3.103 \(\int \frac{\text{Chi}(d (a+b \log (c x^n)))}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]

[Out]

(CoshIntegral[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n) - Sinh[d*(a + b*Log[c*x^n])]/(b*d*n)

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Rubi [A]  time = 0.0328463, antiderivative size = 55, normalized size of antiderivative = 1., 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 = {6529} \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Int[CoshIntegral[d*(a + b*Log[c*x^n])]/x,x]

[Out]

(CoshIntegral[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n) - Sinh[d*(a + b*Log[c*x^n])]/(b*d*n)

Rule 6529

Int[CoshIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*CoshIntegral[a + b*x])/b, x] - Simp[Sinh[a
+ b*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\text{Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{Chi}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \text{Chi}(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n}\\ &=\frac{\text{Chi}\left (a d+b d \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\sinh \left (a d+b d \log \left (c x^n\right )\right )}{b d n}\\ \end{align*}

Mathematica [A]  time = 0.0782421, size = 96, normalized size = 1.75 \[ \frac{a \text{Chi}\left (a d+b \log \left (c x^n\right ) d\right )}{b n}+\frac{\log \left (c x^n\right ) \text{Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n}-\frac{\sinh (a d) \cosh \left (b d \log \left (c x^n\right )\right )}{b d n}-\frac{\cosh (a d) \sinh \left (b d \log \left (c x^n\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Integrate[CoshIntegral[d*(a + b*Log[c*x^n])]/x,x]

[Out]

(a*CoshIntegral[a*d + b*d*Log[c*x^n]])/(b*n) + (CoshIntegral[d*(a + b*Log[c*x^n])]*Log[c*x^n])/n - (Cosh[b*d*L
og[c*x^n]]*Sinh[a*d])/(b*d*n) - (Cosh[a*d]*Sinh[b*d*Log[c*x^n]])/(b*d*n)

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Maple [A]  time = 0.079, size = 73, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ){\it Chi} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) }{n}}+{\frac{{\it Chi} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) a}{bn}}-{\frac{\sinh \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) }{bdn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Chi(d*(a+b*ln(c*x^n)))/x,x)

[Out]

1/n*ln(c*x^n)*Chi(a*d+b*d*ln(c*x^n))+1/n/b*Chi(a*d+b*d*ln(c*x^n))*a-1/n/b/d*sinh(a*d+b*d*ln(c*x^n))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")

[Out]

integrate(Chi((b*log(c*x^n) + a)*d)/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{Chi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")

[Out]

integral(cosh_integral(b*d*log(c*x^n) + a*d)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Chi}\left (a d + b d \log{\left (c x^{n} \right )}\right )}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(d*(a+b*ln(c*x**n)))/x,x)

[Out]

Integral(Chi(a*d + b*d*log(c*x**n))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")

[Out]

integrate(Chi((b*log(c*x^n) + a)*d)/x, x)