3.1.54 \(\int x^2 S(d (a+b \log (c x^n))) \, dx\) [54]

Optimal. Leaf size=231 \[ \left (\frac {1}{12}-\frac {i}{12}\right ) e^{-\frac {3 a}{b n}+\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )+\left (\frac {1}{12}-\frac {i}{12}\right ) e^{-\frac {3 a}{b n}-\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )+\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[Out]

(1/12-1/12*I)*exp(-3*a/b/n+9/2*I/b^2/d^2/n^2/Pi)*x^3*erf((1/2+1/2*I)*(3/n+I*a*b*d^2*Pi+I*b^2*d^2*Pi*ln(c*x^n))
/b/d/Pi^(1/2))/((c*x^n)^(3/n))+(1/12-1/12*I)*exp(-3*a/b/n-9/2*I/b^2/d^2/n^2/Pi)*x^3*erfi((1/2+1/2*I)*(3/n-I*a*
b*d^2*Pi-I*b^2*d^2*Pi*ln(c*x^n))/b/d/Pi^(1/2))/((c*x^n)^(3/n))+1/3*x^3*FresnelS(d*(a+b*ln(c*x^n)))

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Rubi [A]
time = 0.28, antiderivative size = 231, 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*} \left (\frac {1}{12}-\frac {i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}+\frac {9 i}{2 \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 {3}{n}\right )}{\sqrt {\pi } b d}\right )+\left (\frac {1}{12}-\frac {i}{12}\right ) x^3 \left (c x^n\right )^{-3/n} e^{-\frac {3 a}{b n}-\frac {9 i}{2 \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 {3}{n}\right )}{\sqrt {\pi } b d}\right )+\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1/12 - I/12)*E^((-3*a)/(b*n) + ((9*I)/2)/(b^2*d^2*n^2*Pi))*x^3*Erf[((1/2 + I/2)*(3/n + I*a*b*d^2*Pi + I*b^2*
d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(3/n) + ((1/12 - I/12)*E^((-3*a)/(b*n) - ((9*I)/2)/(b^2*d^2*n^2*P
i))*x^3*Erfi[((1/2 + I/2)*(3/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(3/n) + (x^
3*FresnelS[d*(a + b*Log[c*x^n])])/3

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 x^2 S\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} (b d n) \int x^2 \sin \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} (i b d n) \int e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} x^2 \, dx+\frac {1}{6} (i b d n) \int e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2} x^2 \, dx\\ &=\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} (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 ) x^2 \, dx+\frac {1}{6} (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 ) x^2 \, dx\\ &=\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} (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 ) x^2 \left (c x^n\right )^{-i a b d^2 \pi } \, dx+\frac {1}{6} (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 ) x^2 \left (c x^n\right )^{i a b d^2 \pi } \, dx\\ &=\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} \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}{6} \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}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} \left (i b d x^3 \left (c x^n\right )^{-i a b d^2 \pi -\frac {3-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 (3-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}{6} \left (i b d x^3 \left (c x^n\right )^{i a b d^2 \pi -\frac {3+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 (3+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}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} \left (i b d e^{-\frac {3 a}{b n}-\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-i a b d^2 \pi -\frac {3-i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {i \left (\frac {3-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}{6} \left (i b d e^{-\frac {3 a}{b n}+\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{i a b d^2 \pi -\frac {3+i a b d^2 n \pi }{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {i \left (\frac {3+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}{12}-\frac {i}{12}\right ) e^{-\frac {3 a}{b n}+\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{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}{12}-\frac {i}{12}\right ) e^{-\frac {3 a}{b n}-\frac {9 i}{2 b^2 d^2 n^2 \pi }} x^3 \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {3}{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 )+\frac {1}{3} x^3 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end {align*}

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Mathematica [A]
time = 4.49, size = 319, normalized size = 1.38 \begin {gather*} \frac {1}{12} x^3 \left (4 S\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt {2} e^{\frac {1}{2} \left (-\frac {6 a}{b n}-\frac {9 i}{b^2 d^2 n^2 \pi }-i a^2 d^2 \pi +2 i a b d^2 \pi \left (n \log (x)-\log \left (c x^n\right )\right )-i b^2 d^2 \pi \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )} \left (c x^n\right )^{-3/n} \left (e^{\frac {9 i}{b^2 d^2 n^2 \pi }} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-3 i+a b d^2 n \pi +b^2 d^2 n \pi \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }}\right )+i \text {Erfi}\left (\frac {(-1)^{3/4} \left (3 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 ) \left (\cos \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )+i \sin \left (\frac {1}{2} d^2 \pi \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(4*FresnelS[d*(a + b*Log[c*x^n])] + ((-1)^(1/4)*Sqrt[2]*E^(((-6*a)/(b*n) - (9*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)*(E^((9*I)/(
b^2*d^2*n^2*Pi))*Erfi[((1/2 + I/2)*(-3*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)*(3*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]))/(c*x^n)^(3/n)))/12

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Maple [F]
time = 0.48, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {S}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*FresnelS(d*(a+b*ln(c*x^n))),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(x^2*fresnel_sin(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

integrate(x^2*fresnel_sin((b*log(c*x^n) + a)*d), 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. 448 vs. \(2 (187) = 374\).
time = 0.38, size = 448, normalized size = 1.94 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {S}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \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 + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{6} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \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 - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} - \frac {9 i}{2 \, \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 + 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{6} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {3 \, \log \left (c\right )}{n} - \frac {3 \, a}{b n} + \frac {9 i}{2 \, \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 - 3 i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*fresnel_sin(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

1/3*x^3*fresnel_sin(b*d*log(c*x^n) + a*d) - 1/6*I*pi*sqrt(b^2*d^2*n^2)*e^(-3*log(c)/n - 3*a/(b*n) - 9/2*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 + 3*I)*sqrt(b^2*d^2*n^2)
/(pi*b^2*d^2*n^2)) + 1/6*I*pi*sqrt(b^2*d^2*n^2)*e^(-3*log(c)/n - 3*a/(b*n) + 9/2*I/(pi*b^2*d^2*n^2))*fresnel_c
os((pi*b^2*d^2*n^2*log(x) + pi*b^2*d^2*n*log(c) + pi*a*b*d^2*n - 3*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 1/
6*pi*sqrt(b^2*d^2*n^2)*e^(-3*log(c)/n - 3*a/(b*n) - 9/2*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 + 3*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2)) - 1/6*pi*sqrt(b^2*d^2*n^2)*e^
(-3*log(c)/n - 3*a/(b*n) + 9/2*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 - 3*I)*sqrt(b^2*d^2*n^2)/(pi*b^2*d^2*n^2))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} S\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\mathrm {FresnelS}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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