3.1.51 \(\int \frac {\cos (a+b \sqrt [3]{x})}{\sqrt {x}} \, dx\) [51]

Optimal. Leaf size=99 \[ -\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{b^{3/2}}+\frac {3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

[Out]

3*x^(1/6)*sin(a+b*x^(1/3))/b-3/2*cos(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)-3/
2*FresnelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/b^(3/2)

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Rubi [A]
time = 0.09, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3497, 3377, 3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [6]{x}\right )}{b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{b^{3/2}}+\frac {3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^(1/3)]/Sqrt[x],x]

[Out]

(-3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/b^(3/2) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi
]*x^(1/6)]*Sin[a])/b^(3/2) + (3*x^(1/6)*Sin[a + b*x^(1/3)])/b

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 3386

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

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

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 3497

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Module[{k = Denominator[n]}, D
ist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}
, x] && IntegerQ[p] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}} \, dx &=3 \text {Subst}\left (\int \sqrt {x} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {3 \text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {(3 \cos (a)) \text {Subst}\left (\int \frac {\sin (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{2 b}-\frac {(3 \sin (a)) \text {Subst}\left (\int \frac {\cos (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {(3 \cos (a)) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{b}-\frac {(3 \sin (a)) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{b}\\ &=-\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{b^{3/2}}+\frac {3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 94, normalized size = 0.95 \begin {gather*} -\frac {3 \left (\sqrt {2 \pi } \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )+\sqrt {2 \pi } \text {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)-2 \sqrt {b} \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )\right )}{2 b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^(1/3)]/Sqrt[x],x]

[Out]

(-3*(Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] + Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*
Sin[a] - 2*Sqrt[b]*x^(1/6)*Sin[a + b*x^(1/3)]))/(2*b^(3/2))

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Maple [A]
time = 0.05, size = 64, normalized size = 0.65

method	result	size
derivativedivides	\(\frac {3 x^{\frac {1}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{2 b^{\frac {3}{2}}}\)	\(64\)
default	\(\frac {3 x^{\frac {1}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{2 b^{\frac {3}{2}}}\)	\(64\)
meijerg	\(\frac {3 \cos \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (\frac {x^{\frac {1}{6}} \sqrt {2}\, \left (b^{2}\right )^{\frac {3}{4}} \sin \left (b \,x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }\, b}-\frac {\left (b^{2}\right )^{\frac {3}{4}} \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{2 b^{\frac {3}{2}}}\right )}{\left (b^{2}\right )^{\frac {3}{4}}}-\frac {3 \sin \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {x^{\frac {1}{6}} \sqrt {2}\, \sqrt {b}\, \cos \left (b \,x^{\frac {1}{3}}\right )}{2 \sqrt {\pi }}+\frac {\FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{2}\right )}{b^{\frac {3}{2}}}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b*x^(1/3))/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

3*x^(1/6)*sin(a+b*x^(1/3))/b-3/2/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))+s
in(a)*FresnelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2)))

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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 73, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x^{\frac {1}{6}}\right ) + {\left (\left (i - 1\right ) \, \cos \left (a\right ) - \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 8 \, b^{2} x^{\frac {1}{6}} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{8 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^(1/3))/x^(1/2),x, algorithm="maxima")

[Out]

3/8*(sqrt(2)*sqrt(pi)*((-(I + 1)*cos(a) + (I - 1)*sin(a))*erf(sqrt(I*b)*x^(1/6)) + ((I - 1)*cos(a) - (I + 1)*s
in(a))*erf(sqrt(-I*b)*x^(1/6)))*b^(3/2) + 8*b^2*x^(1/6)*sin(b*x^(1/3) + a))/b^3

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Fricas [A]
time = 0.38, size = 78, normalized size = 0.79 \begin {gather*} -\frac {3 \, {\left (\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - 2 \, b x^{\frac {1}{6}} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^(1/3))/x^(1/2),x, algorithm="fricas")

[Out]

-3/2*(sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_sin(sqrt(2)*x^(1/6)*sqrt(b/pi)) + sqrt(2)*pi*sqrt(b/pi)*fresnel_cos
(sqrt(2)*x^(1/6)*sqrt(b/pi))*sin(a) - 2*b*x^(1/6)*sin(b*x^(1/3) + a))/b^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (a + b \sqrt [3]{x} \right )}}{\sqrt {x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x**(1/3))/x**(1/2),x)

[Out]

Integral(cos(a + b*x**(1/3))/sqrt(x), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.49, size = 143, normalized size = 1.44 \begin {gather*} -\frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{4 \, b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {3 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{4 \, b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {3 i \, x^{\frac {1}{6}} e^{\left (i \, b x^{\frac {1}{3}} + i \, a\right )}}{2 \, b} + \frac {3 i \, x^{\frac {1}{6}} e^{\left (-i \, b x^{\frac {1}{3}} - i \, a\right )}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^(1/3))/x^(1/2),x, algorithm="giac")

[Out]

-3/4*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x^(1/6)*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(b*(-I*b/abs(b) + 1)*
sqrt(abs(b))) + 3/4*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x^(1/6)*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(-I*a)/(b*(I*
b/abs(b) + 1)*sqrt(abs(b))) - 3/2*I*x^(1/6)*e^(I*b*x^(1/3) + I*a)/b + 3/2*I*x^(1/6)*e^(-I*b*x^(1/3) - I*a)/b

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\cos \left (a+b\,x^{1/3}\right )}{\sqrt {x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^(1/3))/x^(1/2),x)

[Out]

int(cos(a + b*x^(1/3))/x^(1/2), x)

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