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

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

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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 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 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 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 \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*}

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Mathematica [A]  time = 0.44, size = 148, normalized size = 0.68 \[ \frac {2 \sqrt {b} \sqrt [6]{x} \left (63 \left (16 b^2 x^{2/3}-15\right ) \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+4 b \sqrt [3]{x} \left (9 \left (16 b^2 x^{2/3}-35\right ) \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+64 b^3 x\right )\right )+945 \sqrt {\pi } \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-945 \sqrt {\pi } \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{1536 b^{9/2}} \]

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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fricas [A]  time = 0.73, size = 144, normalized size = 0.66 \[ \frac {512 \, b^{5} x^{\frac {3}{2}} - 2016 \, b^{3} x^{\frac {5}{6}} + 945 \, \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 945 \, \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) + 252 \, {\left (16 \, b^{3} x^{\frac {5}{6}} - 15 \, b x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 144 \, {\left (16 \, b^{4} x^{\frac {7}{6}} - 35 \, b^{2} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) + 1890 \, b x^{\frac {1}{6}}}{1536 \, b^{5}} \]

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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giac [C]  time = 0.45, size = 176, normalized size = 0.81 \[ \frac {1}{3} \, x^{\frac {3}{2}} - \frac {3 \, {\left (64 i \, b^{3} x^{\frac {7}{6}} - 112 \, b^{2} x^{\frac {5}{6}} - 140 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (2 i \, b x^{\frac {1}{3}} + 2 i \, a\right )}}{512 \, b^{4}} - \frac {3 \, {\left (-64 i \, b^{3} x^{\frac {7}{6}} - 112 \, b^{2} x^{\frac {5}{6}} + 140 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (-2 i \, b x^{\frac {1}{3}} - 2 i \, a\right )}}{512 \, b^{4}} - \frac {315 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{1024 \, b^{\frac {9}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {315 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{1024 \, b^{\frac {9}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]

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))

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maple [A]  time = 0.06, size = 145, normalized size = 0.67 \[ \frac {x^{\frac {3}{2}}}{3}+\frac {3 x^{\frac {7}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {21 \left (-\frac {x^{\frac {5}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {5 \sqrt {x}\, \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {15 \left (-\frac {x^{\frac {1}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\sqrt {\pi }\, \left (\cos \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}}{b}\right )}{4 b} \]

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 = 1.24, size = 137, normalized size = 0.63 \[ \frac {4096 \, b^{6} x^{\frac {3}{2}} - 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (945 i - 945\right ) \, \cos \left (2 \, a\right ) + \left (945 i + 945\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (945 i + 945\right ) \, \cos \left (2 \, a\right ) - \left (945 i - 945\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 1008 \, {\left (16 \, b^{4} x^{\frac {5}{6}} - 15 \, b^{2} x^{\frac {1}{6}}\right )} \cos \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right ) + 576 \, {\left (16 \, b^{5} x^{\frac {7}{6}} - 35 \, b^{3} \sqrt {x}\right )} \sin \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right )}{12288 \, b^{6}} \]

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^6*x^(3/2) - 4^(1/4)*sqrt(2)*sqrt(pi)*(((945*I - 945)*cos(2*a) + (945*I + 945)*sin(2*a))*erf(sq
rt(2*I*b)*x^(1/6)) + (-(945*I + 945)*cos(2*a) - (945*I - 945)*sin(2*a))*erf(sqrt(-2*I*b)*x^(1/6)))*b^(3/2) + 1
008*(16*b^4*x^(5/6) - 15*b^2*x^(1/6))*cos(2*b*x^(1/3) + 2*a) + 576*(16*b^5*x^(7/6) - 35*b^3*sqrt(x))*sin(2*b*x
^(1/3) + 2*a))/b^6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*cos(a + b*x^(1/3))^2,x)

[Out]

int(x^(1/2)*cos(a + b*x^(1/3))^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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