Optimal. Leaf size=94 \[ -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}-\frac {\text {FresnelC}(b x)}{6 x^6}+\frac {1}{90} b^6 \pi ^3 S(b x)+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 94, 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, 3469,
3468, 3432} \begin {gather*} \frac {1}{90} \pi ^3 b^6 S(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{30 x^5}+\frac {\pi ^2 b^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x}+\frac {\pi b^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x^3}-\frac {\text {FresnelC}(b x)}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3468
Rule 3469
Rule 6562
Rubi steps
\begin {align*} \int \frac {C(b x)}{x^7} \, dx &=-\frac {C(b x)}{6 x^6}+\frac {1}{6} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}-\frac {C(b x)}{6 x^6}-\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}-\frac {C(b x)}{6 x^6}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {1}{90} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}-\frac {C(b x)}{6 x^6}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}+\frac {1}{90} \left (b^7 \pi ^3\right ) \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}-\frac {C(b x)}{6 x^6}+\frac {1}{90} b^6 \pi ^3 S(b x)+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 74, normalized size = 0.79 \begin {gather*} \frac {1}{90} \left (\frac {b \left (-3+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}-\frac {15 \text {FresnelC}(b x)}{x^6}+b^6 \pi ^3 S(b x)+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 87, normalized size = 0.93
method | result | size |
meijerg | \(-\frac {b \hypergeom \left (\left [-\frac {5}{4}, \frac {1}{4}\right ], \left [-\frac {1}{4}, \frac {1}{2}, \frac {5}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{5 x^{5}}\) | \(26\) |
derivativedivides | \(b^{6} \left (-\frac {\FresnelC \left (b x \right )}{6 b^{6} x^{6}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\mathrm {S}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(87\) |
default | \(b^{6} \left (-\frac {\FresnelC \left (b x \right )}{6 b^{6} x^{6}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\mathrm {S}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.52, size = 61, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {5}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{6}}{192 \, x^{5}} - \frac {\operatorname {C}\left (b x\right )}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 79, normalized size = 0.84 \begin {gather*} \frac {\pi ^{3} \sqrt {b^{2}} b^{5} x^{6} \operatorname {S}\left (\sqrt {b^{2}} x\right ) + \pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{2} b^{5} x^{5} - 3 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 15 \, \operatorname {C}\left (b x\right )}{90 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.81, size = 56, normalized size = 0.60 \begin {gather*} \frac {b \Gamma \left (- \frac {5}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{4} \\ - \frac {1}{4}, \frac {1}{2}, \frac {5}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 x^{5} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {5}{4}\right )} \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^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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