Optimal. Leaf size=228 \[ \frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{\frac {2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{-\frac {2 \left (-\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
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Rubi [A] time = 0.52, antiderivative size = 228, 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 = {6471, 4617, 2278, 2274, 15, 2276, 2234, 2204, 2205} \[ \frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{\frac {2 \pi a b d^2 n+2 i}{\pi b^2 d^2 n^2}} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{-\frac {2 \left (-\pi a b d^2 n+i\right )}{\pi b^2 d^2 n^2}} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
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 \frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=-\frac {S\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 {S\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 {S\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 {\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^3} \, dx-\frac {1}{4} (i b d n) \int \frac {\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^3} \, dx\\ &=-\frac {S\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 {\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 ) \left (c x^n\right )^{-i a b d^2 \pi }}{x^3} \, dx-\frac {1}{4} (i b d n) \int \frac {\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 ) \left (c x^n\right )^{i a b d^2 \pi }}{x^3} \, dx\\ &=-\frac {S\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 {S\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 ) \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 )}{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 ) \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 )}{4 x^2}\\ &=-\frac {S\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 ) \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 )}{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 ) \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 )}{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 {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 4.51, size = 200, normalized size = 0.88 \[ -\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}-\frac {\sqrt [4]{-1} \left (e^{\frac {4 i}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\sqrt [4]{-1} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+2 i\right )}{\sqrt {2 \pi } b d n}\right )+i \text {erfi}\left (\frac {(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-2 i\right )}{\sqrt {2 \pi } b d n}\right )\right ) \exp \left (\frac {2 \left (\frac {a n}{b}-\frac {i}{\pi b^2 d^2}+n \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{n^2}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {S}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]
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 \[ \int \frac {{\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]
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 \[ \int \frac {S\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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