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

Optimal. Leaf size=54 \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac{\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]

[Out]

Cos[d*(a + b*Log[c*x^n])]/(b*d*n) + ((a + b*Log[c*x^n])*SinIntegral[d*(a + b*Log[c*x^n])])/(b*n)

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Rubi [A]  time = 0.0309975, antiderivative size = 54, 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 = {6499} \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac{\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[d*(a + b*Log[c*x^n])]/(b*d*n) + ((a + b*Log[c*x^n])*SinIntegral[d*(a + b*Log[c*x^n])])/(b*n)

Rule 6499

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

Rubi steps

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

Mathematica [A]  time = 0.0638797, size = 95, normalized size = 1.76 \[ \frac{\log \left (c x^n\right ) \text{Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n}+\frac{a \text{Si}\left (a d+b \log \left (c x^n\right ) d\right )}{b n}-\frac{\sin (a d) \sin \left (b d \log \left (c x^n\right )\right )}{b d n}+\frac{\cos (a d) \cos \left (b d \log \left (c x^n\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[a*d]*Cos[b*d*Log[c*x^n]])/(b*d*n) - (Sin[a*d]*Sin[b*d*Log[c*x^n]])/(b*d*n) + (Log[c*x^n]*SinIntegral[d*(a
+ b*Log[c*x^n])])/n + (a*SinIntegral[a*d + b*d*Log[c*x^n]])/(b*n)

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Maple [A]  time = 0.075, size = 72, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ){\it Si} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) }{n}}+{\frac{{\it Si} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) a}{bn}}+{\frac{\cos \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(Si(d*(a+b*ln(c*x^n)))/x,x)

[Out]

1/n*ln(c*x^n)*Si(a*d+b*d*ln(c*x^n))+1/n/b*Si(a*d+b*d*ln(c*x^n))*a+1/n/b/d*cos(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 Si}\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(Si(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")

[Out]

integrate(Si((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{Si}\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(Si(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")

[Out]

integral(sin_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{Si}{\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(Si(d*(a+b*ln(c*x**n)))/x,x)

[Out]

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

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Giac [A]  time = 1.57132, size = 80, normalized size = 1.48 \begin{align*} \frac{{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname{Si}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*d*n*log(x) + b*d*log(c) + a*d)*sin_integral(b*d*n*log(x) + b*d*log(c) + a*d) + cos(b*d*n*log(x) + b*d*log(
c) + a*d))/(b*d*n)