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

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

[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

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Rubi [A]  time = 0.113156, antiderivative size = 99, normalized size of antiderivative = 1., 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 = {3416, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{b^{3/2}}+\frac{3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

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 3416

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]

Rule 3296

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 3306

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 3305

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 3351

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

Rule 3304

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 3352

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

Rubi steps

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

Mathematica [A]  time = 0.152974, size = 94, normalized size = 0.95 \[ -\frac{3 \left (\sqrt{2 \pi } \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )+\sqrt{2 \pi } \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )-2 \sqrt{b} \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )\right )}{2 b^{3/2}} \]

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 = 0.032, size = 64, normalized size = 0.7 \begin{align*} 3\,{\frac{\sqrt [6]{x}\sin \left ( a+b\sqrt [3]{x} \right ) }{b}}-{\frac{3\,\sqrt{2}\sqrt{\pi }}{2} \left ( \cos \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \right ) +\sin \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \right ) \right ){b}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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]  time = 1.95197, size = 346, normalized size = 3.49 \begin{align*} \frac{3 \,{\left (8 \, x^{\frac{1}{6}}{\left | b \right |} \sin \left (b x^{\frac{1}{3}} + a\right ) + \sqrt{\pi }{\left ({\left ({\left (-i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \operatorname{erf}\left (\sqrt{i \, b} x^{\frac{1}{6}}\right ) +{\left ({\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \operatorname{erf}\left (\sqrt{-i \, b} x^{\frac{1}{6}}\right )\right )} \sqrt{{\left | b \right |}}\right )}}{8 \, b{\left | b \right |}} \end{align*}

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*(8*x^(1/6)*abs(b)*sin(b*x^(1/3) + a) + sqrt(pi)*(((-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/
2*arctan2(0, b)) - sin(1/4*pi + 1/2*arctan2(0, b)) + sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (cos(1/4*pi +
1/2*arctan2(0, b)) + cos(-1/4*pi + 1/2*arctan2(0, b)) - I*sin(1/4*pi + 1/2*arctan2(0, b)) + I*sin(-1/4*pi + 1/
2*arctan2(0, b)))*sin(a))*erf(sqrt(I*b)*x^(1/6)) + ((I*cos(1/4*pi + 1/2*arctan2(0, b)) + I*cos(-1/4*pi + 1/2*a
rctan2(0, b)) - sin(1/4*pi + 1/2*arctan2(0, b)) + sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (cos(1/4*pi + 1/2
*arctan2(0, b)) + cos(-1/4*pi + 1/2*arctan2(0, b)) + I*sin(1/4*pi + 1/2*arctan2(0, b)) - I*sin(-1/4*pi + 1/2*a
rctan2(0, b)))*sin(a))*erf(sqrt(-I*b)*x^(1/6)))*sqrt(abs(b)))/(b*abs(b))

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Fricas [A]  time = 1.92261, size = 250, normalized size = 2.53 \begin{align*} -\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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + b \sqrt [3]{x} \right )}}{\sqrt{x}}\, dx \end{align*}

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]  time = 1.16036, size = 193, normalized size = 1.95 \begin{align*} -\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{align*}

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