Optimal. Leaf size=46 \[ \frac {\sqrt {2 \pi } S\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {f}} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3386, 3432}
\begin {gather*} \frac {\sqrt {2 \pi } S\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3386
Rule 3432
Rubi steps
\begin {align*} \int \frac {\sin (f x)}{\sqrt {d x}} \, dx &=\frac {2 \text {Subst}\left (\int \sin \left (\frac {f x^2}{d}\right ) \, dx,x,\sqrt {d x}\right )}{d}\\ &=\frac {\sqrt {2 \pi } S\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {f}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.01, size = 59, normalized size = 1.28 \begin {gather*} \frac {-\sqrt {-i f x} \Gamma \left (\frac {1}{2},-i f x\right )-\sqrt {i f x} \Gamma \left (\frac {1}{2},i f x\right )}{2 f \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 42, normalized size = 0.91
method | result | size |
meijerg | \(\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {x}\, \sqrt {f}}{\sqrt {\pi }}\right )}{\sqrt {d x}\, \sqrt {f}}\) | \(33\) |
derivativedivides | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\) | \(42\) |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 67, normalized size = 1.46 \begin {gather*} \frac {\sqrt {2} {\left (\left (i + 1\right ) \, \sqrt {\pi } \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {i \, f}{d}}\right ) - \left (i - 1\right ) \, \sqrt {\pi } \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {i \, f}{d}}\right )\right )}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 38, normalized size = 0.83 \begin {gather*} \frac {\sqrt {2} \pi \sqrt {\frac {f}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.58, size = 54, normalized size = 1.17 \begin {gather*} \frac {3 \sqrt {2} \sqrt {\pi } S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {3}{4}\right )}{4 \sqrt {d} \sqrt {f} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 4.74, size = 136, normalized size = 2.96 \begin {gather*} -\frac {\frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d f} \sqrt {d x} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}} - \frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d f} \sqrt {d x} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}}{2 \, d} \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 {\sin \left (f\,x\right )}{\sqrt {d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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