Optimal. Leaf size=263 \[ -\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {5 b^9 \pi ^4 \text {CosIntegral}\left (b^2 \pi x^2\right )}{2016}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} b^8 \pi ^4 \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2},x\right ) \]
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Rubi [A]
time = 0.33, 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 {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx &=-\frac {b}{144 x^8}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}-\frac {1}{18} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^9} \, dx+\frac {1}{9} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^8} \, dx\\ &=-\frac {b}{144 x^8}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}-\frac {1}{36} b \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^5} \, dx,x,x^2\right )+\frac {1}{126} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac {1}{63} \left (b^4 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}+\frac {1}{252} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac {1}{144} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac {1}{630} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{315} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx\\ &=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {\left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )}{1260}+\frac {1}{756} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{432} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {\left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx}{1890}+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{3780}-\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{2520}-\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1512}-\frac {1}{864} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx-\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{3780}-\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{2520}-\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1512}-\frac {1}{864} \left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {5 b^9 \pi ^4 \text {Ci}\left (b^2 \pi x^2\right )}{2016}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{10}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [A]
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^{10}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^{10}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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