Optimal. Leaf size=242 \[ -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{420 x^3}+\frac {1}{840} b^8 \pi ^4 \text {FresnelC}(b x)^2-\frac {\text {FresnelC}(b x)^2}{8 x^8}+\frac {b^3 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]
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Rubi [A]
time = 0.27, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6566, 6592,
6600, 6576, 30, 3456, 3461, 3378, 3380, 3460} \begin {gather*} \frac {1}{840} \pi ^4 b^8 \text {FresnelC}(b x)^2+\frac {\pi ^2 b^6}{1680 x^2}-\frac {b \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{28 x^7}-\frac {b^2}{336 x^6}-\frac {b^2 \cos \left (\pi b^2 x^2\right )}{336 x^6}+\frac {1}{280} \pi ^3 b^8 \text {Si}\left (b^2 \pi x^2\right )-\frac {\pi ^3 b^7 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{420 x}+\frac {\pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{336 x^2}+\frac {\pi ^2 b^5 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{420 x^3}+\frac {\pi b^4 \sin \left (\pi b^2 x^2\right )}{420 x^4}+\frac {\pi b^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{140 x^5}-\frac {\text {FresnelC}(b x)^2}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3378
Rule 3380
Rule 3456
Rule 3460
Rule 3461
Rule 6566
Rule 6576
Rule 6592
Rule 6600
Rubi steps
\begin {align*} \int \frac {C(b x)^2}{x^9} \, dx &=-\frac {C(b x)^2}{8 x^8}+\frac {1}{4} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^8} \, dx\\ &=-\frac {b^2}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}-\frac {C(b x)^2}{8 x^8}+\frac {1}{56} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac {1}{28} \left (b^3 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac {b^2}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}+\frac {1}{112} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac {1}{280} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{140} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^4} \, dx\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {1}{560} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{336} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{840} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac {1}{420} \left (b^7 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {\left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1680}-\frac {\left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1120}-\frac {1}{672} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{840} \left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{420} \left (b^9 \pi ^4\right ) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )}{1680}+\frac {\left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1680}+\frac {\left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1120}+\frac {1}{672} \left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{420} \left (b^8 \pi ^4\right ) \text {Subst}(\int x \, dx,x,C(b x))\\ &=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}+\frac {1}{840} b^8 \pi ^4 C(b x)^2-\frac {C(b x)^2}{8 x^8}+\frac {b^3 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 242, normalized size = 1.00 \begin {gather*} -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{28 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{420 x^3}+\frac {1}{840} b^8 \pi ^4 \text {FresnelC}(b x)^2-\frac {\text {FresnelC}(b x)^2}{8 x^8}+\frac {b^3 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac {b^7 \pi ^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\FresnelC \left (b x \right )^{2}}{x^{9}}\, dx\]
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 = 0.38, size = 178, normalized size = 0.74 \begin {gather*} \frac {3 \, \pi ^{3} b^{8} x^{8} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 2 \, \pi ^{2} b^{6} x^{6} + 5 \, {\left (\pi ^{2} b^{6} x^{6} - b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (2 \, \pi b^{4} x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \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 {C^{2}\left (b x\right )}{x^{9}}\, 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 {FresnelC}\left (b\,x\right )}^2}{x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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