\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2)}{\operatorname {FresnelC}(b x)^3} \, dx\) [178]

Optimal result

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

[Out]

-1/2/b/FresnelC(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 = {6576, 30} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelC}(b x)^3} \, dx=-\frac {1}{2 b \operatorname {FresnelC}(b x)^2} \]

[In]

Int[Cos[(b^2*Pi*x^2)/2]/FresnelC[b*x]^3,x]

[Out]

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

Rule 30

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

Rule 6576

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[Pi*(b/(2*d)), Subst[Int[x^n, x], x, Fresne
lC[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 {FresnelC}(b x)\right )}{b} \\ & = -\frac {1}{2 b \operatorname {FresnelC}(b x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelC}(b x)^3} \, dx=-\frac {1}{2 b \operatorname {FresnelC}(b x)^2} \]

[In]

Integrate[Cos[(b^2*Pi*x^2)/2]/FresnelC[b*x]^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelC}(b x)^3} \, dx=-\frac {1}{2 \, b \operatorname {C}\left (b x\right )^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelC}(b x)^3} \, dx=\begin {cases} - \frac {1}{2 b C^{2}\left (b x\right )} & \text {for}\: b \neq 0 \\\tilde {\infty } x & \text {otherwise} \end {cases} \]

[In]

integrate(cos(1/2*b**2*pi*x**2)/fresnelc(b*x)**3,x)

[Out]

Piecewise((-1/(2*b*fresnelc(b*x)**2), Ne(b, 0)), (zoo*x, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelC}(b x)^3} \, dx=-\frac {1}{2 \, b \operatorname {C}\left (b x\right )^{2}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{\operatorname {FresnelC}(b x)^3} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{\operatorname {C}\left (b x\right )^{3}} \,d x } \]

[In]

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

[Out]

integrate(cos(1/2*pi*b^2*x^2)/fresnel_cos(b*x)^3, x)

Mupad [F(-1)]

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

[In]

int(cos((Pi*b^2*x^2)/2)/FresnelC(b*x)^3,x)

[Out]

int(cos((Pi*b^2*x^2)/2)/FresnelC(b*x)^3, x)