Integrand size = 12, antiderivative size = 87 \[ \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx=-\frac {4 f \cos (f x)}{3 d^2 \sqrt {d x}}-\frac {4 f^{3/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}} \]
-2/3*sin(f*x)/d/(d*x)^(3/2)-4/3*f^(3/2)*FresnelS(f^(1/2)*2^(1/2)/Pi^(1/2)* (d*x)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/d^(5/2)-4/3*f*cos(f*x)/d^2/(d*x)^(1/ 2)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.28 \[ \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx=\frac {2 f x^{5/2} \left (-\frac {e^{i f x}-\sqrt {-i f x} \Gamma \left (\frac {1}{2},-i f x\right )}{\sqrt {x}}+\frac {-e^{-i f x}+\sqrt {i f x} \Gamma \left (\frac {1}{2},i f x\right )}{\sqrt {x}}\right )}{3 (d x)^{5/2}}-\frac {2 x \sin (f x)}{3 (d x)^{5/2}} \]
(2*f*x^(5/2)*(-((E^(I*f*x) - Sqrt[(-I)*f*x]*Gamma[1/2, (-I)*f*x])/Sqrt[x]) + (-E^((-I)*f*x) + Sqrt[I*f*x]*Gamma[1/2, I*f*x])/Sqrt[x]))/(3*(d*x)^(5/2 )) - (2*x*Sin[f*x])/(3*(d*x)^(5/2))
Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3778, 3042, 3778, 25, 3042, 3786, 3832}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (f x)}{(d x)^{5/2}}dx\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {2 f \int \frac {\cos (f x)}{(d x)^{3/2}}dx}{3 d}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 f \int \frac {\sin \left (f x+\frac {\pi }{2}\right )}{(d x)^{3/2}}dx}{3 d}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {2 f \left (\frac {2 f \int -\frac {\sin (f x)}{\sqrt {d x}}dx}{d}-\frac {2 \cos (f x)}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 f \left (-\frac {2 f \int \frac {\sin (f x)}{\sqrt {d x}}dx}{d}-\frac {2 \cos (f x)}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 f \left (-\frac {2 f \int \frac {\sin (f x)}{\sqrt {d x}}dx}{d}-\frac {2 \cos (f x)}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {2 f \left (-\frac {4 f \int \sin (f x)d\sqrt {d x}}{d^2}-\frac {2 \cos (f x)}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {2 f \left (-\frac {2 \sqrt {2 \pi } \sqrt {f} \operatorname {FresnelS}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cos (f x)}{d \sqrt {d x}}\right )}{3 d}-\frac {2 \sin (f x)}{3 d (d x)^{3/2}}\) |
(2*f*((-2*Cos[f*x])/(d*Sqrt[d*x]) - (2*Sqrt[f]*Sqrt[2*Pi]*FresnelS[(Sqrt[f ]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/d^(3/2)))/(3*d) - (2*Sin[f*x])/(3*d*(d*x )^(3/2))
3.1.64.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {2}\, f^{\frac {3}{2}} \left (-\frac {16 \sqrt {2}\, \cos \left (f x \right )}{3 \sqrt {\pi }\, \sqrt {x}\, \sqrt {f}}-\frac {8 \sqrt {2}\, \sin \left (f x \right )}{3 \sqrt {\pi }\, x^{\frac {3}{2}} f^{\frac {3}{2}}}-\frac {32 \,\operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {x}\, \sqrt {f}}{\sqrt {\pi }}\right )}{3}\right )}{8 \left (d x \right )^{\frac {5}{2}}}\) | \(73\) |
derivativedivides | \(\frac {-\frac {2 \sin \left (f x \right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {4 f \left (-\frac {\cos \left (f x \right )}{\sqrt {d x}}-\frac {f \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\right )}{3 d}}{d}\) | \(79\) |
default | \(\frac {-\frac {2 \sin \left (f x \right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {4 f \left (-\frac {\cos \left (f x \right )}{\sqrt {d x}}-\frac {f \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\right )}{3 d}}{d}\) | \(79\) |
1/8*Pi^(1/2)/(d*x)^(5/2)*x^(5/2)*2^(1/2)*f^(3/2)*(-16/3/Pi^(1/2)/x^(1/2)*2 ^(1/2)/f^(1/2)*cos(f*x)-8/3/Pi^(1/2)/x^(3/2)*2^(1/2)/f^(3/2)*sin(f*x)-32/3 *FresnelS(1/Pi^(1/2)*2^(1/2)*x^(1/2)*f^(1/2)))
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79 \[ \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} \pi d f x^{2} \sqrt {\frac {f}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right ) + {\left (2 \, f x \cos \left (f x\right ) + \sin \left (f x\right )\right )} \sqrt {d x}\right )}}{3 \, d^{3} x^{2}} \]
-2/3*(2*sqrt(2)*pi*d*f*x^2*sqrt(f/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x)*sq rt(f/(pi*d))) + (2*f*x*cos(f*x) + sin(f*x))*sqrt(d*x))/(d^3*x^2)
Time = 11.44 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx=\frac {\sqrt {2} \sqrt {\pi } f^{\frac {3}{2}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {f \cos {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {\sin {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{6 d^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} \]
sqrt(2)*sqrt(pi)*f**(3/2)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma (-1/4)/(3*d**(5/2)*gamma(3/4)) + f*cos(f*x)*gamma(-1/4)/(3*d**(5/2)*sqrt(x )*gamma(3/4)) + sin(f*x)*gamma(-1/4)/(6*d**(5/2)*x**(3/2)*gamma(3/4))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.44 \[ \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx=-\frac {\left (f x\right )^{\frac {3}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, f x\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, f x\right )\right )}}{4 \, \left (d x\right )^{\frac {3}{2}} d} \]
-1/4*(f*x)^(3/2)*(-(I + 1)*sqrt(2)*gamma(-3/2, I*f*x) + (I - 1)*sqrt(2)*ga mma(-3/2, -I*f*x))/((d*x)^(3/2)*d)
\[ \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx=\int { \frac {\sin \left (f x\right )}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin (f x)}{(d x)^{5/2}} \, dx=\int \frac {\sin \left (f\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]