Optimal. Leaf size=278 \[ \frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac {b^8 \pi ^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}-\frac {b^9 \pi ^5 S(b x)^2}{1890}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {83 b^9 \pi ^4 \text {Si}\left (b^2 \pi x^2\right )}{30240} \]
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Rubi [A]
time = 0.34, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6599, 6591,
6575, 30, 3456, 3461, 3378, 3380, 3460} \begin {gather*} -\frac {\pi ^5 b^9 S(b x)^2}{1890}-\frac {\pi ^3 b^7}{3780 x^2}+\frac {\pi b^3}{756 x^6}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{63 x^7}-\frac {b \sin \left (\pi b^2 x^2\right )}{144 x^8}+\frac {83 \pi ^4 b^9 \text {Si}\left (b^2 \pi x^2\right )}{30240}-\frac {\pi ^4 b^8 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{945 x}+\frac {5 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{2016 x^2}-\frac {\pi ^3 b^6 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{945 x^3}+\frac {67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{30240 x^4}+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac {11 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3024 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3378
Rule 3380
Rule 3456
Rule 3460
Rule 3461
Rule 6575
Rule 6591
Rule 6599
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^{10}} \, dx &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {1}{18} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^9} \, dx-\frac {1}{9} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx\\ &=\frac {b^3 \pi }{756 x^6}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}+\frac {1}{36} b \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^5} \, dx,x,x^2\right )+\frac {1}{126} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac {1}{63} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^6} \, dx\\ &=\frac {b^3 \pi }{756 x^6}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {1}{252} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \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 {\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac {1}{630} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx+\frac {1}{315} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=\frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}-\frac {\left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \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 {\sin \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 {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {\left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx}{1890}+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=\frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac {b^8 \pi ^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\cos \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 {\cos \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 {\cos \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 {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {\left (b^9 \pi ^4\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx}{1890}-\frac {1}{945} \left (b^{10} \pi ^5\right ) \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac {b^8 \pi ^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {b^9 \pi ^4 \text {Si}\left (b^2 \pi x^2\right )}{3780}+\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\sin \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 {\sin \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 {\sin \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 {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{945} \left (b^9 \pi ^5\right ) \text {Subst}(\int x \, dx,x,S(b x))\\ &=\frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac {b^8 \pi ^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}-\frac {b^9 \pi ^5 S(b x)^2}{1890}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {83 b^9 \pi ^4 \text {Si}\left (b^2 \pi x^2\right )}{30240}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 278, normalized size = 1.00 \begin {gather*} \frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{315 x^5}-\frac {b^8 \pi ^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{945 x}-\frac {b^9 \pi ^5 S(b x)^2}{1890}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {83 b^9 \pi ^4 \text {Si}\left (b^2 \pi x^2\right )}{30240} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{10}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.37, size = 203, normalized size = 0.73 \begin {gather*} -\frac {16 \, \pi ^{5} b^{9} x^{9} \operatorname {S}\left (b x\right )^{2} - 83 \, \pi ^{4} b^{9} x^{9} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) + 83 \, \pi ^{3} b^{7} x^{7} - 150 \, \pi b^{3} x^{3} - 10 \, {\left (15 \, \pi ^{3} b^{7} x^{7} - 22 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 32 \, {\left (\pi ^{4} b^{8} x^{8} - 3 \, \pi ^{2} b^{4} x^{4} + 105\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 2 \, {\left ({\left (67 \, \pi ^{2} b^{5} x^{5} - 210 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 16 \, {\left (\pi ^{3} b^{6} x^{6} - 15 \, \pi b^{2} x^{2}\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{30240 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
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^{10}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \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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