Optimal. Leaf size=227 \[ \left (\frac{1}{8}+\frac{i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{\frac{-2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{2}{n}\right )}{\sqrt{\pi } b d}\right )-\left (\frac{1}{8}+\frac{i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{-\frac{2 \left (\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{2}{n}\right )}{\sqrt{\pi } b d}\right )+\frac{1}{2} x^2 \text{FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.407877, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6472, 4618, 2278, 2274, 15, 2276, 2234, 2204, 2205} \[ \left (\frac{1}{8}+\frac{i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{\frac{-2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{2}{n}\right )}{\sqrt{\pi } b d}\right )-\left (\frac{1}{8}+\frac{i}{8}\right ) x^2 \left (c x^n\right )^{-2/n} e^{-\frac{2 \left (\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{2}{n}\right )}{\sqrt{\pi } b d}\right )+\frac{1}{2} x^2 \text{FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 6472
Rule 4618
Rule 2278
Rule 2274
Rule 15
Rule 2276
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x C\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{1}{2} x^2 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} (b d n) \int x \cos \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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{4} (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} (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{4} (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} (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{4} (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} (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{4} \left (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 (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{4} \left (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 (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{4} \left (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 (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end{align*}
Mathematica [A] time = 6.64992, size = 318, normalized size = 1.4 \[ \frac{1}{8} x^2 \left (4 \text{FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt{2} \left (c x^n\right )^{-2/n} \left (i e^{\frac{4 i}{\pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-2 i\right )}{\sqrt{\pi } b d n}\right )+\text{Erfi}\left (\frac{(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+2 i\right )}{\sqrt{2 \pi } b d n}\right )\right ) \exp \left (-\frac{1}{2} i \pi a^2 d^2+i \pi a b d^2 \left (n \log (x)-\log \left (c x^n\right )\right )-\frac{2 a}{b n}-\frac{1}{2} i \pi b^2 d^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2-\frac{2 i}{\pi b^2 d^2 n^2}\right ) \left (\cos \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )+i \sin \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.539, size = 0, normalized size = 0. \begin{align*} \int x{\it FresnelC} \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{\rm fresnelc}\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{\rm fresnelc}\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 C\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{\rm fresnelc}\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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