Integrand size = 17, antiderivative size = 131 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {i e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}+\frac {i e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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Time = 0.19 (sec) , antiderivative size = 131, 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 = {6661, 12, 4585, 2347, 2209} \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {i e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}+\frac {i e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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Rule 12
Rule 2209
Rule 2347
Rule 4585
Rule 6661
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b d n) \int \frac {\sin \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 {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b n) \int \frac {\sin \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 {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {1}{2} \left (i b e^{-i a d} n x^{i b d n} \left (c x^n\right )^{-i b d}\right ) \int \frac {x^{-2-i b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac {1}{2} \left (i b e^{i a d} n x^{-i b d n} \left (c x^n\right )^{i b d}\right ) \int \frac {x^{-2+i b d n}}{a+b \log \left (c x^n\right )} \, dx \\ & = -\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (i b e^{-i a d} \left (c x^n\right )^{-i b d-\frac {-1-i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-1-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 x}-\frac {\left (i b e^{i a d} \left (c x^n\right )^{i b d-\frac {-1+i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-1+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 x} \\ & = -\frac {i e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}+\frac {i e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {i e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (\operatorname {ExpIntegralEi}\left (-\frac {i (-i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {i (i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )-2 \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x} \]
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\[\int \frac {\operatorname {Si}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08 \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {{\left (-i \, x {\rm Ei}\left (\frac {i \, a b d n + {\left (i \, b^{2} d n - b\right )} \log \left (c\right ) + {\left (i \, b^{2} d n^{2} - b n\right )} \log \left (x\right ) - a}{b n}\right ) + i \, x {\rm Ei}\left (\frac {-i \, a b d n + {\left (-i \, b^{2} d n - b\right )} \log \left (c\right ) + {\left (-i \, b^{2} d n^{2} - b n\right )} \log \left (x\right ) - a}{b n}\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - 2 \, \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, x} \]
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\[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\operatorname {Si}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {Si}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]
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