\(\int x^{3/2} \cos (a+b \sqrt [3]{x}) \, dx\) [49]

Optimal result

Integrand size = 16, antiderivative size = 235 \[ \int x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx=\frac {135135 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {405405 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{64 b^{15/2}}+\frac {405405 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{64 b^{15/2}}-\frac {405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

[Out]

-3861/8*x^(7/6)*cos(a+b*x^(1/3))/b^4+39/2*x^(11/6)*cos(a+b*x^(1/3))/b^2-405405/64*x^(1/6)*sin(a+b*x^(1/3))/b^7
+27027/16*x^(5/6)*sin(a+b*x^(1/3))/b^5-429/4*x^(3/2)*sin(a+b*x^(1/3))/b^3+3*x^(13/6)*sin(a+b*x^(1/3))/b+405405
/128*cos(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(15/2)+405405/128*FresnelC(x^(1/6)*b
^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/b^(15/2)+135135/32*cos(a+b*x^(1/3))*x^(1/2)/b^6

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3497, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx=\frac {405405 \sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{64 b^{15/2}}+\frac {405405 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{64 b^{15/2}}-\frac {405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac {135135 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

[In]

Int[x^(3/2)*Cos[a + b*x^(1/3)],x]

[Out]

(135135*Sqrt[x]*Cos[a + b*x^(1/3)])/(32*b^6) - (3861*x^(7/6)*Cos[a + b*x^(1/3)])/(8*b^4) + (39*x^(11/6)*Cos[a
+ b*x^(1/3)])/(2*b^2) + (405405*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/(64*b^(15/2)) + (40540
5*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/(64*b^(15/2)) - (405405*x^(1/6)*Sin[a + b*x^(1/3)])/
(64*b^7) + (27027*x^(5/6)*Sin[a + b*x^(1/3)])/(16*b^5) - (429*x^(3/2)*Sin[a + b*x^(1/3)])/(4*b^3) + (3*x^(13/6
)*Sin[a + b*x^(1/3)])/b

Rule 3377

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 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*} \text {integral}& = 3 \text {Subst}\left (\int x^{13/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {39 \text {Subst}\left (\int x^{11/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{2 b} \\ & = \frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {429 \text {Subst}\left (\int x^{9/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{4 b^2} \\ & = \frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {3861 \text {Subst}\left (\int x^{7/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{8 b^3} \\ & = -\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {27027 \text {Subst}\left (\int x^{5/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{16 b^4} \\ & = -\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {135135 \text {Subst}\left (\int x^{3/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{32 b^5} \\ & = \frac {135135 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {405405 \text {Subst}\left (\int \sqrt {x} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{64 b^6} \\ & = \frac {135135 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {405405 \text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{128 b^7} \\ & = \frac {135135 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {(405405 \cos (a)) \text {Subst}\left (\int \frac {\sin (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{128 b^7}+\frac {(405405 \sin (a)) \text {Subst}\left (\int \frac {\cos (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{128 b^7} \\ & = \frac {135135 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {(405405 \cos (a)) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{64 b^7}+\frac {(405405 \sin (a)) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{64 b^7} \\ & = \frac {135135 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac {3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {405405 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{64 b^{15/2}}+\frac {405405 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{64 b^{15/2}}-\frac {405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.70 \[ \int x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx=\frac {405405 \sqrt {2 \pi } \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )+405405 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)+6 \sqrt {b} \sqrt [6]{x} \left (26 \left (3465 b \sqrt [3]{x}-396 b^3 x+16 b^5 x^{5/3}\right ) \cos \left (a+b \sqrt [3]{x}\right )+\left (-135135+36036 b^2 x^{2/3}-2288 b^4 x^{4/3}+64 b^6 x^2\right ) \sin \left (a+b \sqrt [3]{x}\right )\right )}{128 b^{15/2}} \]

[In]

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

[Out]

(405405*Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] + 405405*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]
*x^(1/6)]*Sin[a] + 6*Sqrt[b]*x^(1/6)*(26*(3465*b*x^(1/3) - 396*b^3*x + 16*b^5*x^(5/3))*Cos[a + b*x^(1/3)] + (-
135135 + 36036*b^2*x^(2/3) - 2288*b^4*x^(4/3) + 64*b^6*x^2)*Sin[a + b*x^(1/3)]))/(128*b^(15/2))

