Optimal. Leaf size=65 \[ \frac {\cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )}{b d n \pi }+\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A]
time = 0.03, 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 = {6553}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 6553
Rubi steps
\begin {align*} \int \frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int S(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\text {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] Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(65)=130\).
time = 0.07, size = 164, normalized size = 2.52 \begin {gather*} \frac {\cos \left (\frac {1}{2} a^2 d^2 \pi \right ) \cos \left (a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} b^2 d^2 \pi \log ^2\left (c x^n\right )\right )}{b d n \pi }+\frac {a S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \log \left (c x^n\right )}{n}-\frac {\sin \left (\frac {1}{2} a^2 d^2 \pi \right ) \sin \left (a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} b^2 d^2 \pi \log ^2\left (c x^n\right )\right )}{b d n \pi } \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.44, size = 63, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\mathrm {S}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {\cos \left (\frac {\pi \left (a d +b d \ln \left (c \,x^{n}\right )\right )^{2}}{2}\right )}{\pi }}{n b d}\) | \(63\) |
default | \(\frac {\mathrm {S}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {\cos \left (\frac {\pi \left (a d +b d \ln \left (c \,x^{n}\right )\right )^{2}}{2}\right )}{\pi }}{n b d}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 81, normalized size = 1.25 \begin {gather*} \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {S}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) + \frac {\cos \left (\frac {1}{2} \, \pi b^{2} d^{2} \log \left (c x^{n}\right )^{2} + \pi a b d^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi }}{b d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 119, normalized size = 1.83 \begin {gather*} \frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \cos \left (\frac {1}{2} \, \pi b^{2} d^{2} n^{2} \log \left (x\right )^{2} + \pi b^{2} d^{2} n \log \left (c\right ) \log \left (x\right ) + \frac {1}{2} \, \pi b^{2} d^{2} \log \left (c\right )^{2} + \pi a b d^{2} n \log \left (x\right ) + \pi a b d^{2} \log \left (c\right ) + \frac {1}{2} \, \pi a^{2} d^{2}\right )}{\pi b d n} \end {gather*}
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 \begin {gather*} \int \frac {S\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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