Optimal. Leaf size=167 \[ \frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{ExpIntegralEi}\left (\frac{(-b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}-\frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{ExpIntegralEi}\left (\frac{(b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}+\frac{(e x)^{m+1} \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.303797, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6555, 12, 5539, 2310, 2178} \[ \frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m-b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}-\frac{x (e x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}+\frac{(e x)^{m+1} \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6555
Rule 12
Rule 5539
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int (e x)^m \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{(e x)^{1+m} \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac{(b d n) \int \frac{(e x)^m \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}{1+m}\\ &=\frac{(e x)^{1+m} \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac{(b n) \int \frac{(e x)^m \sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx}{1+m}\\ &=\frac{(e x)^{1+m} \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac{\left (b e^{-a d} n x^{-m+b d n} (e x)^m \left (c x^n\right )^{-b d}\right ) \int \frac{x^{m-b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)}-\frac{\left (b e^{a d} n x^{-m-b d n} (e x)^m \left (c x^n\right )^{b d}\right ) \int \frac{x^{m+b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)}\\ &=\frac{(e x)^{1+m} \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac{\left (b e^{-a d} x (e x)^m \left (c x^n\right )^{-b d-\frac{1+m-b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}-\frac{\left (b e^{a d} x (e x)^m \left (c x^n\right )^{b d-\frac{1+m+b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}-\frac{e^{-\frac{a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}+\frac{(e x)^{1+m} \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 2.91851, size = 120, normalized size = 0.72 \[ \frac{(e x)^m \left (x^{-m} \exp \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (\text{ExpIntegralEi}\left (\frac{(-b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text{ExpIntegralEi}\left (\frac{(b d n+m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 x \text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m}{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m}{\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.
[In]
[Out]