\(\int \frac {\sin (\frac {1}{2} b^2 \pi x^2)}{\operatorname {FresnelS}(b x)^3} \, dx\) [69]

Optimal result

Integrand size = 19, antiderivative size = 13 \[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=-\frac {1}{2 b \operatorname {FresnelS}(b x)^2} \]

[Out]

-1/2/b/FresnelS(b*x)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6575, 30} \[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=-\frac {1}{2 b \operatorname {FresnelS}(b x)^2} \]

[In]

Int[Sin[(b^2*Pi*x^2)/2]/FresnelS[b*x]^3,x]

[Out]

-1/2*1/(b*FresnelS[b*x]^2)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6575

Int[FresnelS[(b_.)*(x_)]^(n_.)*Sin[(d_.)*(x_)^2], x_Symbol] :> Dist[Pi*(b/(2*d)), Subst[Int[x^n, x], x, Fresne
lS[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2/4)*b^4]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\operatorname {FresnelS}(b x)\right )}{b} \\ & = -\frac {1}{2 b \operatorname {FresnelS}(b x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=-\frac {1}{2 b \operatorname {FresnelS}(b x)^2} \]

[In]

Integrate[Sin[(b^2*Pi*x^2)/2]/FresnelS[b*x]^3,x]

[Out]

-1/2*1/(b*FresnelS[b*x]^2)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(-\frac {1}{2 b \operatorname {FresnelS}\left (b x \right )^{2}}\)	\(12\)
default	\(-\frac {1}{2 b \operatorname {FresnelS}\left (b x \right )^{2}}\)	\(12\)

[In]

int(sin(1/2*b^2*Pi*x^2)/FresnelS(b*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2/b/FresnelS(b*x)^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=-\frac {1}{2 \, b \operatorname {S}\left (b x\right )^{2}} \]

[In]

integrate(sin(1/2*b^2*pi*x^2)/fresnel_sin(b*x)^3,x, algorithm="fricas")

[Out]

-1/2/(b*fresnel_sin(b*x)^2)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=\begin {cases} - \frac {1}{2 b S^{2}\left (b x\right )} & \text {for}\: b \neq 0 \\\text {NaN} & \text {otherwise} \end {cases} \]

[In]

integrate(sin(1/2*b**2*pi*x**2)/fresnels(b*x)**3,x)

[Out]

Piecewise((-1/(2*b*fresnels(b*x)**2), Ne(b, 0)), (nan, True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=-\frac {1}{2 \, b \operatorname {S}\left (b x\right )^{2}} \]

[In]

integrate(sin(1/2*b^2*pi*x^2)/fresnel_sin(b*x)^3,x, algorithm="maxima")

[Out]

-1/2/(b*fresnel_sin(b*x)^2)

Giac [F]

\[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=\int { \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{\operatorname {S}\left (b x\right )^{3}} \,d x } \]

[In]

integrate(sin(1/2*b^2*pi*x^2)/fresnel_sin(b*x)^3,x, algorithm="giac")

[Out]

integrate(sin(1/2*pi*b^2*x^2)/fresnel_sin(b*x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelS}(b x)^3} \, dx=\int \frac {\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{{\mathrm {FresnelS}\left (b\,x\right )}^3} \,d x \]

[In]

int(sin((Pi*b^2*x^2)/2)/FresnelS(b*x)^3,x)

[Out]

int(sin((Pi*b^2*x^2)/2)/FresnelS(b*x)^3, x)