Integrand size = 19, antiderivative size = 172 \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)} \]
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Time = 0.29 (sec) , antiderivative size = 172, 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 = {6662, 12, 4586, 2347, 2209} \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{m+1} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m-i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}-\frac {x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)} \]
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Rule 12
Rule 2209
Rule 2347
Rule 4586
Rule 6662
Rubi steps \begin{align*} \text {integral}& = \frac {(e x)^{1+m} \operatorname {CosIntegral}\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 \cos \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} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b n) \int \frac {(e x)^m \cos \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} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (b e^{-i a d} n x^{-m+i b d n} (e x)^m \left (c x^n\right )^{-i b d}\right ) \int \frac {x^{m-i b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)}-\frac {\left (b e^{i a d} n x^{-m-i b d n} (e x)^m \left (c x^n\right )^{i b d}\right ) \int \frac {x^{m+i b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)} \\ & = \frac {(e x)^{1+m} \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (b e^{-i a d} x (e x)^m \left (c x^n\right )^{-i b d-\frac {1+m-i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(1+m-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}-\frac {\left (b e^{i a d} x (e x)^m \left (c x^n\right )^{i b d-\frac {1+m+i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(1+m+i 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} \operatorname {CosIntegral}\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-i 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+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)} \\ \end{align*}
Time = 2.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.72 \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (2 x \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} \left (\operatorname {ExpIntegralEi}\left (\frac {(1+m-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(1+m+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right )}{2 (1+m)} \]
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\[\int \left (e x \right )^{m} \operatorname {Ci}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (164) = 328\).
Time = 0.29 (sec) , antiderivative size = 689, normalized size of antiderivative = 4.01 \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {\pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \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 + i \, m + 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}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \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 - i \, m - 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}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} - \frac {i \, m}{\pi b^{2} d^{2} n^{2}} - \frac {i}{2 \, \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 + i \, m + 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}} e^{\left (m \log \left (e\right ) - \frac {m \log \left (c\right )}{n} - \frac {a m}{b n} - \frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i \, m^{2}}{2 \, \pi b^{2} d^{2} n^{2}} + \frac {i \, m}{\pi b^{2} d^{2} n^{2}} + \frac {i}{2 \, \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 - i \, m - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, {\left (m + 1\right )}} \]
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\[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \operatorname {Ci}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {C}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int (e x)^m \operatorname {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]
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