Optimal. Leaf size=124 \[ x \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 124, normalized size of antiderivative = 1.00, 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 12
Rule 2178
Rule 2310
Rule 4496
Rule 6524
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*}
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Mathematica [A] time = 1.93, size = 98, normalized size = 0.79 \[ x \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \left (\text {Ei}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text {Ei}\left (\frac {(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Ci}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \Ci \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Ci}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {Ci}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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