3.1.59 \(\int \frac {S(d (a+b \log (c x^n)))}{x^3} \, dx\) [59]

Optimal. Leaf size=228 \[ \frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{\frac {2 i+2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{-\frac {2 \left (i-a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x^2}-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

[Out]

(1/8-1/8*I)*exp((2*I+2*a*b*d^2*n*Pi)/b^2/d^2/n^2/Pi)*(c*x^n)^(2/n)*erf((1/2+1/2*I)*(2/n-I*a*b*d^2*Pi-I*b^2*d^2
*Pi*ln(c*x^n))/b/d/Pi^(1/2))/x^2+(1/8-1/8*I)*(c*x^n)^(2/n)*erfi((1/2+1/2*I)*(2/n+I*a*b*d^2*Pi+I*b^2*d^2*Pi*ln(
c*x^n))/b/d/Pi^(1/2))/exp(2*(I-a*b*d^2*n*Pi)/b^2/d^2/n^2/Pi)/x^2-1/2*FresnelS(d*(a+b*ln(c*x^n)))/x^2

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Rubi [A]
time = 0.26, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6606, 4713, 2314, 2308, 2266, 2235, 2236} \begin {gather*} \frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{\frac {2 \pi a b d^2 n+2 i}{\pi b^2 d^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 {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (c x^n\right )^{2/n} e^{-\frac {2 \left (-\pi a b d^2 n+i\right )}{\pi b^2 d^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 {2}{n}\right )}{\sqrt {\pi } b d}\right )}{x^2}-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[FresnelS[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

((1/8 - I/8)*E^((2*I + 2*a*b*d^2*n*Pi)/(b^2*d^2*n^2*Pi))*(c*x^n)^(2/n)*Erf[((1/2 + I/2)*(2/n - I*a*b*d^2*Pi -
I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x^2 + ((1/8 - I/8)*(c*x^n)^(2/n)*Erfi[((1/2 + I/2)*(2/n + I*a*b*d^2
*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*(I - a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*x^2) - FresnelS[
d*(a + b*Log[c*x^n])]/(2*x^2)

Rule 2235

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 2236

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]

Rule 2266

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 2308

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^(m + 1)/(h*n*(c*(d + e*x)^n)^((m + 1)/n)), Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Lo
g[F]*x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]

Rule 2314

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.))^2*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^m*((c*(d + e*x)^n)^(2*a*b*f*Log[F])/(d + e*x)^(m + 2*a*b*f*n*Log[F])), Int[(d + e*x)^(m + 2
*a*b*f*n*Log[F])*F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x]
&& EqQ[e*g - d*h, 0]

Rule 4713

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 6606

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/2)*(d*(a + b*Log[
c*x^n]))^2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{2} (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^3} \, dx\\ &=-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} (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^3} \, dx-\frac {1}{4} (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^3} \, dx\\ &=-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} (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^3} \, dx-\frac {1}{4} (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^3} \, dx\\ &=-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} (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^3} \, dx-\frac {1}{4} (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^3} \, dx\\ &=-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {1}{4} \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^{-3-i a b d^2 n \pi } \, dx-\frac {1}{4} \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^{-3+i a b d^2 n \pi } \, dx\\ &=-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (i b d \left (c x^n\right )^{-i a b d^2 \pi -\frac {-2-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {1}{2} i a^2 d^2 \pi +\frac {\left (-2-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 )}{4 x^2}-\frac {\left (i b d \left (c x^n\right )^{i a b d^2 \pi -\frac {-2+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {\left (-2+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 )}{4 x^2}\\ &=-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (i b d e^{-\frac {2 \left (i-a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{-i a b d^2 \pi -\frac {-2-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {i \left (\frac {-2-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 )}{4 x^2}-\frac {\left (i b d e^{\frac {2 i+2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{i a b d^2 \pi -\frac {-2+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {i \left (\frac {-2+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 )}{4 x^2}\\ &=\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{\frac {2 i+2 a b d^2 n \pi }{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{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^2}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) e^{-\frac {2 \left (i-a b d^2 n \pi \right )}{b^2 d^2 n^2 \pi }} \left (c x^n\right )^{2/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2}{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^2}-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\\ \end {align*}

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Mathematica [A]
time = 2.59, size = 200, normalized size = 0.88 \begin {gather*} -\frac {\sqrt [4]{-1} e^{\frac {2 \left (\frac {a n}{b}-\frac {i}{b^2 d^2 \pi }+n \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{n^2}} \left (i \text {Erfi}\left (\frac {(-1)^{3/4} \left (-2 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )+e^{\frac {4 i}{b^2 d^2 n^2 \pi }} \text {Erfi}\left (\frac {\sqrt [4]{-1} \left (2 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {2 \pi }}\right )\right )}{4 \sqrt {2}}-\frac {S\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[FresnelS[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

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

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Maple [F]
time = 0.41, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {S}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(d*(a+b*ln(c*x^n)))/x^3,x)

[Out]

int(FresnelS(d*(a+b*ln(c*x^n)))/x^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")

[Out]

integrate(fresnel_sin((b*log(c*x^n) + a)*d)/x^3, x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (183) = 366\).
time = 0.39, size = 460, normalized size = 2.02 \begin {gather*} \frac {-i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + i \, \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \pi \sqrt {b^{2} d^{2} n^{2}} x^{2} e^{\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} - \frac {2 i}{\pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - 2 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - 2 \, \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")

[Out]

1/4*(-I*pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) + 2*I/(pi*b^2*d^2*n^2))*fresnel_cos((pi*b^2*d^2*n^2
*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n + 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + I*pi*sqrt(b^2*d^2*n^
2)*x^2*e^(2*log(c)/n + 2*a/(b*n) - 2*I/(pi*b^2*d^2*n^2))*fresnel_cos((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log
(c) + pi*a*b*d^2*n - 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(
b*n) + 2*I/(pi*b^2*d^2*n^2))*fresnel_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n + 2*I)*sq
rt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) + pi*sqrt(b^2*d^2*n^2)*x^2*e^(2*log(c)/n + 2*a/(b*n) - 2*I/(pi*b^2*d^2*n^2))
*fresnel_sin((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - 2*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*
n^2)) - 2*fresnel_sin(b*d*log(c*x^n) + a*d))/x^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {S\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(d*(a+b*ln(c*x**n)))/x**3,x)

[Out]

Integral(fresnels(a*d + b*d*log(c*x**n))/x**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnel_sin(d*(a+b*log(c*x^n)))/x^3,x, algorithm="giac")

[Out]

integrate(fresnel_sin((b*log(c*x^n) + a)*d)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(d*(a + b*log(c*x^n)))/x^3,x)

[Out]

int(FresnelS(d*(a + b*log(c*x^n)))/x^3, x)

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