Optimal. Leaf size=133 \[ \frac{1}{3} x^3 \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{ExpIntegralEi}\left (\frac{(3-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{6} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{ExpIntegralEi}\left (\frac{(3+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
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Rubi [A] time = 0.251172, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6527, 12, 4498, 2310, 2178} \[ \frac{1}{3} x^3 \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Ei}\left (\frac{(3-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{6} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Ei}\left (\frac{(i b d n+3) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
Antiderivative was successfully verified.
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Rule 6527
Rule 12
Rule 4498
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int x^2 \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{1}{3} x^3 \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{3} (b d n) \int \frac{x^2 \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )} \, dx\\ &=\frac{1}{3} x^3 \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{3} (b n) \int \frac{x^2 \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx\\ &=\frac{1}{3} x^3 \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} \left (b e^{-i a d} n x^{i b d n} \left (c x^n\right )^{-i b d}\right ) \int \frac{x^{2-i b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac{1}{6} \left (b e^{i a d} n x^{-i b d n} \left (c x^n\right )^{i b d}\right ) \int \frac{x^{2+i b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=\frac{1}{3} x^3 \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} \left (b e^{-i a d} x^3 \left (c x^n\right )^{-i b d-\frac{3-i b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(3-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )-\frac{1}{6} \left (b e^{i a d} x^3 \left (c x^n\right )^{i b d-\frac{3+i b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(3+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )\\ &=\frac{1}{3} x^3 \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{6} e^{-\frac{3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \text{Ei}\left (\frac{(3-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{6} e^{-\frac{3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \text{Ei}\left (\frac{(3+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\\ \end{align*}
Mathematica [A] time = 1.74331, size = 102, normalized size = 0.77 \[ \frac{1}{6} x^3 \left (2 \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \left (\text{ExpIntegralEi}\left (\frac{(3-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text{ExpIntegralEi}\left (\frac{(3+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\it Ci} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Ci}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{Ci}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{Ci}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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