Optimal. Leaf size=87 \[ -\frac {4 f \cos (f x)}{3 d^2 \sqrt {d x}}-\frac {4 f^{3/2} \sqrt {2 \pi } S\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3378, 3386,
3432} \begin {gather*} -\frac {4 \sqrt {2 \pi } f^{3/2} S\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {4 f \cos (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3386
Rule 3432
Rubi steps
\begin {align*} \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx &=-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}+\frac {(2 f) \int \frac {\cos (f x)}{(d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {4 f \cos (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}-\frac {\left (4 f^2\right ) \int \frac {\sin (f x)}{\sqrt {d x}} \, dx}{3 d^2}\\ &=-\frac {4 f \cos (f x)}{3 d^2 \sqrt {d x}}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}-\frac {\left (8 f^2\right ) \text {Subst}\left (\int \sin \left (\frac {f x^2}{d}\right ) \, dx,x,\sqrt {d x}\right )}{3 d^3}\\ &=-\frac {4 f \cos (f x)}{3 d^2 \sqrt {d x}}-\frac {4 f^{3/2} \sqrt {2 \pi } S\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.06, size = 111, normalized size = 1.28 \begin {gather*} \frac {2 f x^{5/2} \left (-\frac {e^{i f x}-\sqrt {-i f x} \Gamma \left (\frac {1}{2},-i f x\right )}{\sqrt {x}}+\frac {-e^{-i f x}+\sqrt {i f x} \Gamma \left (\frac {1}{2},i f x\right )}{\sqrt {x}}\right )}{3 (d x)^{5/2}}-\frac {2 x \sin (f x)}{3 (d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 79, normalized size = 0.91
method | result | size |
meijerg | \(\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {2}\, f^{\frac {3}{2}} \left (-\frac {16 \sqrt {2}\, \cos \left (f x \right )}{3 \sqrt {\pi }\, \sqrt {x}\, \sqrt {f}}-\frac {8 \sqrt {2}\, \sin \left (f x \right )}{3 \sqrt {\pi }\, x^{\frac {3}{2}} f^{\frac {3}{2}}}-\frac {32 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {x}\, \sqrt {f}}{\sqrt {\pi }}\right )}{3}\right )}{8 \left (d x \right )^{\frac {5}{2}}}\) | \(73\) |
derivativedivides | \(\frac {-\frac {2 \sin \left (f x \right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {4 f \left (-\frac {\cos \left (f x \right )}{\sqrt {d x}}-\frac {f \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\right )}{3 d}}{d}\) | \(79\) |
default | \(\frac {-\frac {2 \sin \left (f x \right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {4 f \left (-\frac {\cos \left (f x \right )}{\sqrt {d x}}-\frac {f \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\right )}{3 d}}{d}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.59, size = 38, normalized size = 0.44 \begin {gather*} -\frac {\left (f x\right )^{\frac {3}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, f x\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, f x\right )\right )}}{4 \, \left (d x\right )^{\frac {3}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 69, normalized size = 0.79 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {2} \pi d f x^{2} \sqrt {\frac {f}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right ) + {\left (2 \, f x \cos \left (f x\right ) + \sin \left (f x\right )\right )} \sqrt {d x}\right )}}{3 \, d^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 17.75, size = 114, normalized size = 1.31 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } f^{\frac {3}{2}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {f \cos {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {\sin {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{6 d^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{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 {\sin \left (f\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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