Optimal. Leaf size=109 \[ -\frac {b}{12 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{12 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac {1}{6} b^3 \pi ^2 S(b x)^2-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{6} b^3 \pi \text {Si}\left (b^2 \pi x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6591, 6599,
6575, 30, 3456, 3461, 3378, 3380} \begin {gather*} -\frac {1}{6} \pi ^2 b^3 S(b x)^2-\frac {\pi b^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x}-\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}+\frac {b \cos \left (\pi b^2 x^2\right )}{12 x^2}+\frac {1}{6} \pi b^3 \text {Si}\left (b^2 \pi x^2\right )-\frac {b}{12 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 3378
Rule 3380
Rule 3456
Rule 3461
Rule 6575
Rule 6591
Rule 6599
Rubi steps
\begin {align*} \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx &=-\frac {b}{12 x^2}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {1}{6} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac {1}{3} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=-\frac {b}{12 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac {1}{12} b \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{6} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx-\frac {1}{3} \left (b^4 \pi ^2\right ) \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b}{12 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{12 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{12} b^3 \pi \text {Si}\left (b^2 \pi x^2\right )+\frac {1}{12} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{3} \left (b^3 \pi ^2\right ) \text {Subst}(\int x \, dx,x,S(b x))\\ &=-\frac {b}{12 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{12 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac {1}{6} b^3 \pi ^2 S(b x)^2-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{6} b^3 \pi \text {Si}\left (b^2 \pi x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 109, normalized size = 1.00 \begin {gather*} -\frac {b}{12 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{12 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac {1}{6} b^3 \pi ^2 S(b x)^2-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac {1}{6} b^3 \pi \text {Si}\left (b^2 \pi x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 98, normalized size = 0.90 \begin {gather*} -\frac {\pi ^{2} b^{3} x^{3} \operatorname {S}\left (b x\right )^{2} - \pi b^{3} x^{3} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) + 2 \, \pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + b x + 2 \, \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\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 [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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \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]
________________________________________________________________________________________