Optimal. Leaf size=122 \[ \frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{ExpIntegralEi}\left (-\frac{(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{ExpIntegralEi}\left (-\frac{(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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Rubi [A] time = 0.250168, antiderivative size = 122, 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 = {6555, 12, 5539, 2310, 2178} \[ \frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
Antiderivative was successfully verified.
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Rule 6555
Rule 12
Rule 5539
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b d n) \int \frac{\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d x^2 \left (a+b \log \left (c x^n\right )\right )} \, dx\\ &=-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b n) \int \frac{\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2 \left (a+b \log \left (c x^n\right )\right )} \, dx\\ &=-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}-\frac{1}{2} \left (b e^{-a d} n x^{b d n} \left (c x^n\right )^{-b d}\right ) \int \frac{x^{-2-b d n}}{a+b \log \left (c x^n\right )} \, dx+\frac{1}{2} \left (b e^{a d} n x^{-b d n} \left (c x^n\right )^{b d}\right ) \int \frac{x^{-2+b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}-\frac{\left (b e^{-a d} \left (c x^n\right )^{-b d-\frac{-1-b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(-1-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 x}+\frac{\left (b e^{a d} \left (c x^n\right )^{b d-\frac{-1+b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(-1+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 x}\\ &=\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}\\ \end{align*}
Mathematica [A] time = 1.82148, size = 146, normalized size = 1.2 \[ \frac{1}{2} \exp \left (-\frac{(b d n-1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left (\text{ExpIntegralEi}\left (\frac{(b d n-1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text{ExpIntegralEi}\left (-\frac{(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \left (\sinh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )+\cosh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Shi} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{2}}}\, 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 \frac{{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{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 (\frac{\operatorname{Shi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, 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 \frac{\operatorname{Shi}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x^{2}}\, 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 \frac{{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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