3.58 \(\int \frac{S(d (a+b \log (c x^n)))}{x^2} \, dx\)

Optimal. Leaf size=217 \[ \frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \left (c x^n\right )^{\frac{1}{n}} e^{\frac{2 a b n+\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )}{x}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \left (c x^n\right )^{\frac{1}{n}} e^{\frac{2 a b n-\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )}{x}-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.511188, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {6471, 4617, 2278, 2274, 15, 2276, 2234, 2204, 2205} \[ \frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \left (c x^n\right )^{\frac{1}{n}} e^{\frac{2 a b n+\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )}{x}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) \left (c x^n\right )^{\frac{1}{n}} e^{\frac{2 a b n-\frac{i}{\pi d^2}}{2 b^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}\right )}{\sqrt{\pi } b d}\right )}{x}-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6471

Int[FresnelS[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((e*x)^(m + 1)
*FresnelS[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[(e*x)^m*Sin[(Pi*(d*(a + b*Log[c*x
^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rule 4617

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

Rule 2278

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*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, e, m, 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 \frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b d n) \int \frac{\sin \left (\frac{1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )}{x^2} \, dx\\ &=-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{1}{2} (i b d n) \int \frac{e^{-\frac{1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx-\frac{1}{2} (i b d n) \int \frac{e^{\frac{1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx\\ &=-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{1}{2} (i b d n) \int \frac{\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 )}{x^2} \, dx-\frac{1}{2} (i b d n) \int \frac{\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 )}{x^2} \, dx\\ &=-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{1}{2} (i b d n) \int \frac{\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 }}{x^2} \, dx-\frac{1}{2} (i b d n) \int \frac{\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 }}{x^2} \, dx\\ &=-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\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^{-2-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^{-2+i a b d^2 n \pi } \, dx\\ &=-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{\left (i b d \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 )}{2 x}-\frac{\left (i b d \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 )}{2 x}\\ &=-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{\left (i b d e^{\frac{2 a b n-\frac{i}{d^2 \pi }}{2 b^2 n^2}} \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 )}{2 x}-\frac{\left (i b d e^{\frac{2 a b n+\frac{i}{d^2 \pi }}{2 b^2 n^2}} \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 )}{2 x}\\ &=\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) e^{\frac{2 a b n+\frac{i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{\frac{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 )}{x}+\frac{\left (\frac{1}{4}-\frac{i}{4}\right ) e^{\frac{2 a b n-\frac{i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{\frac{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}-\frac{S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}\\ \end{align*}

Mathematica [A]  time = 3.96764, size = 195, normalized size = 0.9 \[ -\frac{4 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt{2} \left (c x^n\right )^{\frac{1}{n}} e^{\frac{2 a b n-\frac{i}{\pi d^2}}{2 b^2 n^2}} \left (e^{\frac{i}{\pi b^2 d^2 n^2}} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+i\right )}{\sqrt{\pi } b d n}\right )+i \text{Erfi}\left (\frac{(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-i\right )}{\sqrt{2 \pi } b d n}\right )\right )}{4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.517, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(d*(a+b*ln(c*x^n)))/x^2,x)

[Out]

int(FresnelS(d*(a+b*ln(c*x^n)))/x^2,x)

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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^{2}}\,{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^2,x, algorithm="maxima")

[Out]

integrate(fresnels((b*log(c*x^n) + a)*d)/x^2, 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^{2}}, 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^2,x, algorithm="fricas")

[Out]

integral(fresnels(b*d*log(c*x^n) + a*d)/x^2, 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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(d*(a+b*ln(c*x**n)))/x**2,x)

[Out]

Integral(fresnels(a*d + b*d*log(c*x**n))/x**2, 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^{2}}\,{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^2,x, algorithm="giac")

[Out]

integrate(fresnels((b*log(c*x^n) + a)*d)/x^2, x)