\(\int \frac {\sin (f x)}{(d x)^{3/2}} \, dx\) [63]

Optimal result

Integrand size = 12, antiderivative size = 64 \[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=\frac {2 \sqrt {f} \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sin (f x)}{d \sqrt {d x}} \]

[Out]

2*FresnelC(f^(1/2)*2^(1/2)/Pi^(1/2)*(d*x)^(1/2)/d^(1/2))*f^(1/2)*2^(1/2)*Pi^(1/2)/d^(3/2)-2*sin(f*x)/d/(d*x)^(
1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3378, 3385, 3433} \[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=\frac {2 \sqrt {2 \pi } \sqrt {f} \operatorname {FresnelC}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sin (f x)}{d \sqrt {d x}} \]

[In]

Int[Sin[f*x]/(d*x)^(3/2),x]

[Out]

(2*Sqrt[f]*Sqrt[2*Pi]*FresnelC[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*Sin[f*x])/(d*Sqrt[d*x])

Rule 3378

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] - Dist[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]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[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]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rubi steps \begin{align*} \text {integral}& = -\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) \text {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 } \operatorname {FresnelC}\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*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=\frac {x \left (-i \sqrt {-i f x} \Gamma \left (\frac {1}{2},-i f x\right )+i \sqrt {i f x} \Gamma \left (\frac {1}{2},i f x\right )-2 \sin (f x)\right )}{(d x)^{3/2}} \]

[In]

Integrate[Sin[f*x]/(d*x)^(3/2),x]

[Out]

(x*((-I)*Sqrt[(-I)*f*x]*Gamma[1/2, (-I)*f*x] + I*Sqrt[I*f*x]*Gamma[1/2, I*f*x] - 2*Sin[f*x]))/(d*x)^(3/2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86

method	result	size
meijerg	\(\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {2}\, \sqrt {f}\, \left (-\frac {4 \sqrt {2}\, \sin \left (f x \right )}{\sqrt {\pi }\, \sqrt {x}\, \sqrt {f}}+8 \,\operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {x}\, \sqrt {f}}{\sqrt {\pi }}\right )\right )}{4 \left (d x \right )^{\frac {3}{2}}}\)	\(55\)
derivativedivides	\(\frac {-\frac {2 \sin \left (f x \right )}{\sqrt {d x}}+\frac {2 f \sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}}{d}\)	\(60\)
default	\(\frac {-\frac {2 \sin \left (f x \right )}{\sqrt {d x}}+\frac {2 f \sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}}{d}\)	\(60\)

[In]

int(sin(f*x)/(d*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4*Pi^(1/2)/(d*x)^(3/2)*x^(3/2)*2^(1/2)*f^(1/2)*(-4/Pi^(1/2)*2^(1/2)/x^(1/2)/f^(1/2)*sin(f*x)+8*FresnelC(1/Pi
^(1/2)*2^(1/2)*x^(1/2)*f^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {2} \pi d x \sqrt {\frac {f}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right ) - \sqrt {d x} \sin \left (f x\right )\right )}}{d^{2} x} \]

[In]

integrate(sin(f*x)/(d*x)^(3/2),x, algorithm="fricas")

[Out]

2*(sqrt(2)*pi*d*x*sqrt(f/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d))) - sqrt(d*x)*sin(f*x))/(d^2*x)

Sympy [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.25 \[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=\frac {\sqrt {2} \sqrt {\pi } \sqrt {f} C\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{2 d^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} - \frac {\sin {\left (f x \right )} \Gamma \left (\frac {1}{4}\right )}{2 d^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate(sin(f*x)/(d*x)**(3/2),x)

[Out]

sqrt(2)*sqrt(pi)*sqrt(f)*fresnelc(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma(1/4)/(2*d**(3/2)*gamma(5/4)) - sin(f
*x)*gamma(1/4)/(2*d**(3/2)*sqrt(x)*gamma(5/4))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.59 \[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=-\frac {\sqrt {f x} {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, f x\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, f x\right )\right )}}{4 \, \sqrt {d x} d} \]

[In]

integrate(sin(f*x)/(d*x)^(3/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(f*x)*((I - 1)*sqrt(2)*gamma(-1/2, I*f*x) - (I + 1)*sqrt(2)*gamma(-1/2, -I*f*x))/(sqrt(d*x)*d)

Giac [F]

\[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=\int { \frac {\sin \left (f x\right )}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(sin(f*x)/(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sin(f*x)/(d*x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (f x)}{(d x)^{3/2}} \, dx=\int \frac {\sin \left (f\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]

[In]

int(sin(f*x)/(d*x)^(3/2),x)

[Out]

int(sin(f*x)/(d*x)^(3/2), x)