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {3 x^{\frac {13}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{b}-\frac {39 \left (-\frac {x^{\frac {11}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {11 x^{\frac {3}{2}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {99 \left (-\frac {x^{\frac {7}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {7 x^{\frac {5}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {35 \left (-\frac {\sqrt {x}\, \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {3 x^{\frac {1}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}}{b}\right )}{4 b}}{b}\right )}{4 b}}{b}\right )}{b}\)	\(196\)
default	\(\frac {3 x^{\frac {13}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{b}-\frac {39 \left (-\frac {x^{\frac {11}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {11 x^{\frac {3}{2}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {99 \left (-\frac {x^{\frac {7}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {7 x^{\frac {5}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {35 \left (-\frac {\sqrt {x}\, \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {3 x^{\frac {1}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}}{b}\right )}{4 b}}{b}\right )}{4 b}}{b}\right )}{b}\)	\(196\)
meijerg	\(\frac {192 \sqrt {2}\, \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {\sqrt {x}\, \sqrt {2}\, \left (b^{2}\right )^{\frac {15}{4}} \left (3120 x^{\frac {4}{3}} b^{4}-77220 x^{\frac {2}{3}} b^{2}+675675\right ) \cos \left (b \,x^{\frac {1}{3}}\right )}{61440 \sqrt {\pi }\, b^{6}}-\frac {x^{\frac {1}{6}} \sqrt {2}\, \left (b^{2}\right )^{\frac {15}{4}} \left (-960 x^{2} b^{6}+34320 x^{\frac {4}{3}} b^{4}-540540 x^{\frac {2}{3}} b^{2}+2027025\right ) \sin \left (b \,x^{\frac {1}{3}}\right )}{122880 \sqrt {\pi }\, b^{7}}+\frac {135135 \left (b^{2}\right )^{\frac {15}{4}} \operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{8192 b^{\frac {15}{2}}}\right )}{\left (b^{2}\right )^{\frac {15}{4}}}-\frac {192 \sqrt {2}\, \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {x^{\frac {1}{6}} \sqrt {2}\, \sqrt {b}\, \left (-1088 x^{2} b^{6}+38896 x^{\frac {4}{3}} b^{4}-612612 x^{\frac {2}{3}} b^{2}+2297295\right ) \cos \left (b \,x^{\frac {1}{3}}\right )}{139264 \sqrt {\pi }}+\frac {\sqrt {x}\, \sqrt {2}\, b^{\frac {3}{2}} \left (3536 x^{\frac {4}{3}} b^{4}-87516 x^{\frac {2}{3}} b^{2}+765765\right ) \sin \left (b \,x^{\frac {1}{3}}\right )}{69632 \sqrt {\pi }}-\frac {135135 \,\operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{8192}\right )}{b^{\frac {15}{2}}}\)	\(250\)

[In]

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

[Out]

3*x^(13/6)*sin(a+b*x^(1/3))/b-39/b*(-1/2/b*x^(11/6)*cos(a+b*x^(1/3))+11/2/b*(1/2/b*x^(3/2)*sin(a+b*x^(1/3))-9/
2/b*(-1/2/b*x^(7/6)*cos(a+b*x^(1/3))+7/2/b*(1/2/b*x^(5/6)*sin(a+b*x^(1/3))-5/2/b*(-1/2/b*x^(1/2)*cos(a+b*x^(1/
3))+3/2/b*(1/2*x^(1/6)*sin(a+b*x^(1/3))/b-1/4/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2
)/Pi^(1/2))+sin(a)*FresnelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2)))))))))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.62 \[ \int x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx=\frac {3 \, {\left (135135 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + 135135 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) + 52 \, {\left (16 \, b^{6} x^{\frac {11}{6}} - 396 \, b^{4} x^{\frac {7}{6}} + 3465 \, b^{2} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) - 2 \, {\left (2288 \, b^{5} x^{\frac {3}{2}} - 36036 \, b^{3} x^{\frac {5}{6}} - {\left (64 \, b^{7} x^{2} - 135135 \, b\right )} x^{\frac {1}{6}}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{128 \, b^{8}} \]

[In]

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

[Out]

