Optimal. Leaf size=77 \[ -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^4}-\frac {1}{80} b^5 \pi ^2 \text {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\text {FresnelC}(b x)}{5 x^5}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{40 x^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3461,
3378, 3383} \begin {gather*} -\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{20 x^4}-\frac {1}{80} \pi ^2 b^5 \text {CosIntegral}\left (\frac {1}{2} \pi b^2 x^2\right )+\frac {\pi b^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{40 x^2}-\frac {\text {FresnelC}(b x)}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3383
Rule 3461
Rule 6562
Rubi steps
\begin {align*} \int \frac {C(b x)}{x^6} \, dx &=-\frac {C(b x)}{5 x^5}+\frac {1}{5} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac {C(b x)}{5 x^5}+\frac {1}{10} b \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^4}-\frac {C(b x)}{5 x^5}-\frac {1}{40} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^4}-\frac {C(b x)}{5 x^5}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{40 x^2}-\frac {1}{80} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^4}-\frac {1}{80} b^5 \pi ^2 \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {C(b x)}{5 x^5}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{40 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 77, normalized size = 1.00 \begin {gather*} -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^4}-\frac {1}{80} b^5 \pi ^2 \text {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\text {FresnelC}(b x)}{5 x^5}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{40 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 71, normalized size = 0.92
method | result | size |
derivativedivides | \(b^{5} \left (-\frac {\FresnelC \left (b x \right )}{5 b^{5} x^{5}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{20 b^{4} x^{4}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}+\frac {\pi \cosineIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{20}\right )\) | \(71\) |
default | \(b^{5} \left (-\frac {\FresnelC \left (b x \right )}{5 b^{5} x^{5}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{20 b^{4} x^{4}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}+\frac {\pi \cosineIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{20}\right )\) | \(71\) |
meijerg | \(\frac {\pi ^{\frac {5}{2}} b^{5} \left (\frac {\pi ^{\frac {3}{2}} x^{4} b^{4} \hypergeom \left (\left [1, 1, \frac {9}{4}\right ], \left [2, \frac {5}{2}, 3, \frac {13}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{54}-\frac {8 \left (-\frac {19}{5}+2 \gamma -2 \ln \left (2\right )+4 \ln \left (x \right )+2 \ln \left (\pi \right )+4 \ln \left (b \right )\right )}{5 \sqrt {\pi }}-\frac {64}{\pi ^{\frac {5}{2}} x^{4} b^{4}}\right )}{256}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.31, size = 46, normalized size = 0.60 \begin {gather*} \frac {1}{80} \, {\left (\pi ^{2} \Gamma \left (-2, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \pi ^{2} \Gamma \left (-2, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{5} - \frac {\operatorname {C}\left (b x\right )}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 85, normalized size = 1.10 \begin {gather*} -\frac {\pi ^{2} b^{5} x^{5} \operatorname {Ci}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \pi ^{2} b^{5} x^{5} \operatorname {Ci}\left (-\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, \pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 8 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 32 \, \operatorname {C}\left (b x\right )}{160 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.34, size = 65, normalized size = 0.84 \begin {gather*} \frac {\pi ^{4} b^{9} x^{4} \Gamma \left (\frac {9}{4}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {9}{4} \\ 2, \frac {5}{2}, 3, \frac {13}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{6144 \Gamma \left (\frac {13}{4}\right )} - \frac {\pi ^{2} b^{5} \log {\left (b^{4} x^{4} \right )}}{160} - \frac {b}{4 x^{4}} \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 = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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