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*ln(c*x^n)))/b/d/n+(a+b*ln(c*x^n))*Si(d*(a+b*ln(c*x^n)))/b/n

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Rubi [A]  time = 0.03, antiderivative size = 54, 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 = {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*}

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Mathematica [A]  time = 0.08, 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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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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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giac [A]  time = 4.00, size = 59, normalized size = 1.09 \[ \frac {{\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )} \operatorname {Si}\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) + \cos \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )}{b d n} \]

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)

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maple [A]  time = 0.04, size = 72, normalized size = 1.33 \[ \frac {\ln \left (c \,x^{n}\right ) \Si \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n}+\frac {\Si \left (a d +b d \ln \left (c \,x^{n}\right )\right ) a}{n b}+\frac {\cos \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Si}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \frac {\mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}+\frac {\cos \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,d\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinint(d*(a + b*log(c*x^n)))/x,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Si}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

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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