Integrand size = 17, antiderivative size = 135 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \]
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Time = 0.19 (sec) , antiderivative size = 135, 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 = {6662, 12, 4586, 2347, 2209} \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}+\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(i b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \]
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Rule 12
Rule 2209
Rule 2347
Rule 4586
Rule 6662
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (b d n) \int \frac {\cos \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 {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (b n) \int \frac {\cos \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 {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} \left (b e^{-i a d} n x^{i b d n} \left (c x^n\right )^{-i b d}\right ) \int \frac {x^{-3-i b d n}}{a+b \log \left (c x^n\right )} \, dx+\frac {1}{4} \left (b e^{i a d} n x^{-i b d n} \left (c x^n\right )^{i b d}\right ) \int \frac {x^{-3+i b d n}}{a+b \log \left (c x^n\right )} \, dx \\ & = -\frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (b e^{-i a d} \left (c x^n\right )^{-i b d-\frac {-2-i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-2-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2}+\frac {\left (b e^{i a d} \left (c x^n\right )^{i b d-\frac {-2+i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-2+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2} \\ & = -\frac {\operatorname {CosIntegral}\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-i 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+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2} \\ \end{align*}
Time = 1.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {-2 \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {i (-2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {i (2 i+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{4 x^2} \]
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\[\int \frac {\operatorname {Ci}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (121) = 242\).
Time = 0.29 (sec) , antiderivative size = 460, normalized size of antiderivative = 3.41 \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \]
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\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\operatorname {Ci}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]
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