\(\int x^{3/2} \cos ^2(a+b \sqrt [3]{x}) \, dx\) [55]

Optimal result

Integrand size = 18, antiderivative size = 310 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {405405 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {405405 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{32768 b^{15/2}}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7} \]

[Out]

3861/256*x^(7/6)/b^4-39/16*x^(11/6)/b^2+1/5*x^(5/2)-3861/128*x^(7/6)*cos(a+b*x^(1/3))^2/b^4+39/8*x^(11/6)*cos(
a+b*x^(1/3))^2/b^2+27027/512*x^(5/6)*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/b^5-429/32*x^(3/2)*cos(a+b*x^(1/3))*sin
(a+b*x^(1/3))/b^3+3/2*x^(13/6)*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/b-405405/16384*x^(1/6)*sin(2*a+2*b*x^(1/3))/b
^7+405405/32768*cos(2*a)*FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(15/2)+405405/32768*FresnelC(2*x^(1/6
)*b^(1/2)/Pi^(1/2))*sin(2*a)*Pi^(1/2)/b^(15/2)-135135/4096*x^(1/2)/b^6+135135/2048*cos(a+b*x^(1/3))^2*x^(1/2)/
b^6

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3497, 3392, 30, 3393, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\frac {405405 \sqrt {\pi } \sin (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {405405 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5} \]

[In]

Int[x^(3/2)*Cos[a + b*x^(1/3)]^2,x]

[Out]

(-135135*Sqrt[x])/(4096*b^6) + (3861*x^(7/6))/(256*b^4) - (39*x^(11/6))/(16*b^2) + x^(5/2)/5 + (135135*Sqrt[x]
*Cos[a + b*x^(1/3)]^2)/(2048*b^6) - (3861*x^(7/6)*Cos[a + b*x^(1/3)]^2)/(128*b^4) + (39*x^(11/6)*Cos[a + b*x^(
1/3)]^2)/(8*b^2) + (405405*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(32768*b^(15/2)) + (40540
5*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/(32768*b^(15/2)) + (27027*x^(5/6)*Cos[a + b*x^(1/3
)]*Sin[a + b*x^(1/3)])/(512*b^5) - (429*x^(3/2)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(32*b^3) + (3*x^(13/6)*
Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(2*b) - (405405*x^(1/6)*Sin[2*(a + b*x^(1/3))])/(16384*b^7)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 3392

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 3393

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 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 ^2(a+b x) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {3}{2} \text {Subst}\left (\int x^{13/2} \, dx,x,\sqrt [3]{x}\right )-\frac {429 \text {Subst}\left (\int x^{9/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{16 b^2} \\ & = \frac {x^{5/2}}{5}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {27027 \text {Subst}\left (\int x^{5/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{256 b^4}-\frac {429 \text {Subst}\left (\int x^{9/2} \, dx,x,\sqrt [3]{x}\right )}{32 b^2} \\ & = -\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \text {Subst}\left (\int \sqrt {x} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{4096 b^6}+\frac {27027 \text {Subst}\left (\int x^{5/2} \, dx,x,\sqrt [3]{x}\right )}{512 b^4} \\ & = \frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \text {Subst}\left (\int \left (\frac {\sqrt {x}}{2}+\frac {1}{2} \sqrt {x} \cos (2 a+2 b x)\right ) \, dx,x,\sqrt [3]{x}\right )}{4096 b^6} \\ & = -\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \text {Subst}\left (\int \sqrt {x} \cos (2 a+2 b x) \, dx,x,\sqrt [3]{x}\right )}{8192 b^6} \\ & = -\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {405405 \text {Subst}\left (\int \frac {\sin (2 a+2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{32768 b^7} \\ & = -\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {(405405 \cos (2 a)) \text {Subst}\left (\int \frac {\sin (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{32768 b^7}+\frac {(405405 \sin (2 a)) \text {Subst}\left (\int \frac {\cos (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{32768 b^7} \\ & = -\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {(405405 \cos (2 a)) \text {Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{16384 b^7}+\frac {(405405 \sin (2 a)) \text {Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{16384 b^7} \\ & = -\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {405405 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {405405 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{32768 b^{15/2}}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.56 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\frac {2027025 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+2027025 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+2 \sqrt {b} \sqrt [6]{x} \left (16384 b^7 x^{7/3}+780 \left (3465 b \sqrt [3]{x}-1584 b^3 x+256 b^5 x^{5/3}\right ) \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+15 \left (-135135+144144 b^2 x^{2/3}-36608 b^4 x^{4/3}+4096 b^6 x^2\right ) \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )\right )}{163840 b^{15/2}} \]

