Optimal. Leaf size=55 \[ \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6528} \[ \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 6528
Rubi steps
\begin {align*} \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {Shi}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \text {Shi}(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n}\\ &=-\frac {\cosh \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (a d+b d \log \left (c x^n\right )\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 96, normalized size = 1.75 \[ \frac {\log \left (c x^n\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n}+\frac {a \text {Shi}\left (a d+b \log \left (c x^n\right ) d\right )}{b n}-\frac {\sinh (a d) \sinh \left (b d \log \left (c x^n\right )\right )}{b d n}-\frac {\cosh (a d) \cosh \left (b d \log \left (c x^n\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Shi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x}, x\right ) \]
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 \[ \int \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 73, normalized size = 1.33 \[ \frac {\ln \left (c \,x^{n}\right ) \Shi \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n}+\frac {\Shi \left (a d +b d \ln \left (c \,x^{n}\right )\right ) a}{n b}-\frac {\cosh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d} \]
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 \[ \int \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{a\,d}\,{\left (c\,x^n\right )}^{b\,d}}{2\,b\,d\,n}-\frac {{\mathrm {e}}^{-a\,d}}{2\,b\,d\,n\,{\left (c\,x^n\right )}^{b\,d}} \]
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 \[ \int \frac {\operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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