Optimal. Leaf size=65 \[ \frac {\cos \left (\frac {1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )\right )^2\right )}{\pi b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6418} \[ \frac {\cos \left (\frac {1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )\right )^2\right )}{\pi b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 6418
Rubi steps
\begin {align*} \int \frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int S(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int S(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n}\\ &=\frac {\cos \left (\frac {1}{2} \pi \left (a d+b d \log \left (c x^n\right )\right )^2\right )}{b d n \pi }+\frac {S\left (a d+b d \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end {align*}
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Mathematica [B] time = 0.13, size = 164, normalized size = 2.52 \[ -\frac {\sin \left (\frac {1}{2} \pi a^2 d^2\right ) \sin \left (\pi a b d^2 \log \left (c x^n\right )+\frac {1}{2} \pi b^2 d^2 \log ^2\left (c x^n\right )\right )}{\pi b d n}+\frac {\cos \left (\frac {1}{2} \pi a^2 d^2\right ) \cos \left (\pi a b d^2 \log \left (c x^n\right )+\frac {1}{2} \pi b^2 d^2 \log ^2\left (c x^n\right )\right )}{\pi b d n}+\frac {a S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac {\log \left (c x^n\right ) S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x}, x\right ) \]
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 \[ \int \frac {{\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 80, normalized size = 1.23 \[ \frac {\ln \left (c \,x^{n}\right ) \mathrm {S}\left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n}+\frac {\mathrm {S}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) a}{n b}+\frac {\cos \left (\frac {\pi \left (a d +b d \ln \left (c \,x^{n}\right )\right )^{2}}{2}\right )}{n b d \pi } \]
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 \[ \int \frac {{\rm fresnels}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x} \,d x \]
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 \[ \int \frac {S\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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