Integrand size = 17, antiderivative size = 130 \[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \]
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Time = 0.19 (sec) , antiderivative size = 130, 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 = {6691, 12, 5651, 2347, 2209} \[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \]
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Rule 12
Rule 2209
Rule 2347
Rule 5651
Rule 6691
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (b d n) \int \frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx \\ & = -\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (b n) \int \frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx \\ & = -\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} \left (b e^{-a d} n x^{b d n} \left (c x^n\right )^{-b d}\right ) \int \frac {x^{-3-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^{-3+b d n}}{a+b \log \left (c x^n\right )} \, dx \\ & = -\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (b e^{-a d} \left (c x^n\right )^{-b d-\frac {-2-b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-2-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2}+\frac {\left (b e^{a d} \left (c x^n\right )^{b d-\frac {-2+b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-2+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2} \\ & = -\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \\ \end{align*}
Time = 1.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12 \[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} e^{-\frac {(-2+b d n) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (\operatorname {ExpIntegralEi}\left (\frac {(-2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (-\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \left (\cosh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\sinh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right ) \]
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\[\int \frac {\operatorname {Chi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]
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\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\operatorname {Chi}\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x^{3}}\, dx \]
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\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {coshint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]
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