Optimal. Leaf size=231 \[ \left (\frac{1}{12}-\frac{i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac{3 a}{b n}+\frac{9 i}{2 \pi b^2 d^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{3}{n}\right )}{\sqrt{\pi } b d}\right )+\left (\frac{1}{12}-\frac{i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac{3 a}{b n}-\frac{9 i}{2 \pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{3}{n}\right )}{\sqrt{\pi } b d}\right )+\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.570596, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {6471, 4617, 2278, 2274, 15, 2276, 2234, 2204, 2205} \[ \left (\frac{1}{12}-\frac{i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac{3 a}{b n}+\frac{9 i}{2 \pi b^2 d^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{3}{n}\right )}{\sqrt{\pi } b d}\right )+\left (\frac{1}{12}-\frac{i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac{3 a}{b n}-\frac{9 i}{2 \pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{3}{n}\right )}{\sqrt{\pi } b d}\right )+\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 6471
Rule 4617
Rule 2278
Rule 2274
Rule 15
Rule 2276
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{3} (b d n) \int x^2 \sin \left (\frac{1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} (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^2 \, dx+\frac{1}{6} (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^2 \, dx\\ &=\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} (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^2 \, dx+\frac{1}{6} (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^2 \, dx\\ &=\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} (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^2 \left (c x^n\right )^{-i a b d^2 \pi } \, dx+\frac{1}{6} (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^2 \left (c x^n\right )^{i a b d^2 \pi } \, dx\\ &=\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} \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^{2-i a b d^2 n \pi } \, dx+\frac{1}{6} \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^{2+i a b d^2 n \pi } \, dx\\ &=\frac{1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} \left (i b d x^3 \left (c x^n\right )^{-i a b d^2 \pi -\frac{3-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 (3-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}{6} \left (i b d x^3 \left (c x^n\right )^{i a b d^2 \pi -\frac{3+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 (3+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}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} \left (i b d e^{-\frac{3 a}{b n}-\frac{9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-i a b d^2 \pi -\frac{3-i a b d^2 n \pi }{n}}\right ) \operatorname{Subst}\left (\int \exp \left (\frac{i \left (\frac{3-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}{6} \left (i b d e^{-\frac{3 a}{b n}+\frac{9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{i a b d^2 \pi -\frac{3+i a b d^2 n \pi }{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-\frac{i \left (\frac{3+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}{12}-\frac{i}{12}\right ) e^{-\frac{3 a}{b n}+\frac{9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text{erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\frac{3}{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}{12}-\frac{i}{12}\right ) e^{-\frac{3 a}{b n}-\frac{9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text{erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\frac{3}{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}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end{align*}
Mathematica [A] time = 6.88797, size = 319, normalized size = 1.38 \[ \frac{1}{12} x^3 \left (4 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt{2} \left (c x^n\right )^{-3/n} \left (e^{\frac{9 i}{\pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-3 i\right )}{\sqrt{\pi } b d n}\right )+i \text{Erfi}\left (\frac{(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+3 i\right )}{\sqrt{2 \pi } b d n}\right )\right ) \exp \left (\frac{1}{2} \left (-i \pi a^2 d^2+2 i \pi a b d^2 \left (n \log (x)-\log \left (c x^n\right )\right )-\frac{6 a}{b n}-i \pi b^2 d^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2-\frac{9 i}{\pi b^2 d^2 n^2}\right )\right ) \left (\cos \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )+i \sin \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.701, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\it FresnelS} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2}{\rm fresnels}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} S\left (a d + b d \log{\left (c x^{n} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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