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

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

[Out]

8*b*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/x^(1/6)-8*b^(3/2)*cos(2*a)*FresnelC(2*x^(1/6)*b^(1/2)/Pi^(1/2))*Pi^(1/2)
+8*b^(3/2)*FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))*sin(2*a)*Pi^(1/2)-2*cos(a+b*x^(1/3))^2/x^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 30

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

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 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 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 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 0.25, size = 116, normalized size = 1.00 \[ \frac {8 \sqrt {\pi } b^{3/2} \sqrt {x} \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+4 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-\cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-1}{\sqrt {x}}-8 \sqrt {\pi } b^{3/2} \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \]

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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fricas [A]  time = 0.87, size = 100, normalized size = 0.86 \[ -\frac {2 \, {\left (4 \, \pi b x \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 4 \, \pi b x \sqrt {\frac {b}{\pi }} \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 4 \, b x^{\frac {5}{6}} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) + \sqrt {x} \cos \left (b x^{\frac {1}{3}} + a\right )^{2}\right )}}{x} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x^{\frac {1}{3}} + a\right )^{2}}{x^{\frac {3}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.06, size = 87, normalized size = 0.75 \[ -\frac {1}{\sqrt {x}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{\sqrt {x}}-4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right ) \]

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 = 1.30, size = 87, normalized size = 0.75 \[ -\frac {\sqrt {2} {\left ({\left (\left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, 2 i \, b x^{\frac {1}{3}}\right ) - \left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (2 \, a\right ) + {\left (\left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, 2 i \, b x^{\frac {1}{3}}\right ) - \left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (2 \, a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b x^{\frac {1}{3}} + 4}{4 \, \sqrt {x}} \]

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*I - 3)*sqrt(2)*gamma(-3/2, 2*I*b*x^(1/3)) - (3*I + 3)*sqrt(2)*gamma(-3/2, -2*I*b*x^(1/3)))*
cos(2*a) + ((3*I + 3)*sqrt(2)*gamma(-3/2, 2*I*b*x^(1/3)) - (3*I - 3)*sqrt(2)*gamma(-3/2, -2*I*b*x^(1/3)))*sin(
2*a))*sqrt(b*x^(1/3))*b*x^(1/3) + 4)/sqrt(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{x^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^(1/3))^2/x^(3/2),x)

[Out]

int(cos(a + b*x^(1/3))^2/x^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {3}{2}}}\, dx \]

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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