3.52 \(\int \frac{\cos (a+b \sqrt [3]{x})}{x^{3/2}} \, dx\)

Optimal. Leaf size=110 \[ -4 \sqrt{2 \pi } b^{3/2} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )+4 \sqrt{2 \pi } b^{3/2} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}} \]

[Out]

(-2*Cos[a + b*x^(1/3)])/Sqrt[x] - 4*b^(3/2)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] + 4*b^(3/2)
*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a] + (4*b*Sin[a + b*x^(1/3)])/x^(1/6)

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Rubi [A]  time = 0.142877, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3416, 3297, 3306, 3305, 3351, 3304, 3352} \[ -4 \sqrt{2 \pi } b^{3/2} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )+4 \sqrt{2 \pi } b^{3/2} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^(1/3)]/x^(3/2),x]

[Out]

(-2*Cos[a + b*x^(1/3)])/Sqrt[x] - 4*b^(3/2)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] + 4*b^(3/2)
*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a] + (4*b*Sin[a + b*x^(1/3)])/x^(1/6)

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 3297

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 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 )}{x^{3/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}-(2 b) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (4 b^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )+\left (4 b^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (8 b^2 \cos (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )+\left (8 b^2 \sin (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}-4 b^{3/2} \sqrt{2 \pi } \cos (a) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )+4 b^{3/2} \sqrt{2 \pi } S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right ) \sin (a)+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\\ \end{align*}

Mathematica [A]  time = 0.248194, size = 110, normalized size = 1. \[ -4 \sqrt{2 \pi } b^{3/2} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )+4 \sqrt{2 \pi } b^{3/2} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )+\frac{4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\frac{2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^(1/3)]/x^(3/2),x]

[Out]

(-2*Cos[a + b*x^(1/3)])/Sqrt[x] - 4*b^(3/2)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] + 4*b^(3/2)
*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a] + (4*b*Sin[a + b*x^(1/3)])/x^(1/6)

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Maple [A]  time = 0.03, size = 78, normalized size = 0.7 \begin{align*} -2\,{\frac{\cos \left ( a+b\sqrt [3]{x} \right ) }{\sqrt{x}}}-4\,b \left ( -{\frac{\sin \left ( a+b\sqrt [3]{x} \right ) }{\sqrt [6]{x}}}+\sqrt{b}\sqrt{2}\sqrt{\pi } \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt [6]{x}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt [6]{x}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(a+b*x^(1/3))/x^(1/2)-4*b*(-1/x^(1/6)*sin(a+b*x^(1/3))+b^(1/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelC(x^(1/6)
*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))))

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Maxima [C]  time = 1.85966, size = 358, normalized size = 3.25 \begin{align*} -\frac{3 \,{\left ({\left ({\left (\Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) - i \, \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) +{\left ({\left (-i \, \Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (-\frac{3}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \sqrt{x^{\frac{1}{3}}{\left | b \right |}}{\left | b \right |}}{4 \, x^{\frac{1}{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.02239, size = 300, normalized size = 2.73 \begin{align*} -\frac{2 \,{\left (2 \, \sqrt{2} \pi b x \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) - 2 \, \sqrt{2} \pi b x \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) - 2 \, b x^{\frac{5}{6}} \sin \left (b x^{\frac{1}{3}} + a\right ) + \sqrt{x} \cos \left (b x^{\frac{1}{3}} + a\right )\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*sqrt(2)*pi*b*x*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x^(1/6)*sqrt(b/pi)) - 2*sqrt(2)*pi*b*x*sqrt(b/pi)*f
resnel_sin(sqrt(2)*x^(1/6)*sqrt(b/pi))*sin(a) - 2*b*x^(5/6)*sin(b*x^(1/3) + a) + sqrt(x)*cos(b*x^(1/3) + a))/x

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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 )}}{x^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x^{\frac{1}{3}} + a\right )}{x^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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