Optimal. Leaf size=124 \[ x \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{ExpIntegralEi}\left (\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{ExpIntegralEi}\left (\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
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Rubi [A] time = 0.228058, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6524, 12, 4496, 2310, 2178} \[ x \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
Antiderivative was successfully verified.
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Rule 6524
Rule 12
Rule 4496
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-(b d n) \int \frac{\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\\ &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-(b n) \int \frac{\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx\\ &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\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^{-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^{i b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} \left (b e^{-i a d} x \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 )-\frac{1}{2} \left (b e^{i a d} x \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 )\\ &=x \text{Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} e^{-\frac{a}{b n}} x \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{1}{2} e^{-\frac{a}{b n}} x \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\\ \end{align*}
Mathematica [A] time = 1.65783, size = 98, normalized size = 0.79 \[ x \text{CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \left (\text{ExpIntegralEi}\left (\frac{(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text{ExpIntegralEi}\left (\frac{(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{\it Ci} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, 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{\rm Ci}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{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 (\operatorname{Ci}\left (b d \log \left (c x^{n}\right ) + a d\right ), 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 \operatorname{Ci}{\left (d \left (a + b \log{\left (c x^{n} \right )}\right ) \right )}\, 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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