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

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

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 169, normalized size of antiderivative = 1.00, 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 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 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 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 \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*}

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Mathematica [A]  time = 0.35, size = 141, normalized size = 0.83 \[ \frac {6 \sqrt {b} \sqrt [6]{x} \left (2 b \sqrt [3]{x} \left (4 b^2 x^{2/3}-35\right ) \sin \left (a+b \sqrt [3]{x}\right )+7 \left (4 b^2 x^{2/3}-15\right ) \cos \left (a+b \sqrt [3]{x}\right )\right )+315 \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-315 \sqrt {2 \pi } \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{16 b^{9/2}} \]

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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fricas [A]  time = 0.77, size = 118, normalized size = 0.70 \[ \frac {3 \, {\left (105 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 105 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a) + 14 \, {\left (4 \, b^{3} x^{\frac {5}{6}} - 15 \, b x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) + 4 \, {\left (4 \, b^{4} x^{\frac {7}{6}} - 35 \, b^{2} \sqrt {x}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{16 \, b^{5}} \]

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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giac [C]  time = 0.50, size = 193, normalized size = 1.14 \[ -\frac {3 \, {\left (8 i \, b^{3} x^{\frac {7}{6}} - 28 \, b^{2} x^{\frac {5}{6}} - 70 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (i \, b x^{\frac {1}{3}} + i \, a\right )}}{16 \, b^{4}} - \frac {3 \, {\left (-8 i \, b^{3} x^{\frac {7}{6}} - 28 \, b^{2} x^{\frac {5}{6}} + 70 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (-i \, b x^{\frac {1}{3}} - i \, a\right )}}{16 \, b^{4}} - \frac {315 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{32 \, b^{4} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {315 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{32 \, b^{4} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} \]

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)))

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maple [A]  time = 0.03, size = 131, normalized size = 0.78 \[ \frac {3 x^{\frac {7}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{b}-\frac {21 \left (-\frac {x^{\frac {5}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {5 \sqrt {x}\, \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {15 \left (-\frac {x^{\frac {1}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}\right )}{4 b}}{b}\right )}{b} \]

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 = 1.24, size = 111, normalized size = 0.66 \[ -\frac {3 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (105 i - 105\right ) \, \cos \relax (a) + \left (105 i + 105\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (105 i + 105\right ) \, \cos \relax (a) - \left (105 i - 105\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} - 56 \, {\left (4 \, b^{4} x^{\frac {5}{6}} - 15 \, b^{2} x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) - 16 \, {\left (4 \, b^{5} x^{\frac {7}{6}} - 35 \, b^{3} \sqrt {x}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{64 \, b^{6}} \]

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(2)*sqrt(pi)*(((105*I - 105)*cos(a) + (105*I + 105)*sin(a))*erf(sqrt(I*b)*x^(1/6)) + (-(105*I + 105
)*cos(a) - (105*I - 105)*sin(a))*erf(sqrt(-I*b)*x^(1/6)))*b^(3/2) - 56*(4*b^4*x^(5/6) - 15*b^2*x^(1/6))*cos(b*
x^(1/3) + a) - 16*(4*b^5*x^(7/6) - 35*b^3*sqrt(x))*sin(b*x^(1/3) + a))/b^6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*cos(a + b*x^(1/3)),x)

[Out]

int(x^(1/2)*cos(a + b*x^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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