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

Optimal. Leaf size=169 \[ \frac{315 \sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{8 b^{9/2}}-\frac{315 \sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{8 b^{9/2}}+\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}-\frac{315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

[Out]

(-315*x^(1/6)*Cos[a + b*x^(1/3)])/(8*b^4) + (21*x^(5/6)*Cos[a + b*x^(1/3)])/(2*b^2) + (315*Sqrt[Pi/2]*Cos[a]*F
resnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/(8*b^(9/2)) - (315*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a]
)/(8*b^(9/2)) - (105*Sqrt[x]*Sin[a + b*x^(1/3)])/(4*b^3) + (3*x^(7/6)*Sin[a + b*x^(1/3)])/b

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Rubi [A]  time = 0.196254, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{315 \sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{8 b^{9/2}}-\frac{315 \sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{8 b^{9/2}}+\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}-\frac{315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*Cos[a + b*x^(1/3)],x]

[Out]

(-315*x^(1/6)*Cos[a + b*x^(1/3)])/(8*b^4) + (21*x^(5/6)*Cos[a + b*x^(1/3)])/(2*b^2) + (315*Sqrt[Pi/2]*Cos[a]*F
resnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/(8*b^(9/2)) - (315*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a]
)/(8*b^(9/2)) - (105*Sqrt[x]*Sin[a + b*x^(1/3)])/(4*b^3) + (3*x^(7/6)*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 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^{7/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{21 \operatorname{Subst}\left (\int x^{5/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{105 \operatorname{Subst}\left (\int x^{3/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{4 b^2}\\ &=\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{315 \operatorname{Subst}\left (\int \sqrt{x} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{8 b^3}\\ &=-\frac{315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{315 \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{16 b^4}\\ &=-\frac{315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{(315 \cos (a)) \operatorname{Subst}\left (\int \frac{\cos (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{16 b^4}-\frac{(315 \sin (a)) \operatorname{Subst}\left (\int \frac{\sin (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{16 b^4}\\ &=-\frac{315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{(315 \cos (a)) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{8 b^4}-\frac{(315 \sin (a)) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{8 b^4}\\ &=-\frac{315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac{315 \sqrt{\frac{\pi }{2}} \cos (a) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{8 b^{9/2}}-\frac{315 \sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{8 b^{9/2}}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.362793, size = 141, normalized size = 0.83 \[ \frac{6 \sqrt{b} \sqrt [6]{x} \left (2 b \sqrt [3]{x} \left (4 b^2 x^{2/3}-35\right ) \sin \left (a+b \sqrt [3]{x}\right )+7 \left (4 b^2 x^{2/3}-15\right ) \cos \left (a+b \sqrt [3]{x}\right )\right )+315 \sqrt{2 \pi } \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )-315 \sqrt{2 \pi } \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{16 b^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(315*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] - 315*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/
6)]*Sin[a] + 6*Sqrt[b]*x^(1/6)*(7*(-15 + 4*b^2*x^(2/3))*Cos[a + b*x^(1/3)] + 2*b*(-35 + 4*b^2*x^(2/3))*x^(1/3)
*Sin[a + b*x^(1/3)]))/(16*b^(9/2))

