Integrand size = 19, antiderivative size = 167 \[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)} \]
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Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6691, 12, 5651, 2347, 2209} \[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{m+1} \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m-b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}-\frac {x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)} \]
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Rule 12
Rule 2209
Rule 2347
Rule 5651
Rule 6691
Rubi steps \begin{align*} \text {integral}& = \frac {(e x)^{1+m} \text {Chi}\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 \cosh \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 {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b n) \int \frac {(e x)^m \cosh \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 {Chi}\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 {Chi}\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 ) \text {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 ) \text {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 x)^{1+m} \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\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}} \operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)} \\ \end{align*}
Time = 1.37 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.71 \[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (2 x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} \left (\operatorname {ExpIntegralEi}\left (\frac {(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right )}{2 (1+m)} \]
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\[\int \left (e x \right )^{m} \operatorname {Chi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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\[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \operatorname {Chi}\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \]
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\[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {coshint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]
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