[In]

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

[Out]

(2027025*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] + 2027025*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/
6))/Sqrt[Pi]]*Sin[2*a] + 2*Sqrt[b]*x^(1/6)*(16384*b^7*x^(7/3) + 780*(3465*b*x^(1/3) - 1584*b^3*x + 256*b^5*x^(
5/3))*Cos[2*(a + b*x^(1/3))] + 15*(-135135 + 144144*b^2*x^(2/3) - 36608*b^4*x^(4/3) + 4096*b^6*x^2)*Sin[2*(a +
b*x^(1/3))]))/(163840*b^(15/2))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {x^{\frac {5}{2}}}{5}+\frac {3 x^{\frac {13}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {39 \left (-\frac {x^{\frac {11}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {11 x^{\frac {3}{2}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {99 \left (-\frac {x^{\frac {7}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {7 x^{\frac {5}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {35 \left (-\frac {\sqrt {x}\, \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{32 b^{\frac {3}{2}}}}{b}\right )}{16 b}}{b}\right )}{16 b}}{b}\right )}{4 b}\)	\(219\)
default	\(\frac {x^{\frac {5}{2}}}{5}+\frac {3 x^{\frac {13}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {39 \left (-\frac {x^{\frac {11}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {11 x^{\frac {3}{2}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {99 \left (-\frac {x^{\frac {7}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {7 x^{\frac {5}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {35 \left (-\frac {\sqrt {x}\, \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{32 b^{\frac {3}{2}}}}{b}\right )}{16 b}}{b}\right )}{16 b}}{b}\right )}{4 b}\)	\(219\)

[In]

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

[Out]

1/5*x^(5/2)+3/4/b*x^(13/6)*sin(2*a+2*b*x^(1/3))-39/4/b*(-1/4/b*x^(11/6)*cos(2*a+2*b*x^(1/3))+11/4/b*(1/4/b*x^(
3/2)*sin(2*a+2*b*x^(1/3))-9/4/b*(-1/4/b*x^(7/6)*cos(2*a+2*b*x^(1/3))+7/4/b*(1/4/b*x^(5/6)*sin(2*a+2*b*x^(1/3))
-5/4/b*(-1/4/b*x^(1/2)*cos(2*a+2*b*x^(1/3))+3/4/b*(1/4*x^(1/6)*sin(2*a+2*b*x^(1/3))/b-1/8/b^(3/2)*Pi^(1/2)*(co
s(2*a)*FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))+sin(2*a)*FresnelC(2*x^(1/6)*b^(1/2)/Pi^(1/2)))))))))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.59 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=-\frac {399360 \, b^{6} x^{\frac {11}{6}} - 2471040 \, b^{4} x^{\frac {7}{6}} - 2027025 \, \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 2027025 \, \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 3120 \, {\left (256 \, b^{6} x^{\frac {11}{6}} - 1584 \, b^{4} x^{\frac {7}{6}} + 3465 \, b^{2} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 60 \, {\left (36608 \, b^{5} x^{\frac {3}{2}} - 144144 \, b^{3} x^{\frac {5}{6}} - {\left (4096 \, b^{7} x^{2} - 135135 \, b\right )} x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) - 8 \, {\left (4096 \, b^{8} x^{2} - 675675 \, b^{2}\right )} \sqrt {x}}{163840 \, b^{8}} \]

[In]

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

[Out]

-1/163840*(399360*b^6*x^(11/6) - 2471040*b^4*x^(7/6) - 2027025*pi*sqrt(b/pi)*cos(2*a)*fresnel_sin(2*x^(1/6)*sq
rt(b/pi)) - 2027025*pi*sqrt(b/pi)*fresnel_cos(2*x^(1/6)*sqrt(b/pi))*sin(2*a) - 3120*(256*b^6*x^(11/6) - 1584*b
^4*x^(7/6) + 3465*b^2*sqrt(x))*cos(b*x^(1/3) + a)^2 + 60*(36608*b^5*x^(3/2) - 144144*b^3*x^(5/6) - (4096*b^7*x
^2 - 135135*b)*x^(1/6))*cos(b*x^(1/3) + a)*sin(b*x^(1/3) + a) - 8*(4096*b^8*x^2 - 675675*b^2)*sqrt(x))/b^8

