Optimal. Leaf size=69 \[ -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} b^4 \pi ^2 \text {FresnelC}(b x)-\frac {\text {FresnelC}(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x} \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3469,
3468, 3433} \begin {gather*} -\frac {1}{12} \pi ^2 b^4 \text {FresnelC}(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x^3}+\frac {\pi b^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x}-\frac {\text {FresnelC}(b x)}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3433
Rule 3468
Rule 3469
Rule 6562
Rubi steps
\begin {align*} \int \frac {C(b x)}{x^5} \, dx &=-\frac {C(b x)}{4 x^4}+\frac {1}{4} b \int \frac {\cos \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 )}{12 x^3}-\frac {C(b x)}{4 x^4}-\frac {1}{12} \left (b^3 \pi \right ) \int \frac {\sin \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 )}{12 x^3}-\frac {C(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {1}{12} \left (b^5 \pi ^2\right ) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} b^4 \pi ^2 C(b x)-\frac {C(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 69, normalized size = 1.00 \begin {gather*} -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} b^4 \pi ^2 \text {FresnelC}(b x)-\frac {\text {FresnelC}(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 64, normalized size = 0.93
method | result | size |
derivativedivides | \(b^{4} \left (-\frac {\FresnelC \left (b x \right )}{4 b^{4} x^{4}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \FresnelC \left (b x \right )\right )}{12}\right )\) | \(64\) |
default | \(b^{4} \left (-\frac {\FresnelC \left (b x \right )}{4 b^{4} x^{4}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \FresnelC \left (b x \right )\right )}{12}\right )\) | \(64\) |
meijerg | \(\frac {\pi ^{2} b^{4} \left (-\frac {32 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2} x^{3} b^{3}}+\frac {32 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi x b}-\frac {32 \left (x^{4} \pi ^{2} b^{4}+3\right ) \FresnelC \left (b x \right )}{3 \pi ^{2} x^{4} b^{4}}\right )}{128}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 61, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {3}{2}} {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{4}}{64 \, x^{3}} - \frac {\operatorname {C}\left (b x\right )}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 56, normalized size = 0.81 \begin {gather*} \frac {\pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.61, size = 110, normalized size = 1.59 \begin {gather*} \frac {\pi ^{2} b^{4} C\left (b x\right ) \Gamma \left (- \frac {3}{4}\right )}{64 \Gamma \left (\frac {5}{4}\right )} - \frac {\pi b^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {3}{4}\right )}{64 x \Gamma \left (\frac {5}{4}\right )} + \frac {b \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {3}{4}\right )}{64 x^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {3 C\left (b x\right ) \Gamma \left (- \frac {3}{4}\right )}{64 x^{4} \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^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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