Optimal. Leaf size=55 \[ \frac {\text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]
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Rubi [A]
time = 0.02, antiderivative size = 55, 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 = {6635}
\begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right ) \text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 6635
Rubi steps
\begin {align*} \int \frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \text {Ci}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\text {Subst}\left (\int \text {Ci}(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n}\\ &=\frac {\text {Ci}\left (a d+b d \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin \left (a d+b d \log \left (c x^n\right )\right )}{b d n}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 96, normalized size = 1.75 \begin {gather*} \frac {a \text {CosIntegral}\left (a d+b d \log \left (c x^n\right )\right )}{b n}+\frac {\text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \log \left (c x^n\right )}{n}-\frac {\cos \left (b d \log \left (c x^n\right )\right ) \sin (a d)}{b d n}-\frac {\cos (a d) \sin \left (b d \log \left (c x^n\right )\right )}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.97, size = 56, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {\cosineIntegral \left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\sin \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(56\) |
default | \(\frac {\cosineIntegral \left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\sin \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 82, normalized size = 1.49 \begin {gather*} \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac {\sin \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (55) = 110\).
time = 0.40, size = 121, normalized size = 2.20 \begin {gather*} \frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \sin \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 {\operatorname {Ci}{\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*} \frac {\ln \left (c\,x^n\right )\,\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{n}+\frac {a\,\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,d\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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