Integrand size = 17, antiderivative size = 122 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6690, 12, 5650, 2347, 2209} \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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Rule 12
Rule 2209
Rule 2347
Rule 5650
Rule 6690
Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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 ) \text {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}} \operatorname {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}} \operatorname {ExpIntegralEi}\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*}
Time = 1.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.20 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {1}{2} e^{-\frac {(-1+b d n) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (\operatorname {ExpIntegralEi}\left (\frac {(-1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (-\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \left (\cosh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\sinh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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\[\int \frac {\operatorname {Shi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]
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\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]
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