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

Optimal. Leaf size=218 \[ \frac{315 \sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{512 b^{9/2}}-\frac{315 \sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{512 b^{9/2}}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac{21 x^{5/6}}{16 b^2}+\frac{315 \sqrt [6]{x}}{256 b^4}+\frac{x^{3/2}}{3} \]

[Out]

(315*x^(1/6))/(256*b^4) - (21*x^(5/6))/(16*b^2) + x^(3/2)/3 - (315*x^(1/6)*Cos[a + b*x^(1/3)]^2)/(128*b^4) + (
21*x^(5/6)*Cos[a + b*x^(1/3)]^2)/(8*b^2) + (315*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(512
*b^(9/2)) - (315*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/(512*b^(9/2)) - (105*Sqrt[x]*Cos[a
+ b*x^(1/3)]*Sin[a + b*x^(1/3)])/(32*b^3) + (3*x^(7/6)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(2*b)

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Rubi [A]  time = 0.250679, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3416, 3311, 30, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac{315 \sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{512 b^{9/2}}-\frac{315 \sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{512 b^{9/2}}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac{105 \sqrt{x} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac{21 x^{5/6}}{16 b^2}+\frac{315 \sqrt [6]{x}}{256 b^4}+\frac{x^{3/2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*Cos[a + b*x^(1/3)]^2,x]

[Out]

(315*x^(1/6))/(256*b^4) - (21*x^(5/6))/(16*b^2) + x^(3/2)/3 - (315*x^(1/6)*Cos[a + b*x^(1/3)]^2)/(128*b^4) + (
21*x^(5/6)*Cos[a + b*x^(1/3)]^2)/(8*b^2) + (315*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(512
*b^(9/2)) - (315*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/(512*b^(9/2)) - (105*Sqrt[x]*Cos[a
+ b*x^(1/3)]*Sin[a + b*x^(1/3)])/(32*b^3) + (3*x^(7/6)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(2*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 3311

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

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 \sqrt{x} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^{7/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac{3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac{3}{2} \operatorname{Subst}\left (\int x^{7/2} \, dx,x,\sqrt [3]{x}\right )-\frac{105 \operatorname{Subst}\left (\int x^{3/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{16 b^2}\\ &=\frac{x^{3/2}}{3}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{105 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac{315 \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{256 b^4}-\frac{105 \operatorname{Subst}\left (\int x^{3/2} \, dx,x,\sqrt [3]{x}\right )}{32 b^2}\\ &=-\frac{21 x^{5/6}}{16 b^2}+\frac{x^{3/2}}{3}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{105 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac{315 \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 )}{256 b^4}\\ &=\frac{315 \sqrt [6]{x}}{256 b^4}-\frac{21 x^{5/6}}{16 b^2}+\frac{x^{3/2}}{3}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{105 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac{315 \operatorname{Subst}\left (\int \frac{\cos (2 a+2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{512 b^4}\\ &=\frac{315 \sqrt [6]{x}}{256 b^4}-\frac{21 x^{5/6}}{16 b^2}+\frac{x^{3/2}}{3}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{105 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac{(315 \cos (2 a)) \operatorname{Subst}\left (\int \frac{\cos (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{512 b^4}-\frac{(315 \sin (2 a)) \operatorname{Subst}\left (\int \frac{\sin (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{512 b^4}\\ &=\frac{315 \sqrt [6]{x}}{256 b^4}-\frac{21 x^{5/6}}{16 b^2}+\frac{x^{3/2}}{3}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{105 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac{(315 \cos (2 a)) \operatorname{Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{256 b^4}-\frac{(315 \sin (2 a)) \operatorname{Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{256 b^4}\\ &=\frac{315 \sqrt [6]{x}}{256 b^4}-\frac{21 x^{5/6}}{16 b^2}+\frac{x^{3/2}}{3}-\frac{315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac{21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac{315 \sqrt{\pi } \cos (2 a) C\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{512 b^{9/2}}-\frac{315 \sqrt{\pi } S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right ) \sin (2 a)}{512 b^{9/2}}-\frac{105 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.442387, size = 148, normalized size = 0.68 \[ \frac{2 \sqrt{b} \sqrt [6]{x} \left (4 b \sqrt [3]{x} \left (9 \left (16 b^2 x^{2/3}-35\right ) \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+64 b^3 x\right )+63 \left (16 b^2 x^{2/3}-15\right ) \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )\right )+945 \sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-945 \sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{1536 b^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(945*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] - 945*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt
[Pi]]*Sin[2*a] + 2*Sqrt[b]*x^(1/6)*(63*(-15 + 16*b^2*x^(2/3))*Cos[2*(a + b*x^(1/3))] + 4*b*x^(1/3)*(64*b^3*x +
9*(-35 + 16*b^2*x^(2/3))*Sin[2*(a + b*x^(1/3))])))/(1536*b^(9/2))

