3.57 \(\int \frac{S(d (a+b \log (c x^n)))}{x} \, dx\)

Optimal. Leaf size=65 \[ \frac{\cos \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )\right )^2\right )}{\pi b d n}+\frac{\left (a+b \log \left (c x^n\right )\right ) S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

Cos[(d^2*Pi*(a + b*Log[c*x^n])^2)/2]/(b*d*n*Pi) + (FresnelS[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)

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Rubi [A]  time = 0.0411381, antiderivative size = 65, 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 = {6418} \[ \frac{\cos \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )\right )^2\right )}{\pi b d n}+\frac{\left (a+b \log \left (c x^n\right )\right ) S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cos[(d^2*Pi*(a + b*Log[c*x^n])^2)/2]/(b*d*n*Pi) + (FresnelS[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)

Rule 6418

Int[FresnelS[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*FresnelS[a + b*x])/b, x] + Simp[Cos[(Pi*(a + b*
x)^2)/2]/(b*Pi), x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int S(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int S(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n}\\ &=\frac{\cos \left (\frac{1}{2} \pi \left (a d+b d \log \left (c x^n\right )\right )^2\right )}{b d n \pi }+\frac{S\left (a d+b d \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}

Mathematica [B]  time = 0.0931173, size = 164, normalized size = 2.52 \[ -\frac{\sin \left (\frac{1}{2} \pi a^2 d^2\right ) \sin \left (\pi a b d^2 \log \left (c x^n\right )+\frac{1}{2} \pi b^2 d^2 \log ^2\left (c x^n\right )\right )}{\pi b d n}+\frac{\cos \left (\frac{1}{2} \pi a^2 d^2\right ) \cos \left (\pi a b d^2 \log \left (c x^n\right )+\frac{1}{2} \pi b^2 d^2 \log ^2\left (c x^n\right )\right )}{\pi b d n}+\frac{a S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac{\log \left (c x^n\right ) S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(a^2*d^2*Pi)/2]*Cos[a*b*d^2*Pi*Log[c*x^n] + (b^2*d^2*Pi*Log[c*x^n]^2)/2])/(b*d*n*Pi) + (a*FresnelS[d*(a +
b*Log[c*x^n])])/(b*n) + (FresnelS[d*(a + b*Log[c*x^n])]*Log[c*x^n])/n - (Sin[(a^2*d^2*Pi)/2]*Sin[a*b*d^2*Pi*L
og[c*x^n] + (b^2*d^2*Pi*Log[c*x^n]^2)/2])/(b*d*n*Pi)

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Maple [A]  time = 0.216, size = 80, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ){\it FresnelS} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) }{n}}+{\frac{{\it FresnelS} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) a}{bn}}+{\frac{1}{bdn\pi }\cos \left ({\frac{\pi \, \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*ln(c*x^n)*FresnelS(a*d+b*d*ln(c*x^n))+1/n/b*FresnelS(a*d+b*d*ln(c*x^n))*a+1/n/b/d/Pi*cos(1/2*Pi*(a*d+b*d*l
n(c*x^n))^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\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(fresnels(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(fresnels(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{S\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(fresnels(d*(a+b*ln(c*x**n)))/x,x)

[Out]

Integral(fresnels(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 fresnels}\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(fresnels(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")

[Out]

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