3.2.23 \(\int \frac {\sin (a+\frac {b}{x^2})}{x^4} \, dx\) [123]

Optimal. Leaf size=97 \[ \frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)}{2 b^{3/2}} \]

[Out]

1/2*cos(a+b/x^2)/b/x-1/4*cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/x)*2^(1/2)*Pi^(1/2)/b^(3/2)+1/4*FresnelS(b^(
1/2)*2^(1/2)/Pi^(1/2)/x)*sin(a)*2^(1/2)*Pi^(1/2)/b^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3490, 3466, 3435, 3433, 3432} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{x}\right )}{2 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/x^2]/x^4,x]

[Out]

Cos[a + b/x^2]/(2*b*x) - (Sqrt[Pi/2]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x])/(2*b^(3/2)) + (Sqrt[Pi/2]*Fresne
lS[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a])/(2*b^(3/2))

Rule 3432

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]

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]

Rule 3435

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rule 3466

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^(n - 1))*(e*x)^(m - n + 1)*(Cos[c +
d*x^n]/(d*n)), x] + Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3490

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subst[Int[(a + b*Sin[c + d/x^
n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2
]

Rubi steps

\begin {align*} \int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x^4} \, dx &=-\text {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}\\ &=\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\cos (a) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}+\frac {\sin (a) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}\\ &=\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)}{2 b^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.11, size = 89, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {b} \cos \left (a+\frac {b}{x^2}\right )-\sqrt {2 \pi } x \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+\sqrt {2 \pi } x S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)}{4 b^{3/2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/x^2]/x^4,x]

[Out]

(2*Sqrt[b]*Cos[a + b/x^2] - Sqrt[2*Pi]*x*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x] + Sqrt[2*Pi]*x*FresnelS[(Sqrt
[b]*Sqrt[2/Pi])/x]*Sin[a])/(4*b^(3/2)*x)

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 65, normalized size = 0.67

method	result	size
derivativedivides	\(\frac {\cos \left (a +\frac {b}{x^{2}}\right )}{2 b x}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )}{4 b^{\frac {3}{2}}}\)	\(65\)
default	\(\frac {\cos \left (a +\frac {b}{x^{2}}\right )}{2 b x}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )}{4 b^{\frac {3}{2}}}\)	\(65\)
risch	\(-\frac {{\mathrm e}^{i a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-i b}}{x}\right )}{8 b \sqrt {-i b}}-\frac {{\mathrm e}^{-i a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {i b}}{x}\right )}{8 b \sqrt {i b}}+\frac {\cos \left (\frac {a \,x^{2}+b}{x^{2}}\right )}{2 b x}\)	\(82\)
meijerg	\(-\frac {\sqrt {\pi }\, \cos \left (a \right ) \sqrt {2}\, \left (-\frac {\sqrt {2}\, \sqrt {b}\, \cos \left (\frac {b}{x^{2}}\right )}{2 \sqrt {\pi }\, x}+\frac {\FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )}{2}\right )}{2 b^{\frac {3}{2}}}-\frac {\sqrt {\pi }\, \sin \left (a \right ) \sqrt {2}\, \left (b^{2}\right )^{\frac {1}{4}} \left (\frac {\sqrt {2}\, \left (b^{2}\right )^{\frac {3}{4}} \sin \left (\frac {b}{x^{2}}\right )}{2 \sqrt {\pi }\, x b}-\frac {\left (b^{2}\right )^{\frac {3}{4}} \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )}{2 b^{\frac {3}{2}}}\right )}{2 b^{2}}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/x^2)/x^4,x,method=_RETURNVERBOSE)

[Out]

1/2*cos(a+b/x^2)/b/x-1/4/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/x)-sin(a)*FresnelS
(b^(1/2)*2^(1/2)/Pi^(1/2)/x))

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.34, size = 74, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {2} {\left (x^{4}\right )}^{\frac {3}{2}} {\left ({\left (\left (i - 1\right ) \, \Gamma \left (\frac {3}{2}, \frac {i \, b}{x^{2}}\right ) - \left (i + 1\right ) \, \Gamma \left (\frac {3}{2}, -\frac {i \, b}{x^{2}}\right )\right )} \cos \left (a\right ) + {\left (\left (i + 1\right ) \, \Gamma \left (\frac {3}{2}, \frac {i \, b}{x^{2}}\right ) - \left (i - 1\right ) \, \Gamma \left (\frac {3}{2}, -\frac {i \, b}{x^{2}}\right )\right )} \sin \left (a\right )\right )} \left (\frac {b^{2}}{x^{4}}\right )^{\frac {3}{4}}}{8 \, b^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x^2)/x^4,x, algorithm="maxima")

[Out]

-1/8*sqrt(2)*(((I - 1)*gamma(3/2, I*b/x^2) - (I + 1)*gamma(3/2, -I*b/x^2))*cos(a) + ((I + 1)*gamma(3/2, I*b/x^
2) - (I - 1)*gamma(3/2, -I*b/x^2))*sin(a))*(x^4)^(3/2)*(b^2/x^4)^(3/4)/(b^3*x^3)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 85, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) - \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) \sin \left (a\right ) - 2 \, b \cos \left (\frac {a x^{2} + b}{x^{2}}\right )}{4 \, b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x^2)/x^4,x, algorithm="fricas")

[Out]

-1/4*(sqrt(2)*pi*x*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*sqrt(b/pi)/x) - sqrt(2)*pi*x*sqrt(b/pi)*fresnel_sin(s
qrt(2)*sqrt(b/pi)/x)*sin(a) - 2*b*cos((a*x^2 + b)/x^2))/(b^2*x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (a + \frac {b}{x^{2}} \right )}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/x**2)/x**4,x)

[Out]

Integral(sin(a + b/x**2)/x**4, x)

________________________________________________________________________________________

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.

[In]

integrate(sin(a+b/x^2)/x^4,x, algorithm="giac")

[Out]

integrate(sin(a + b/x^2)/x^4, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b/x^2)/x^4,x)

[Out]

int(sin(a + b/x^2)/x^4, x)

________________________________________________________________________________________