Optimal. Leaf size=13 \[ -\frac {1}{2 b S(b x)^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6575, 30}
\begin {gather*} -\frac {1}{2 b S(b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6575
Rubi steps
\begin {align*} \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{S(b x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3} \, dx,x,S(b x)\right )}{b}\\ &=-\frac {1}{2 b S(b x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b S(b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 12, normalized size = 0.92
method | result | size |
derivativedivides | \(-\frac {1}{2 b \mathrm {S}\left (b x \right )^{2}}\) | \(12\) |
default | \(-\frac {1}{2 b \mathrm {S}\left (b x \right )^{2}}\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2 \, b \operatorname {S}\left (b x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2 \, b \operatorname {S}\left (b x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 14, normalized size = 1.08 \begin {gather*} \begin {cases} - \frac {1}{2 b S^{2}\left (b x\right )} & \text {for}\: b \neq 0 \\\text {NaN} & \text {otherwise} \end {cases} \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 = -0.08 \begin {gather*} \int \frac {\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{{\mathrm {FresnelS}\left (b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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