Optimal. Leaf size=127 \[ -\frac {\text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}+\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x} \]
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Rubi [A] time = 0.25, antiderivative size = 127, normalized size of antiderivative = 1.00, 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 12
Rule 2178
Rule 2310
Rule 4498
Rule 6527
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*}
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Mathematica [A] time = 1.92, size = 102, normalized size = 0.80 \[ \frac {-2 \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (\text {Ei}\left (-\frac {i (b d n-i) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text {Ei}\left (\frac {i (b d n+i) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{2 x} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, x\right ) \]
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 = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\Ci \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]
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 \[ \int \frac {{\rm Ci}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {Ci}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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