Optimal. Leaf size=89 \[ \frac {1}{12} b^3 \pi \text {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}-\frac {1}{3} b^2 \pi \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}+\frac {1}{6} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{3} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}+\frac {1}{12} b \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{3} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}-\frac {1}{3} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{12} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=\frac {1}{12} b^3 \pi \text {Ci}\left (b^2 \pi x^2\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{12 x^2}-\frac {1}{3} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
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]
[Out]
________________________________________________________________________________________
Fricas [A]
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]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
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]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________