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

Optimal. Leaf size=116 \[ -8 \sqrt{\pi } b^{3/2} \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+8 \sqrt{\pi } b^{3/2} \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}} \]

[Out]

(-2*Cos[a + b*x^(1/3)]^2)/Sqrt[x] - 8*b^(3/2)*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] + 8*b^(
3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/x^
(1/6)

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Rubi [A]  time = 0.18694, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3416, 3314, 30, 3312, 3306, 3305, 3351, 3304, 3352} \[ -8 \sqrt{\pi } b^{3/2} \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+8 \sqrt{\pi } b^{3/2} \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^(1/3)]^2/x^(3/2),x]

[Out]

(-2*Cos[a + b*x^(1/3)]^2)/Sqrt[x] - 8*b^(3/2)*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] + 8*b^(
3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + (8*b*Cos[a + b*x^(1/3)]*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 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && 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 ^2\left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}+\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )-\left (16 b^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=16 b^2 \sqrt [6]{x}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (16 b^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 a+2 b x)}{2 \sqrt{x}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{\cos (2 a+2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (8 b^2 \cos (2 a)\right ) \operatorname{Subst}\left (\int \frac{\cos (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )+\left (8 b^2 \sin (2 a)\right ) \operatorname{Subst}\left (\int \frac{\sin (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (16 b^2 \cos (2 a)\right ) \operatorname{Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )+\left (16 b^2 \sin (2 a)\right ) \operatorname{Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}}-8 b^{3/2} \sqrt{\pi } \cos (2 a) C\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+8 b^{3/2} \sqrt{\pi } S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right ) \sin (2 a)+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\\ \end{align*}

Mathematica [A]  time = 0.254318, size = 116, normalized size = 1. \[ \frac{8 \sqrt{\pi } b^{3/2} \sqrt{x} \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+4 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-\cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-1}{\sqrt{x}}-8 \sqrt{\pi } b^{3/2} \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-8*b^(3/2)*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] + (-1 - Cos[2*(a + b*x^(1/3))] + 8*b^(3/2)
*Sqrt[Pi]*Sqrt[x]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + 4*b*x^(1/3)*Sin[2*(a + b*x^(1/3))])/Sqrt[x
]

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Maple [A]  time = 0.037, size = 87, normalized size = 0.8 \begin{align*} -{\frac{1}{\sqrt{x}}}-{\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ){\frac{1}{\sqrt{x}}}}-4\,b \left ( -{\frac{\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }{\sqrt [6]{x}}}+2\,\sqrt{b}\sqrt{\pi } \left ( \cos \left ( 2\,a \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\sqrt{\pi }}} \right ) -\sin \left ( 2\,a \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\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))^2/x^(3/2),x)

[Out]

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

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Maxima [C]  time = 1.74449, size = 379, normalized size = 3.27 \begin{align*} -\frac{\sqrt{2}{\left ({\left (3 \,{\left (\Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -2 i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) + 3 \,{\left (\Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -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 (3 i \, \Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) - 3 i \, \Gamma \left (-\frac{3}{2}, -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 (-3 i \, \Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) + 3 i \, \Gamma \left (-\frac{3}{2}, -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 (2 \, a\right ) +{\left ({\left (-3 i \, \Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) + 3 i \, \Gamma \left (-\frac{3}{2}, -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 (-3 i \, \Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) + 3 i \, \Gamma \left (-\frac{3}{2}, -2 i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) + 3 \,{\left (\Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -2 i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) - 3 \,{\left (\Gamma \left (-\frac{3}{2}, 2 i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{3}{2}, -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 (2 \, a\right )\right )} \sqrt{x^{\frac{1}{3}}{\left | b \right |}} x^{\frac{1}{3}}{\left | b \right |} + 4}{4 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.09213, size = 296, normalized size = 2.55 \begin{align*} -\frac{2 \,{\left (4 \, \pi b x \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{C}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) - 4 \, \pi b x \sqrt{\frac{b}{\pi }} \operatorname{S}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (2 \, a\right ) - 4 \, b x^{\frac{5}{6}} \cos \left (b x^{\frac{1}{3}} + a\right ) \sin \left (b x^{\frac{1}{3}} + a\right ) + \sqrt{x} \cos \left (b x^{\frac{1}{3}} + a\right )^{2}\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(4*pi*b*x*sqrt(b/pi)*cos(2*a)*fresnel_cos(2*x^(1/6)*sqrt(b/pi)) - 4*pi*b*x*sqrt(b/pi)*fresnel_sin(2*x^(1/6)
*sqrt(b/pi))*sin(2*a) - 4*b*x^(5/6)*cos(b*x^(1/3) + a)*sin(b*x^(1/3) + a) + sqrt(x)*cos(b*x^(1/3) + a)^2)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\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))**2/x**(3/2),x)

[Out]

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

[Out]

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