Optimal. Leaf size=128 \[ \frac {1}{6} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {(3-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{6} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {(b d n+3) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.27, antiderivative size = 128, 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 = {6555, 12, 5539, 2310, 2178} \[ \frac {1}{6} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {(3-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{6} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {(b d n+3) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2310
Rule 5539
Rule 6555
Rubi steps
\begin {align*} \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} (b d n) \int \frac {x^2 \sinh \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\\ &=\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} (b n) \int \frac {x^2 \sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx\\ &=\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{6} \left (b e^{-a d} n x^{b d n} \left (c x^n\right )^{-b d}\right ) \int \frac {x^{2-b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac {1}{6} \left (b e^{a d} n x^{-b d n} \left (c x^n\right )^{b d}\right ) \int \frac {x^{2+b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{6} \left (b e^{-a d} x^3 \left (c x^n\right )^{-b d-\frac {3-b d n}{n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(3-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )-\frac {1}{6} \left (b e^{a d} x^3 \left (c x^n\right )^{b d-\frac {3+b d n}{n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(3+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )\\ &=\frac {1}{6} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {(3-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{6} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {(3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end {align*}
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Mathematica [A] time = 1.65, size = 98, normalized size = 0.77 \[ \frac {1}{6} x^3 \left (e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (\text {Ei}\left (-\frac {(b d n-3) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text {Ei}\left (\frac {(b d n+3) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {Shi}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{2} \Shi \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
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 \[ \int x^{2} {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
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 \[ \int x^2\,\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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 \[ \int x^{2} \operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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