Optimal. Leaf size=122 \[ \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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Rubi [A]
time = 0.17, antiderivative size = 122, 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 = {6690, 12, 5650,
2347, 2209} \begin {gather*} \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {(1-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 {(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2347
Rule 5650
Rule 6690
Rubi steps
\begin {align*} \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b d n) \int \frac {\sinh \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 {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b n) \int \frac {\sinh \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 {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}-\frac {1}{2} \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}{2} \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 {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}-\frac {\left (b e^{-a d} \left (c x^n\right )^{-b d-\frac {-1-b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-1-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 x}+\frac {\left (b e^{a d} \left (c x^n\right )^{b d-\frac {-1+b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(-1+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 x}\\ &=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {(1-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+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 x}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 1.10, size = 146, normalized size = 1.20 \begin {gather*} \frac {1}{2} e^{-\frac {(-1+b d n) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (\text {Ei}\left (\frac {(-1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text {Ei}\left (-\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \left (\cosh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\sinh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\hyperbolicSineIntegral \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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