Optimal. Leaf size=231 \[ \left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}+\frac {9 i}{2 \pi b^2 d^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {3}{n}\right )}{\sqrt {\pi } b d}\right )-\left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}-\frac {9 i}{2 \pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {3}{n}\right )}{\sqrt {\pi } b d}\right )+\frac {1}{3} x^3 C\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.52, antiderivative size = 231, normalized size of antiderivative = 1.00, 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 = {6472, 4618, 2278, 2274, 15, 2276, 2234, 2204, 2205} \[ \left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}+\frac {9 i}{2 \pi b^2 d^2 n^2}} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {3}{n}\right )}{\sqrt {\pi } b d}\right )-\left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}-\frac {9 i}{2 \pi b^2 d^2 n^2}} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {3}{n}\right )}{\sqrt {\pi } b d}\right )+\frac {1}{3} x^3 \text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 15
Rule 2204
Rule 2205
Rule 2234
Rule 2274
Rule 2276
Rule 2278
Rule 4618
Rule 6472
Rubi steps
\begin {align*} \int x^2 C\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {1}{3} x^3 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} (b d n) \int x^2 \cos \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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} (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} (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} (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} (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} (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} (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} \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^{2-i a b d^2 n \pi } \, dx-\frac {1}{6} \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^{2+i a b d^2 n \pi } \, dx\\ &=\frac {1}{3} x^3 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} \left (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 (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} \left (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 (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 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end {align*}
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Mathematica [A] time = 7.68, size = 318, normalized size = 1.38 \[ \frac {1}{12} x^3 \left (4 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt {2} \left (c x^n\right )^{-3/n} \left (i e^{\frac {9 i}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-3 i\right )}{\sqrt {\pi } b d n}\right )+\text {erfi}\left (\frac {(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+3 i\right )}{\sqrt {2 \pi } b d n}\right )\right ) \exp \left (\frac {1}{2} \left (-i \pi a^2 d^2+2 i \pi a b d^2 \left (n \log (x)-\log \left (c x^n\right )\right )-\frac {6 a}{b n}-i \pi b^2 d^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2-\frac {9 i}{\pi b^2 d^2 n^2}\right )\right ) \left (\cos \left (\frac {1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )+i \sin \left (\frac {1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} {\rm fresnelc}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]
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 \[ \int x^{2} {\rm fresnelc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x^{2} \FresnelC \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
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 \[ \int x^{2} {\rm fresnelc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
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 \[ \int x^2\,\mathrm {FresnelC}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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 \[ \int x^{2} C\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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