Optimal. Leaf size=227 \[ \left (\frac {1}{8}-\frac {i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{\frac {-2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{-\frac {2 \left (\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )+\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.44, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6471, 4617, 2278, 2274, 15, 2276, 2234, 2204, 2205} \[ \left (\frac {1}{8}-\frac {i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{\frac {-2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{-\frac {2 \left (\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )+\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 15
Rule 2204
Rule 2205
Rule 2234
Rule 2274
Rule 2276
Rule 2278
Rule 4617
Rule 6471
Rubi steps
\begin {align*} \int x S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} (b d n) \int x \sin \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} (i b d n) \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} x \, dx+\frac {1}{4} (i b d n) \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} x \, dx\\ &=\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} (i b d n) \int \exp \left (-\frac {1}{2} i a^2 d^2 \pi -i a b d^2 \pi \log \left (c x^n\right )-\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x \, dx+\frac {1}{4} (i b d n) \int \exp \left (\frac {1}{2} i a^2 d^2 \pi +i a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x \, dx\\ &=\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} (i b d n) \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 \left (c x^n\right )^{-i a b d^2 \pi } \, dx+\frac {1}{4} (i b d n) \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 \left (c x^n\right )^{i a b d^2 \pi } \, dx\\ &=\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\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^{1-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^{1+i a b d^2 n \pi } \, dx\\ &=\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} \left (i b d x^2 \left (c x^n\right )^{-i a b d^2 \pi -\frac {2-i a b d^2 n \pi }{n}}\right ) \operatorname {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 )+\frac {1}{4} \left (i b d x^2 \left (c x^n\right )^{i a b d^2 \pi -\frac {2+i a b d^2 n \pi }{n}}\right ) \operatorname {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 )\\ &=\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} \left (i b d e^{-\frac {2 \left (i+a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} x^2 \left (c x^n\right )^{-i a b d^2 \pi -\frac {2-i a b d^2 n \pi }{n}}\right ) \operatorname {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 )+\frac {1}{4} \left (i b d e^{\frac {2 i-2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} x^2 \left (c x^n\right )^{i a b d^2 \pi -\frac {2+i a b d^2 n \pi }{n}}\right ) \operatorname {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 )\\ &=\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 }} x^2 \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 )+\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 }} x^2 \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 )+\frac {1}{2} x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end {align*}
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Mathematica [A] time = 7.91, size = 319, normalized size = 1.41 \[ \frac {1}{8} x^2 \left (4 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt {2} \left (c x^n\right )^{-2/n} \left (e^{\frac {4 i}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-2 i\right )}{\sqrt {\pi } b d n}\right )+i \text {erfi}\left (\frac {(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+2 i\right )}{\sqrt {2 \pi } b d n}\right )\right ) \exp \left (-\frac {1}{2} i \pi a^2 d^2+i \pi a b d^2 \left (n \log (x)-\log \left (c x^n\right )\right )-\frac {2 a}{b n}-\frac {1}{2} i \pi b^2 d^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2-\frac {2 i}{\pi b^2 d^2 n^2}\right ) \left (\cos \left (\frac {1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )+i \sin \left (\frac {1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x {\rm fresnels}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int x \,\mathrm {S}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x S\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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