Optimal. Leaf size=64 \[ \frac {2 \sqrt {2 \pi } \sqrt {f} C\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sin (f x)}{d \sqrt {d x}} \]
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Rubi [A] time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3297, 3304, 3352} \[ \frac {2 \sqrt {2 \pi } \sqrt {f} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {f} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sin (f x)}{d \sqrt {d x}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3304
Rule 3352
Rubi steps
\begin {align*} \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx &=-\frac {2 \sin (f x)}{d \sqrt {d x}}+\frac {(2 f) \int \frac {\cos (f x)}{\sqrt {d x}} \, dx}{d}\\ &=-\frac {2 \sin (f x)}{d \sqrt {d x}}+\frac {(4 f) \operatorname {Subst}\left (\int \cos \left (\frac {f x^2}{d}\right ) \, dx,x,\sqrt {d x}\right )}{d^2}\\ &=\frac {2 \sqrt {f} \sqrt {2 \pi } C\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sin (f x)}{d \sqrt {d x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 64, normalized size = 1.00 \[ \frac {x \left (-2 \sin (f x)-i \sqrt {-i f x} \Gamma \left (\frac {1}{2},-i f x\right )+i \sqrt {i f x} \Gamma \left (\frac {1}{2},i f x\right )\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 57, normalized size = 0.89 \[ \frac {2 \, {\left (\sqrt {2} \pi d x \sqrt {\frac {f}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right ) - \sqrt {d x} \sin \left (f x\right )\right )}}{d^{2} x} \]
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 \[ \int \frac {\sin \left (f x\right )}{\left (d x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.94 \[ \frac {-\frac {2 \sin \left (f x \right )}{\sqrt {d x}}+\frac {2 f \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x}\, f}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.39, size = 38, normalized size = 0.59 \[ -\frac {\sqrt {f x} {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, f x\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, f x\right )\right )}}{4 \, \sqrt {d x} d} \]
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 \[ \int \frac {\sin \left (f\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.42, size = 80, normalized size = 1.25 \[ \frac {\sqrt {2} \sqrt {\pi } \sqrt {f} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{2 d^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} - \frac {\sin {\left (f x \right )} \Gamma \left (\frac {1}{4}\right )}{2 d^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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