Optimal. Leaf size=87 \[ -\frac{4 \sqrt{2 \pi } f^{3/2} S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0926116, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3297, 3305, 3351} \[ -\frac{4 \sqrt{2 \pi } f^{3/2} S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3305
Rule 3351
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 ) \operatorname{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*}
Mathematica [C] time = 0.0849214, size = 111, normalized size = 1.28 \[ -\frac{2 x \sin (f x)}{3 (d x)^{5/2}}+\frac{2 f x^{5/2} \left (\frac{\sqrt{i f x} \text{Gamma}\left (\frac{1}{2},i f x\right )-e^{-i f x}}{\sqrt{x}}-\frac{e^{i f x}-\sqrt{-i f x} \text{Gamma}\left (\frac{1}{2},-i f x\right )}{\sqrt{x}}\right )}{3 (d x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 79, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ( -1/3\,{\frac{\sin \left ( fx \right ) }{ \left ( dx \right ) ^{3/2}}}+2/3\,{\frac{f}{d} \left ( -{\frac{\cos \left ( fx \right ) }{\sqrt{dx}}}-{\frac{f\sqrt{2}\sqrt{\pi }}{d}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{dx}f}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.17412, size = 231, normalized size = 2.66 \begin{align*} \frac{\left (\frac{d x{\left | f \right |}}{{\left | d \right |}}\right )^{\frac{3}{2}}{\left ({\left (-i \, \Gamma \left (-\frac{3}{2}, i \, f x\right ) + i \, \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \cos \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) +{\left (-i \, \Gamma \left (-\frac{3}{2}, i \, f x\right ) + i \, \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \cos \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) +{\left (\Gamma \left (-\frac{3}{2}, i \, f x\right ) + \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \sin \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) -{\left (\Gamma \left (-\frac{3}{2}, i \, f x\right ) + \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \sin \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right )\right )}}{4 \, \left (d x\right )^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.28062, size = 189, normalized size = 2.17 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 145.307, size = 114, normalized size = 1.31 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x\right )}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]