Optimal. Leaf size=263 \[ -\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {5 b^9 \pi ^4 \text {CosIntegral}\left (b^2 \pi x^2\right )}{2016}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} b^8 \pi ^4 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^2},x\right ) \]
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Rubi [A]
time = 0.34, 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 ) \text {FresnelC}(b x)}{x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^{10}} \, dx &=-\frac {b}{144 x^8}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{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 {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx\\ &=-\frac {b}{144 x^8}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{9 x^9}+\frac {b^2 \pi C(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 {\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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{315 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\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 {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{315 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{315 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{315 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{315 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.23, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{x^{10}}\, dx\]
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 {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\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 {FresnelC}\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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