Optimal. Leaf size=52 \[ -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^2}-\frac {\text {FresnelC}(b x)}{3 x^3}-\frac {1}{12} b^3 \pi \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 52, 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, 3461,
3378, 3380} \begin {gather*} -\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^2}-\frac {1}{12} \pi b^3 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\text {FresnelC}(b x)}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3461
Rule 6562
Rubi steps
\begin {align*} \int \frac {C(b x)}{x^4} \, dx &=-\frac {C(b x)}{3 x^3}+\frac {1}{3} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac {C(b x)}{3 x^3}+\frac {1}{6} b \text {Subst}\left (\int \frac {\cos \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 )}{6 x^2}-\frac {C(b x)}{3 x^3}-\frac {1}{12} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \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 )}{6 x^2}-\frac {C(b x)}{3 x^3}-\frac {1}{12} b^3 \pi \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 52, normalized size = 1.00 \begin {gather*} -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x^2}-\frac {\text {FresnelC}(b x)}{3 x^3}-\frac {1}{12} b^3 \pi \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 49, normalized size = 0.94
method | result | size |
meijerg | \(-\frac {b \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {1}{2}, \frac {5}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{2 x^{2}}\) | \(26\) |
derivativedivides | \(b^{3} \left (-\frac {\FresnelC \left (b x \right )}{3 b^{3} x^{3}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 b^{2} x^{2}}-\frac {\pi \sinIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12}\right )\) | \(49\) |
default | \(b^{3} \left (-\frac {\FresnelC \left (b x \right )}{3 b^{3} x^{3}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 b^{2} x^{2}}-\frac {\pi \sinIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12}\right )\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 44, normalized size = 0.85 \begin {gather*} -\frac {1}{24} \, {\left (i \, \pi \Gamma \left (-1, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - i \, \pi \Gamma \left (-1, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{3} - \frac {\operatorname {C}\left (b x\right )}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 44, normalized size = 0.85 \begin {gather*} -\frac {\pi b^{3} x^{3} \operatorname {Si}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, \operatorname {C}\left (b x\right )}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 42, normalized size = 0.81 \begin {gather*} - \frac {b \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2}, \frac {1}{2}, \frac {5}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{8 x^{2} \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 = -0.02 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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