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Maple [A]  time = 0.03, size = 131, normalized size = 0.8 \begin{align*} 3\,{\frac{{x}^{7/6}\sin \left ( a+b\sqrt [3]{x} \right ) }{b}}-21\,{\frac{1}{b} \left ( -1/2\,{\frac{{x}^{5/6}\cos \left ( a+b\sqrt [3]{x} \right ) }{b}}+5/2\,{\frac{1}{b} \left ( 1/2\,{\frac{\sqrt{x}\sin \left ( a+b\sqrt [3]{x} \right ) }{b}}-3/2\,{\frac{1}{b} \left ( -1/2\,{\frac{\cos \left ( a+b\sqrt [3]{x} \right ) \sqrt [6]{x}}{b}}+1/4\,{\frac{\sqrt{2}\sqrt{\pi }}{{b}^{3/2}} \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 ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(7/6)*sin(a+b*x^(1/3))/b-21/b*(-1/2/b*x^(5/6)*cos(a+b*x^(1/3))+5/2/b*(1/2/b*x^(1/2)*sin(a+b*x^(1/3))-3/2/b
*(-1/2/b*x^(1/6)*cos(a+b*x^(1/3))+1/4/b^(3/2)*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 = 2.43454, size = 419, normalized size = 2.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/64*(sqrt(pi)*(((105*cos(1/4*pi + 1/2*arctan2(0, b)) + 105*cos(-1/4*pi + 1/2*arctan2(0, b)) - 105*I*sin(1/4*p
i + 1/2*arctan2(0, b)) + 105*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (105*I*cos(1/4*pi + 1/2*arctan2(0, b
)) + 105*I*cos(-1/4*pi + 1/2*arctan2(0, b)) + 105*sin(1/4*pi + 1/2*arctan2(0, b)) - 105*sin(-1/4*pi + 1/2*arct
an2(0, b)))*sin(a))*erf(sqrt(I*b)*x^(1/6)) + ((105*cos(1/4*pi + 1/2*arctan2(0, b)) + 105*cos(-1/4*pi + 1/2*arc
tan2(0, b)) + 105*I*sin(1/4*pi + 1/2*arctan2(0, b)) - 105*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (-105*I
*cos(1/4*pi + 1/2*arctan2(0, b)) - 105*I*cos(-1/4*pi + 1/2*arctan2(0, b)) + 105*sin(1/4*pi + 1/2*arctan2(0, b)
) - 105*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf(sqrt(-I*b)*x^(1/6)))*sqrt(abs(b)) + 56*(4*b^2*x^(5/6)*ab
s(b) - 15*x^(1/6)*abs(b))*cos(b*x^(1/3) + a) + 16*(4*b^3*x^(7/6)*abs(b) - 35*b*sqrt(x)*abs(b))*sin(b*x^(1/3) +
a))/(b^4*abs(b))

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Fricas [A]  time = 1.7632, size = 366, normalized size = 2.17 \begin{align*} \frac{3 \,{\left (105 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) - 105 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) + 14 \,{\left (4 \, b^{3} x^{\frac{5}{6}} - 15 \, b x^{\frac{1}{6}}\right )} \cos \left (b x^{\frac{1}{3}} + a\right ) + 4 \,{\left (4 \, b^{4} x^{\frac{7}{6}} - 35 \, b^{2} \sqrt{x}\right )} \sin \left (b x^{\frac{1}{3}} + a\right )\right )}}{16 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*(105*sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x^(1/6)*sqrt(b/pi)) - 105*sqrt(2)*pi*sqrt(b/pi)*fre
snel_sin(sqrt(2)*x^(1/6)*sqrt(b/pi))*sin(a) + 14*(4*b^3*x^(5/6) - 15*b*x^(1/6))*cos(b*x^(1/3) + a) + 4*(4*b^4*
x^(7/6) - 35*b^2*sqrt(x))*sin(b*x^(1/3) + a))/b^5

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.15468, size = 261, normalized size = 1.54 \begin{align*} -\frac{3 \,{\left (8 i \, b^{3} x^{\frac{7}{6}} - 28 \, b^{2} x^{\frac{5}{6}} - 70 i \, b \sqrt{x} + 105 \, x^{\frac{1}{6}}\right )} e^{\left (i \, b x^{\frac{1}{3}} + i \, a\right )}}{16 \, b^{4}} - \frac{3 \,{\left (-8 i \, b^{3} x^{\frac{7}{6}} - 28 \, b^{2} x^{\frac{5}{6}} + 70 i \, b \sqrt{x} + 105 \, x^{\frac{1}{6}}\right )} e^{\left (-i \, b x^{\frac{1}{3}} - i \, a\right )}}{16 \, b^{4}} - \frac{315 \, \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 )}}{32 \, b^{4}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} - \frac{315 \, \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 )}}{32 \, b^{4}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*(8*I*b^3*x^(7/6) - 28*b^2*x^(5/6) - 70*I*b*sqrt(x) + 105*x^(1/6))*e^(I*b*x^(1/3) + I*a)/b^4 - 3/16*(-8*I
*b^3*x^(7/6) - 28*b^2*x^(5/6) + 70*I*b*sqrt(x) + 105*x^(1/6))*e^(-I*b*x^(1/3) - I*a)/b^4 - 315/32*sqrt(2)*sqrt
(pi)*erf(-1/2*sqrt(2)*x^(1/6)*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(b^4*(-I*b/abs(b) + 1)*sqrt(abs(b))) - 3
15/32*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x^(1/6)*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(-I*a)/(b^4*(I*b/abs(b) + 1)*
sqrt(abs(b)))