Optimal. Leaf size=128 \[ \frac{1}{4} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{ExpIntegralEi}\left (\frac{(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{4} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{ExpIntegralEi}\left (\frac{(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac{1}{2} x^2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.253044, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6555, 12, 5539, 2310, 2178} \[ \frac{1}{4} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Ei}\left (\frac{(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{4} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Ei}\left (\frac{(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac{1}{2} x^2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 6555
Rule 12
Rule 5539
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int x \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{1}{2} x^2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} (b d n) \int \frac{x \sinh \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}{2} x^2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} (b n) \int \frac{x \sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx\\ &=\frac{1}{2} x^2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{1}{4} \left (b e^{-a d} n x^{b d n} \left (c x^n\right )^{-b d}\right ) \int \frac{x^{1-b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac{1}{4} \left (b e^{a d} n x^{-b d n} \left (c x^n\right )^{b d}\right ) \int \frac{x^{1+b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=\frac{1}{2} x^2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{1}{4} \left (b e^{-a d} x^2 \left (c x^n\right )^{-b d-\frac{2-b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(2-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )-\frac{1}{4} \left (b e^{a d} x^2 \left (c x^n\right )^{b d-\frac{2+b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(2+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )\\ &=\frac{1}{4} e^{-\frac{2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \text{Ei}\left (\frac{(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{4} e^{-\frac{2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \text{Ei}\left (\frac{(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac{1}{2} x^2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end{align*}
Mathematica [A] time = 1.65802, size = 98, normalized size = 0.77 \[ \frac{1}{4} x^2 \left (e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \left (\text{ExpIntegralEi}\left (-\frac{(b d n-2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text{ExpIntegralEi}\left (\frac{(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.108, size = 0, normalized size = 0. \begin{align*} \int x{\it Shi} \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{\rm Shi}\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 \operatorname{Shi}\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 \operatorname{Shi}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Shi}\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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