3.1.40 \(\int \cos (a+\frac {b}{x^2}) \, dx\) [40]

Optimal. Leaf size=79 \[ x \cos \left (a+\frac {b}{x^2}\right )+\sqrt {b} \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+\sqrt {b} \sqrt {2 \pi } \text {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a) \]

[Out]

x*cos(a+b/x^2)+cos(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/x)*b^(1/2)*2^(1/2)*Pi^(1/2)+FresnelC(b^(1/2)*2^(1/2)/P
i^(1/2)/x)*sin(a)*b^(1/2)*2^(1/2)*Pi^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3441, 3469, 3434, 3433, 3432} \begin {gather*} \sqrt {2 \pi } \sqrt {b} \sin (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{x}\right )+\sqrt {2 \pi } \sqrt {b} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+x \cos \left (a+\frac {b}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b/x^2],x]

[Out]

x*Cos[a + b/x^2] + Sqrt[b]*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x] + Sqrt[b]*Sqrt[2*Pi]*FresnelC[(S
qrt[b]*Sqrt[2/Pi])/x]*Sin[a]

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 3434

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

Rule 3441

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

Rule 3469

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

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 80, normalized size = 1.01 \begin {gather*} x \cos (a) \cos \left (\frac {b}{x^2}\right )+\sqrt {b} \sqrt {2 \pi } \left (\cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+\text {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)\right )-x \sin (a) \sin \left (\frac {b}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b/x^2],x]

[Out]

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

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Maple [A]
time = 0.11, size = 57, normalized size = 0.72

method	result	size
derivativedivides	\(x \cos \left (a +\frac {b}{x^{2}}\right )+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )+\sin \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )\)	\(57\)
default	\(x \cos \left (a +\frac {b}{x^{2}}\right )+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )+\sin \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )\)	\(57\)
risch	\(\frac {i {\mathrm e}^{-i a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {i b}}{x}\right )}{2 \sqrt {i b}}-\frac {i {\mathrm e}^{i a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {-i b}}{x}\right )}{2 \sqrt {-i b}}+x \cos \left (\frac {a \,x^{2}+b}{x^{2}}\right )\)	\(74\)
meijerg	\(-\frac {\sqrt {\pi }\, \cos \left (a \right ) \sqrt {2}\, \left (b^{2}\right )^{\frac {1}{4}} \left (-\frac {4 x \sqrt {2}\, \cos \left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }\, \left (b^{2}\right )^{\frac {1}{4}}}-\frac {8 \sqrt {b}\, \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )}{\left (b^{2}\right )^{\frac {1}{4}}}\right )}{8}+\frac {\sqrt {\pi }\, \sin \left (a \right ) \sqrt {2}\, \sqrt {b}\, \left (-\frac {4 \sqrt {2}\, x \sin \left (\frac {b}{x^{2}}\right )}{\sqrt {b}\, \sqrt {\pi }}+8 \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )}{8}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.35, size = 127, normalized size = 1.61 \begin {gather*} \frac {\sqrt {2} {\left (2 \, \sqrt {2} b x^{2} \sqrt {\frac {1}{x^{4}}} \cos \left (\frac {a x^{2} + b}{x^{2}}\right ) + {\left ({\left (\left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} - \left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \cos \left (a\right ) + {\left (-\left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \sin \left (a\right )\right )} b \left (\frac {b^{2}}{x^{4}}\right )^{\frac {1}{4}}\right )} \sqrt {x^{4}}}{4 \, b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*(2*sqrt(2)*b*x^2*sqrt(x^(-4))*cos((a*x^2 + b)/x^2) + (((I + 1)*sqrt(pi)*(erf(sqrt(I*b/x^2)) - 1) -
(I - 1)*sqrt(pi)*(erf(sqrt(-I*b/x^2)) - 1))*cos(a) + (-(I - 1)*sqrt(pi)*(erf(sqrt(I*b/x^2)) - 1) + (I + 1)*sq
rt(pi)*(erf(sqrt(-I*b/x^2)) - 1))*sin(a))*b*(b^2/x^4)^(1/4))*sqrt(x^4)/(b*x)

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Fricas [A]
time = 0.35, size = 73, normalized size = 0.92 \begin {gather*} \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) + \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) \sin \left (a\right ) + x \cos \left (\frac {a x^{2} + b}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_sin(sqrt(2)*sqrt(b/pi)/x) + sqrt(2)*pi*sqrt(b/pi)*fresnel_cos(sqrt(2)*sqr
t(b/pi)/x)*sin(a) + x*cos((a*x^2 + b)/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x**2),x)

[Out]

Integral(cos(a + b/x**2), x)

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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.

[In]

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

[Out]

integrate(cos(a + b/x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b/x^2),x)

[Out]

int(cos(a + b/x^2), x)

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