3.56 \(\int S(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=214 \[ \left (\frac{1}{4}-\frac{i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac{2 a b n-\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )+\left (\frac{1}{4}-\frac{i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac{2 a b n+\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )+x S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[Out]

((1/4 - I/4)*x*Erf[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b
*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) + ((1/4 - I/4)*x*Erfi[((1/2 + I/2)*(n^(-1) - I*a*b*d^2*Pi - I*b^
2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b*n + I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) + x*FresnelS[d*
(a + b*Log[c*x^n])]

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Rubi [A]  time = 0.326252, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {6468, 4615, 2277, 2274, 15, 2276, 2234, 2204, 2205} \[ \left (\frac{1}{4}-\frac{i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac{2 a b n-\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )+\left (\frac{1}{4}-\frac{i}{4}\right ) x \left (c x^n\right )^{-1/n} e^{-\frac{2 a b n+\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )+x S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[FresnelS[d*(a + b*Log[c*x^n])],x]

[Out]

((1/4 - I/4)*x*Erf[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b
*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) + ((1/4 - I/4)*x*Erfi[((1/2 + I/2)*(n^(-1) - I*a*b*d^2*Pi - I*b^
2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b*n + I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) + x*FresnelS[d*
(a + b*Log[c*x^n])]

Rule 6468

Int[FresnelS[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[x*FresnelS[d*(a + b*Log[c*x^n])],
x] - Dist[b*d*n, Int[Sin[(Pi*(d*(a + b*Log[c*x^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 4615

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)], x_Symbol] :> Dist[I/2, Int[E^(-(I*d*(a + b*Log[c*x^n])
^2)), x], x] - Dist[I/2, Int[E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 2277

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)), x_Symbol] :> Int[F^(a^2*d + 2*a*b*d*Log[c*x^n] + b^2
*d*Log[c*x^n]^2), x] /; FreeQ[{F, a, b, c, d, n}, x]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2276

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]^2*(b_.))*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)/(
e*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(a*d*Log[F] + ((m + 1)*x)/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /;
FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-(b d n) \int \sin \left (\frac{1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} (i b d n) \int e^{-\frac{1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} \, dx+\frac{1}{2} (i b d n) \int e^{\frac{1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} \, dx\\ &=x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} (i b d n) \int \exp \left (-\frac{1}{2} i a^2 d^2 \pi -i a b d^2 \pi \log \left (c x^n\right )-\frac{1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) \, dx+\frac{1}{2} (i b d n) \int \exp \left (\frac{1}{2} i a^2 d^2 \pi +i a b d^2 \pi \log \left (c x^n\right )+\frac{1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) \, dx\\ &=x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} (i b d n) \int \exp \left (-\frac{1}{2} i a^2 d^2 \pi -\frac{1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) \left (c x^n\right )^{-i a b d^2 \pi } \, dx+\frac{1}{2} (i b d n) \int \exp \left (\frac{1}{2} i a^2 d^2 \pi +\frac{1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) \left (c x^n\right )^{i a b d^2 \pi } \, dx\\ &=x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} \left (i b d n x^{i a b d^2 n \pi } \left (c x^n\right )^{-i a b d^2 \pi }\right ) \int \exp \left (-\frac{1}{2} i a^2 d^2 \pi -\frac{1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{-i a b d^2 n \pi } \, dx+\frac{1}{2} \left (i b d n x^{-i a b d^2 n \pi } \left (c x^n\right )^{i a b d^2 \pi }\right ) \int \exp \left (\frac{1}{2} i a^2 d^2 \pi +\frac{1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{i a b d^2 n \pi } \, dx\\ &=x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} \left (i b d x \left (c x^n\right )^{-i a b d^2 \pi -\frac{1-i a b d^2 n \pi }{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-\frac{1}{2} i a^2 d^2 \pi +\frac{\left (1-i a b d^2 n \pi \right ) x}{n}-\frac{1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )+\frac{1}{2} \left (i b d x \left (c x^n\right )^{i a b d^2 \pi -\frac{1+i a b d^2 n \pi }{n}}\right ) \operatorname{Subst}\left (\int \exp \left (\frac{1}{2} i a^2 d^2 \pi +\frac{\left (1+i a b d^2 n \pi \right ) x}{n}+\frac{1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )\\ &=x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{2} \left (i b d e^{-\frac{2 a b n+\frac{i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{-i a b d^2 \pi -\frac{1-i a b d^2 n \pi }{n}}\right ) \operatorname{Subst}\left (\int \exp \left (\frac{i \left (\frac{1-i a b d^2 n \pi }{n}-i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )+\frac{1}{2} \left (i b d e^{-\frac{2 a b n-\frac{i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{i a b d^2 \pi -\frac{1+i a b d^2 n \pi }{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-\frac{i \left (\frac{1+i a b d^2 n \pi }{n}+i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )\\ &=\left (\frac{1}{4}-\frac{i}{4}\right ) e^{-\frac{2 a b n-\frac{i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{-1/n} \text{erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\frac{1}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt{\pi }}\right )+\left (\frac{1}{4}-\frac{i}{4}\right ) e^{-\frac{2 a b n+\frac{i}{d^2 \pi }}{2 b^2 n^2}} x \left (c x^n\right )^{-1/n} \text{erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\frac{1}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt{\pi }}\right )+x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end{align*}

Mathematica [A]  time = 6.65872, size = 316, normalized size = 1.48 \[ x S\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{\sqrt [4]{-1} x \left (c x^n\right )^{-1/n} \left (e^{\frac{i}{\pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-i\right )}{\sqrt{\pi } b d n}\right )+i \text{Erfi}\left (\frac{(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+i\right )}{\sqrt{2 \pi } b d n}\right )\right ) \exp \left (\frac{1}{2} \left (-i \pi a^2 d^2+2 i \pi a b d^2 \left (n \log (x)-\log \left (c x^n\right )\right )-\frac{2 a}{b n}-i \pi b^2 d^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2-\frac{i}{\pi b^2 d^2 n^2}\right )\right ) \left (\cos \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )+i \sin \left (\frac{1}{2} \pi d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2\right )\right )}{2 \sqrt{2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

x*FresnelS[d*(a + b*Log[c*x^n])] + ((-1)^(1/4)*E^(((-2*a)/(b*n) - I/(b^2*d^2*n^2*Pi) - I*a^2*d^2*Pi + (2*I)*a*
b*d^2*Pi*(n*Log[x] - Log[c*x^n]) - I*b^2*d^2*Pi*(-(n*Log[x]) + Log[c*x^n])^2)/2)*x*(E^(I/(b^2*d^2*n^2*Pi))*Erf
i[((1/2 + I/2)*(-I + a*b*d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])] + I*Erfi[((-1)^(3/4)*(I + a*b*
d^2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[2*Pi])])*(Cos[(d^2*Pi*(a - b*n*Log[x] + b*Log[c*x^n])^2)/2] +
I*Sin[(d^2*Pi*(a - b*n*Log[x] + b*Log[c*x^n])^2)/2]))/(2*Sqrt[2]*(c*x^n)^n^(-1))

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Maple [F]  time = 0.541, size = 0, normalized size = 0. \begin{align*} \int{\it FresnelS} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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