} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)} \]
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Time = 0.41 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6606, 4713, 2314, 2308, 2266, 2235, 2236} \[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (\frac {i (m+1) \left (2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2 n+i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{m+1}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (-\frac {i (m+1) \left (-2 i \pi a b d^2 n+m+1\right )}{2 \pi b^2 d^2 n^2}\right ) \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2 n-i \pi b^2 d^2 n \log \left (c x^n\right )+m+1\right )}{\sqrt {\pi } b d n}\right )}{m+1}+\frac {(e x)^{m+1} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 2308
Rule 2314
Rule 4713
Rule 6606
Rubi steps \begin{align*} \text {integral}& = \frac {(e x)^{1+m} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b d n) \int (e x)^m \sin \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx}{1+m} \\ & = \frac {(e x)^{1+m} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(i b d n) \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} (e x)^m \, dx}{2 (1+m)}+\frac {(i b d n) \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} (e x)^m \, dx}{2 (1+m)} \\ & = \frac {(e x)^{1+m} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i b d n x^{-m+i a b d^2 n \pi } (e x)^m \left (c x^n\right )^{-i a b d^2 \pi }\right ) \int \exp \left (-\frac {1}{2} i a^2 d^2 \pi -\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{m-i a b d^2 n \pi } \, dx}{2 (1+m)}+\frac {\left (i b d n x^{-m-i a b d^2 n \pi } (e x)^m \left (c x^n\right )^{i a b d^2 \pi }\right ) \int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{m+i a b d^2 n \pi } \, dx}{2 (1+m)} \\ & = \frac {(e x)^{1+m} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i b d x (e x)^m \left (c x^n\right )^{-i a b d^2 \pi -\frac {1+m-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {1}{2} i a^2 d^2 \pi +\frac {\left (1+m-i a b d^2 n \pi \right ) x}{n}-\frac {1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}+\frac {\left (i b d x (e x)^m \left (c x^n\right )^{i a b d^2 \pi -\frac {1+m+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {\left (1+m+i a b d^2 n \pi \right ) x}{n}+\frac {1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)} \\ & = \frac {(e x)^{1+m} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i b d \exp \left (-\frac {i (1+m) \left (1+m-2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{-i a b d^2 \pi -\frac {1+m-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {i \left (\frac {1+m-i a b d^2 n \pi }{n}-i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}+\frac {\left (i b d \exp \left (\frac {i (1+m) \left (1+m+2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{i a b d^2 \pi -\frac {1+m+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {i \left (\frac {1+m+i a b d^2 n \pi }{n}+i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)} \\ & = \frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \exp \left (\frac {i (1+m) \left (1+m+2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \exp \left (-\frac {i (1+m) \left (1+m-2 i a b d^2 n \pi \right )}{2 b^2 d^2 n^2 \pi }\right ) x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+m-i a b d^2 n \pi -i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )}{1+m}+\frac {(e x)^{1+m} \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)} \\ \end{align*}
Time = 3.68 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.87 \[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (-\sqrt [4]{-1} \sqrt {2} e^{-\frac {(1+m) \left (i+i m+2 a b d^2 n \pi +2 b^2 d^2 n \pi \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{2 b^2 d^2 n^2 \pi }} x^{-m} \left (\text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i+i m+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )+e^{\frac {i (1+m)^2}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {(-1)^{3/4} \left (1+m+i a b d^2 n \pi +i b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right )+4 x \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{4 (1+m)} \]
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\[\int \left (e x \right )^{m} \operatorname {FresnelS}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 691 vs. \(2 (310) = 620\).
Time = 0.29 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.47 \[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {-i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i \, m + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i \, m + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + 2 \, x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, {\left (m + 1\right )}} \]
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\[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} S\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \]
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\[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int (e x)^m \operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]
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