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Maple [A]  time = 0.037, size = 145, normalized size = 0.7 \begin{align*}{\frac{1}{3}{x}^{{\frac{3}{2}}}}+{\frac{3}{4\,b}{x}^{{\frac{7}{6}}}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }-{\frac{21}{4\,b} \left ( -{\frac{1}{4\,b}{x}^{{\frac{5}{6}}}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }+{\frac{5}{4\,b} \left ({\frac{1}{4\,b}\sqrt{x}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }-{\frac{3}{4\,b} \left ( -{\frac{1}{4\,b}\sqrt [6]{x}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }+{\frac{\sqrt{\pi }}{8} \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 ){b}^{-{\frac{3}{2}}}} \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))^2,x)

[Out]

1/3*x^(3/2)+3/4/b*x^(7/6)*sin(2*a+2*b*x^(1/3))-21/4/b*(-1/4/b*x^(5/6)*cos(2*a+2*b*x^(1/3))+5/4/b*(1/4/b*x^(1/2
)*sin(2*a+2*b*x^(1/3))-3/4/b*(-1/4/b*x^(1/6)*cos(2*a+2*b*x^(1/3))+1/8/b^(3/2)*Pi^(1/2)*(cos(2*a)*FresnelC(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 = 1.96806, size = 455, normalized size = 2.09 \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))^2,x, algorithm="maxima")

[Out]

1/12288*(4096*b^4*x^(3/2)*abs(b) + sqrt(2)*sqrt(pi)*(((945*cos(1/4*pi + 1/2*arctan2(0, b)) + 945*cos(-1/4*pi +
1/2*arctan2(0, b)) - 945*I*sin(1/4*pi + 1/2*arctan2(0, b)) + 945*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(2*a)
- (945*I*cos(1/4*pi + 1/2*arctan2(0, b)) + 945*I*cos(-1/4*pi + 1/2*arctan2(0, b)) + 945*sin(1/4*pi + 1/2*arct
an2(0, b)) - 945*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(2*a))*erf(sqrt(2*I*b)*x^(1/6)) + ((945*cos(1/4*pi + 1/2
*arctan2(0, b)) + 945*cos(-1/4*pi + 1/2*arctan2(0, b)) + 945*I*sin(1/4*pi + 1/2*arctan2(0, b)) - 945*I*sin(-1/
4*pi + 1/2*arctan2(0, b)))*cos(2*a) - (-945*I*cos(1/4*pi + 1/2*arctan2(0, b)) - 945*I*cos(-1/4*pi + 1/2*arctan
2(0, b)) + 945*sin(1/4*pi + 1/2*arctan2(0, b)) - 945*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(2*a))*erf(sqrt(-2*I
*b)*x^(1/6)))*sqrt(abs(b)) + 1008*(16*b^2*x^(5/6)*abs(b) - 15*x^(1/6)*abs(b))*cos(2*b*x^(1/3) + 2*a) + 576*(16
*b^3*x^(7/6)*abs(b) - 35*b*sqrt(x)*abs(b))*sin(2*b*x^(1/3) + 2*a))/(b^4*abs(b))

