Integrand size = 17, antiderivative size = 228 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{\frac {2 i+2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{-\frac {2 \left (i-a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}-\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
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Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6606, 4713, 2314, 2308, 2266, 2235, 2236} \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{\frac {2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{-\frac {2 \left (-\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}-\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
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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 {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (b d n) \int \frac {\sin \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )}{x^3} \, dx \\ & = -\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} (i b d n) \int \frac {e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^3} \, dx-\frac {1}{4} (i b d n) \int \frac {e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^3} \, dx \\ & = -\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} \left (i b d n x^{i a b d^2 n \pi } \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^{-3-i a b d^2 n \pi } \, dx-\frac {1}{4} \left (i b d n x^{-i a b d^2 n \pi } \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^{-3+i a b d^2 n \pi } \, dx \\ & = -\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (i b d \left (c x^n\right )^{-i a b d^2 \pi -\frac {-2-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 (-2-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 )}{4 x^2}-\frac {\left (i b d \left (c x^n\right )^{i a b d^2 \pi -\frac {-2+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 (-2+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 )}{4 x^2} \\ & = -\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (i b d e^{-\frac {2 \left (i-a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{-i a b d^2 \pi -\frac {-2-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {i \left (\frac {-2-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 )}{4 x^2}-\frac {\left (i b d e^{\frac {2 i+2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{i a b d^2 \pi -\frac {-2+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {i \left (\frac {-2+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 )}{4 x^2} \\ & = \frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{\frac {2 i+2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{-\frac {2 \left (i-a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}-\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \\ \end{align*}
Time = 2.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.88 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\sqrt [4]{-1} e^{\frac {2 \left (\frac {a n}{b}-\frac {i}{b^2 d^2 \pi }+n \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{n^2}} \left (i \text {erfi}\left (\frac {(-1)^{3/4} \left (-2 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )+e^{\frac {4 i}{b^2 d^2 n^2 \pi }} \text {erfi}\left (\frac {\sqrt [4]{-1} \left (2 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right )}{4 \sqrt {2}}-\frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
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\[\int \frac {\operatorname {FresnelS}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (183) = 366\).
Time = 0.28 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.02 \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {-i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \]
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\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {S\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x^{3}}\, dx \]
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\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelS}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]
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