3.1.52 \(\int \frac {\cos (a+b \sqrt [3]{x})}{x^{3/2}} \, dx\) [52]

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

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 = {3497, 3378, 3387, 3386, 3432, 3385, 3433} \begin {gather*} -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}} \end {gather*}

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 3378

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 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 )}{x^{3/2}} \, dx &=3 \text {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) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {\cos (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )+\left (4 b^2 \sin (a)\right ) \text {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 ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )+\left (8 b^2 \sin (a)\right ) \text {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*}

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Mathematica [A]
time = 0.17, size = 110, normalized size = 1.00 \begin {gather*} -\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}-4 b^{3/2} \sqrt {2 \pi } \cos (a) \text {FresnelC}\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 {gather*}

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 = 0.05, size = 78, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{\sqrt {x}}-4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )\)	\(78\)
default	\(-\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{\sqrt {x}}-4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )\)	\(78\)
meijerg	\(\frac {3 \cos \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (b^{2}\right )^{\frac {3}{4}} \left (-\frac {8 \sqrt {2}\, \cos \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, \sqrt {x}\, \left (b^{2}\right )^{\frac {3}{4}}}+\frac {16 \sqrt {2}\, b \sin \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, x^{\frac {1}{6}} \left (b^{2}\right )^{\frac {3}{4}}}-\frac {32 b^{\frac {3}{2}} \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{3 \left (b^{2}\right )^{\frac {3}{4}}}\right )}{8}-\frac {3 \sin \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, b^{\frac {3}{2}} \left (-\frac {16 \sqrt {2}\, \cos \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, x^{\frac {1}{6}} \sqrt {b}}-\frac {8 \sqrt {2}\, \sin \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, \sqrt {x}\, b^{\frac {3}{2}}}-\frac {32 \,\mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{3}\right )}{8}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Result contains complex when optimal does not.
time = 0.58, size = 74, normalized size = 0.67 \begin {gather*} -\frac {3 \, {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (a\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b}{4 \, x^{\frac {1}{6}}} \end {gather*}

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*(((I - 1)*sqrt(2)*gamma(-3/2, I*b*x^(1/3)) - (I + 1)*sqrt(2)*gamma(-3/2, -I*b*x^(1/3)))*cos(a) + ((I + 1)
*sqrt(2)*gamma(-3/2, I*b*x^(1/3)) - (I - 1)*sqrt(2)*gamma(-3/2, -I*b*x^(1/3)))*sin(a))*sqrt(b*x^(1/3))*b/x^(1/
6)

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Fricas [A]
time = 0.36, size = 96, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {3}{2}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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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 )}{x^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^(1/3))/x^(3/2),x)

[Out]

int(cos(a + b*x^(1/3))/x^(3/2), x)

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