Optimal. Leaf size=127 \[ -\frac{\text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{ExpIntegralEi}\left (-\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}+\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{ExpIntegralEi}\left (-\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x} \]
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Rubi [A] time = 0.247361, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6527, 12, 4498, 2310, 2178} \[ -\frac{\text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}+\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x} \]
Antiderivative was successfully verified.
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Rule 6527
Rule 12
Rule 4498
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{\text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=-\frac{\text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b d n) \int \frac{\cos \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{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b n) \int \frac{\cos \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{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{1}{2} \left (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 (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{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{\left (b e^{-i a d} \left (c x^n\right )^{-i b d-\frac{-1-i b d n}{n}}\right ) \operatorname{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 (b e^{i a d} \left (c x^n\right )^{i b d-\frac{-1+i b d n}{n}}\right ) \operatorname{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{\text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{(1-i 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}} \text{Ei}\left (-\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}\\ \end{align*}
Mathematica [A] time = 1.701, size = 102, normalized size = 0.8 \[ \frac{-2 \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (\text{ExpIntegralEi}\left (-\frac{i (b d n-i) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text{ExpIntegralEi}\left (\frac{i (b d n+i) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Ci} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{Ci}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Ci}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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