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Fricas [A]  time = 2.08488, size = 444, normalized size = 2.04 \begin{align*} \frac{512 \, b^{5} x^{\frac{3}{2}} - 2016 \, b^{3} x^{\frac{5}{6}} + 945 \, \pi \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{C}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) - 945 \, \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (2 \, a\right ) + 252 \,{\left (16 \, b^{3} x^{\frac{5}{6}} - 15 \, b x^{\frac{1}{6}}\right )} \cos \left (b x^{\frac{1}{3}} + a\right )^{2} + 144 \,{\left (16 \, b^{4} x^{\frac{7}{6}} - 35 \, b^{2} \sqrt{x}\right )} \cos \left (b x^{\frac{1}{3}} + a\right ) \sin \left (b x^{\frac{1}{3}} + a\right ) + 1890 \, b x^{\frac{1}{6}}}{1536 \, 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))^2,x, algorithm="fricas")

[Out]

1/1536*(512*b^5*x^(3/2) - 2016*b^3*x^(5/6) + 945*pi*sqrt(b/pi)*cos(2*a)*fresnel_cos(2*x^(1/6)*sqrt(b/pi)) - 94
5*pi*sqrt(b/pi)*fresnel_sin(2*x^(1/6)*sqrt(b/pi))*sin(2*a) + 252*(16*b^3*x^(5/6) - 15*b*x^(1/6))*cos(b*x^(1/3)
+ a)^2 + 144*(16*b^4*x^(7/6) - 35*b^2*sqrt(x))*cos(b*x^(1/3) + a)*sin(b*x^(1/3) + a) + 1890*b*x^(1/6))/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))**2,x)

[Out]

Timed out

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Giac [C]  time = 1.17979, size = 238, normalized size = 1.09 \begin{align*} \frac{1}{3} \, x^{\frac{3}{2}} - \frac{3 \,{\left (64 i \, b^{3} x^{\frac{7}{6}} - 112 \, b^{2} x^{\frac{5}{6}} - 140 i \, b \sqrt{x} + 105 \, x^{\frac{1}{6}}\right )} e^{\left (2 i \, b x^{\frac{1}{3}} + 2 i \, a\right )}}{512 \, b^{4}} - \frac{3 \,{\left (-64 i \, b^{3} x^{\frac{7}{6}} - 112 \, b^{2} x^{\frac{5}{6}} + 140 i \, b \sqrt{x} + 105 \, x^{\frac{1}{6}}\right )} e^{\left (-2 i \, b x^{\frac{1}{3}} - 2 i \, a\right )}}{512 \, b^{4}} - \frac{315 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{b} x^{\frac{1}{6}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{1024 \, b^{\frac{9}{2}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}} - \frac{315 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{b} x^{\frac{1}{6}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{1024 \, b^{\frac{9}{2}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^(3/2) - 3/512*(64*I*b^3*x^(7/6) - 112*b^2*x^(5/6) - 140*I*b*sqrt(x) + 105*x^(1/6))*e^(2*I*b*x^(1/3) + 2*
I*a)/b^4 - 3/512*(-64*I*b^3*x^(7/6) - 112*b^2*x^(5/6) + 140*I*b*sqrt(x) + 105*x^(1/6))*e^(-2*I*b*x^(1/3) - 2*I
*a)/b^4 - 315/1024*sqrt(pi)*erf(-sqrt(b)*x^(1/6)*(-I*b/abs(b) + 1))*e^(2*I*a)/(b^(9/2)*(-I*b/abs(b) + 1)) - 31
5/1024*sqrt(pi)*erf(-sqrt(b)*x^(1/6)*(I*b/abs(b) + 1))*e^(-2*I*a)/(b^(9/2)*(I*b/abs(b) + 1))