3/128*(135135*sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_sin(sqrt(2)*x^(1/6)*sqrt(b/pi)) + 135135*sqrt(2)*pi*sqrt(b/
pi)*fresnel_cos(sqrt(2)*x^(1/6)*sqrt(b/pi))*sin(a) + 52*(16*b^6*x^(11/6) - 396*b^4*x^(7/6) + 3465*b^2*sqrt(x))
*cos(b*x^(1/3) + a) - 2*(2288*b^5*x^(3/2) - 36036*b^3*x^(5/6) - (64*b^7*x^2 - 135135*b)*x^(1/6))*sin(b*x^(1/3)
+ a))/b^8

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.58 \[ \int x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx=\frac {3 \, {\left (135135 \, \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 208 \, {\left (16 \, b^{7} x^{\frac {11}{6}} - 396 \, b^{5} x^{\frac {7}{6}} + 3465 \, b^{3} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) + 8 \, {\left (64 \, b^{8} x^{\frac {13}{6}} - 2288 \, b^{6} x^{\frac {3}{2}} + 36036 \, b^{4} x^{\frac {5}{6}} - 135135 \, b^{2} x^{\frac {1}{6}}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{512 \, b^{9}} \]

[In]

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

[Out]

3/512*(135135*sqrt(2)*sqrt(pi)*(((I + 1)*cos(a) - (I - 1)*sin(a))*erf(sqrt(I*b)*x^(1/6)) + (-(I - 1)*cos(a) +
(I + 1)*sin(a))*erf(sqrt(-I*b)*x^(1/6)))*b^(3/2) + 208*(16*b^7*x^(11/6) - 396*b^5*x^(7/6) + 3465*b^3*sqrt(x))*
cos(b*x^(1/3) + a) + 8*(64*b^8*x^(13/6) - 2288*b^6*x^(3/2) + 36036*b^4*x^(5/6) - 135135*b^2*x^(1/6))*sin(b*x^(
1/3) + a))/b^9

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.03 \[ \int x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx=-\frac {3 \, {\left (64 i \, b^{6} x^{\frac {13}{6}} - 416 \, b^{5} x^{\frac {11}{6}} - 2288 i \, b^{4} x^{\frac {3}{2}} + 10296 \, b^{3} x^{\frac {7}{6}} + 36036 i \, b^{2} x^{\frac {5}{6}} - 90090 \, b \sqrt {x} - 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (i \, b x^{\frac {1}{3}} + i \, a\right )}}{128 \, b^{7}} - \frac {3 \, {\left (-64 i \, b^{6} x^{\frac {13}{6}} - 416 \, b^{5} x^{\frac {11}{6}} + 2288 i \, b^{4} x^{\frac {3}{2}} + 10296 \, b^{3} x^{\frac {7}{6}} - 36036 i \, b^{2} x^{\frac {5}{6}} - 90090 \, b \sqrt {x} + 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (-i \, b x^{\frac {1}{3}} - i \, a\right )}}{128 \, b^{7}} + \frac {405405 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{256 \, b^{7} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {405405 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{256 \, b^{7} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} \]

[In]

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

[Out]

-3/128*(64*I*b^6*x^(13/6) - 416*b^5*x^(11/6) - 2288*I*b^4*x^(3/2) + 10296*b^3*x^(7/6) + 36036*I*b^2*x^(5/6) -
90090*b*sqrt(x) - 135135*I*x^(1/6))*e^(I*b*x^(1/3) + I*a)/b^7 - 3/128*(-64*I*b^6*x^(13/6) - 416*b^5*x^(11/6) +
2288*I*b^4*x^(3/2) + 10296*b^3*x^(7/6) - 36036*I*b^2*x^(5/6) - 90090*b*sqrt(x) + 135135*I*x^(1/6))*e^(-I*b*x^
(1/3) - I*a)/b^7 + 405405/256*sqrt(2)*sqrt(pi)*erf(-1/2*I*sqrt(2)*x^(1/6)*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*
a)/(b^7*(I*b/abs(b) + 1)*sqrt(abs(b))) + 405405/256*sqrt(2)*sqrt(pi)*erf(1/2*I*sqrt(2)*x^(1/6)*(-I*b/abs(b) +
1)*sqrt(abs(b)))*e^(-I*a)/(b^7*(-I*b/abs(b) + 1)*sqrt(abs(b)))

Mupad [F(-1)]

Timed out. \[ \int x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx=\int x^{3/2}\,\cos \left (a+b\,x^{1/3}\right ) \,d x \]

[In]

int(x^(3/2)*cos(a + b*x^(1/3)),x)

[Out]

int(x^(3/2)*cos(a + b*x^(1/3)), x)