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.52 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\frac {262144 \, b^{9} x^{\frac {5}{2}} + 2027025 \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \cos \left (2 \, a\right ) - \left (i - 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (2 \, a\right ) + \left (i + 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 12480 \, {\left (256 \, b^{7} x^{\frac {11}{6}} - 1584 \, b^{5} x^{\frac {7}{6}} + 3465 \, b^{3} \sqrt {x}\right )} \cos \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right ) + 240 \, {\left (4096 \, b^{8} x^{\frac {13}{6}} - 36608 \, b^{6} x^{\frac {3}{2}} + 144144 \, b^{4} x^{\frac {5}{6}} - 135135 \, b^{2} x^{\frac {1}{6}}\right )} \sin \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right )}{1310720 \, b^{9}} \]

[In]

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

[Out]

1/1310720*(262144*b^9*x^(5/2) + 2027025*4^(1/4)*sqrt(2)*sqrt(pi)*(((I + 1)*cos(2*a) - (I - 1)*sin(2*a))*erf(sq
rt(2*I*b)*x^(1/6)) + (-(I - 1)*cos(2*a) + (I + 1)*sin(2*a))*erf(sqrt(-2*I*b)*x^(1/6)))*b^(3/2) + 12480*(256*b^
7*x^(11/6) - 1584*b^5*x^(7/6) + 3465*b^3*sqrt(x))*cos(2*b*x^(1/3) + 2*a) + 240*(4096*b^8*x^(13/6) - 36608*b^6*
x^(3/2) + 144144*b^4*x^(5/6) - 135135*b^2*x^(1/6))*sin(2*b*x^(1/3) + 2*a))/b^9

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.72 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\frac {1}{5} \, x^{\frac {5}{2}} - \frac {3 \, {\left (4096 i \, b^{6} x^{\frac {13}{6}} - 13312 \, b^{5} x^{\frac {11}{6}} - 36608 i \, b^{4} x^{\frac {3}{2}} + 82368 \, b^{3} x^{\frac {7}{6}} + 144144 i \, b^{2} x^{\frac {5}{6}} - 180180 \, b \sqrt {x} - 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (2 i \, b x^{\frac {1}{3}} + 2 i \, a\right )}}{32768 \, b^{7}} - \frac {3 \, {\left (-4096 i \, b^{6} x^{\frac {13}{6}} - 13312 \, b^{5} x^{\frac {11}{6}} + 36608 i \, b^{4} x^{\frac {3}{2}} + 82368 \, b^{3} x^{\frac {7}{6}} - 144144 i \, b^{2} x^{\frac {5}{6}} - 180180 \, b \sqrt {x} + 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (-2 i \, b x^{\frac {1}{3}} - 2 i \, a\right )}}{32768 \, b^{7}} + \frac {405405 \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {b} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{65536 \, b^{\frac {15}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} + \frac {405405 \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {b} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{65536 \, b^{\frac {15}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]

[In]

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

[Out]

1/5*x^(5/2) - 3/32768*(4096*I*b^6*x^(13/6) - 13312*b^5*x^(11/6) - 36608*I*b^4*x^(3/2) + 82368*b^3*x^(7/6) + 14
4144*I*b^2*x^(5/6) - 180180*b*sqrt(x) - 135135*I*x^(1/6))*e^(2*I*b*x^(1/3) + 2*I*a)/b^7 - 3/32768*(-4096*I*b^6
*x^(13/6) - 13312*b^5*x^(11/6) + 36608*I*b^4*x^(3/2) + 82368*b^3*x^(7/6) - 144144*I*b^2*x^(5/6) - 180180*b*sqr
t(x) + 135135*I*x^(1/6))*e^(-2*I*b*x^(1/3) - 2*I*a)/b^7 + 405405/65536*sqrt(pi)*erf(-I*sqrt(b)*x^(1/6)*(I*b/ab
s(b) + 1))*e^(2*I*a)/(b^(15/2)*(I*b/abs(b) + 1)) + 405405/65536*sqrt(pi)*erf(I*sqrt(b)*x^(1/6)*(-I*b/abs(b) +
1))*e^(-2*I*a)/(b^(15/2)*(-I*b/abs(b) + 1))

Mupad [F(-1)]

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

[In]

int(x^(3/2)*cos(a + b*x^(1/3))^2,x)

[Out]

int(x^(3/2)*cos(a + b*x^(1/3))^2, x)