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

Optimal. Leaf size=310 \[ \frac{405405 \sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{32768 b^{15/2}}+\frac{405405 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{32768 b^{15/2}}+\frac{39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac{405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac{135135 \sqrt{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac{39 x^{11/6}}{16 b^2}+\frac{3861 x^{7/6}}{256 b^4}-\frac{135135 \sqrt{x}}{4096 b^6}+\frac{x^{5/2}}{5} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.359734, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {3416, 3311, 30, 3312, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac{405405 \sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{32768 b^{15/2}}+\frac{405405 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{32768 b^{15/2}}+\frac{39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac{3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac{405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac{135135 \sqrt{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac{39 x^{11/6}}{16 b^2}+\frac{3861 x^{7/6}}{256 b^4}-\frac{135135 \sqrt{x}}{4096 b^6}+\frac{x^{5/2}}{5} \]

Antiderivative was successfully verified.

[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 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 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 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 x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx &=3 \operatorname{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} \operatorname{Subst}\left (\int x^{13/2} \, dx,x,\sqrt [3]{x}\right )-\frac{429 \operatorname{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 \operatorname{Subst}\left (\int x^{5/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{256 b^4}-\frac{429 \operatorname{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 \operatorname{Subst}\left (\int \sqrt{x} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{4096 b^6}+\frac{27027 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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)) \operatorname{Subst}\left (\int \frac{\sin (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{32768 b^7}+\frac{(405405 \sin (2 a)) \operatorname{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)) \operatorname{Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{16384 b^7}+\frac{(405405 \sin (2 a)) \operatorname{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) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{32768 b^{15/2}}+\frac{405405 \sqrt{\pi } C\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]  time = 0.658827, size = 174, normalized size = 0.56 \[ \frac{2 \sqrt{b} \sqrt [6]{x} \left (15 \left (4096 b^6 x^2-36608 b^4 x^{4/3}+144144 b^2 x^{2/3}-135135\right ) \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+780 \left (256 b^5 x^{5/3}-1584 b^3 x+3465 b \sqrt [3]{x}\right ) \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+16384 b^7 x^{7/3}\right )+2027025 \sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+2027025 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{163840 b^{15/2}} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.04, size = 219, normalized size = 0.7 \begin{align*}{\frac{1}{5}{x}^{{\frac{5}{2}}}}+{\frac{3}{4\,b}{x}^{{\frac{13}{6}}}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }-{\frac{39}{4\,b} \left ( -{\frac{1}{4\,b}{x}^{{\frac{11}{6}}}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }+{\frac{11}{4\,b} \left ({\frac{1}{4\,b}{x}^{{\frac{3}{2}}}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }-{\frac{9}{4\,b} \left ( -{\frac{1}{4\,b}{x}^{{\frac{7}{6}}}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }+{\frac{7}{4\,b} \left ({\frac{1}{4\,b}{x}^{{\frac{5}{6}}}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }-{\frac{5}{4\,b} \left ( -{\frac{1}{4\,b}\sqrt{x}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }+{\frac{3}{4\,b} \left ({\frac{1}{4\,b}\sqrt [6]{x}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }-{\frac{\sqrt{\pi }}{8} \left ( \cos \left ( 2\,a \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\sqrt{\pi }}} \right ) +\sin \left ( 2\,a \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\sqrt{\pi }}} \right ) \right ){b}^{-{\frac{3}{2}}}} \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)))))))))

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Maxima [C]  time = 2.14783, size = 497, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1310720*(262144*b^7*x^(5/2)*abs(b) - sqrt(2)*sqrt(pi)*(((-2027025*I*cos(1/4*pi + 1/2*arctan2(0, b)) - 202702
5*I*cos(-1/4*pi + 1/2*arctan2(0, b)) - 2027025*sin(1/4*pi + 1/2*arctan2(0, b)) + 2027025*sin(-1/4*pi + 1/2*arc
tan2(0, b)))*cos(2*a) - (2027025*cos(1/4*pi + 1/2*arctan2(0, b)) + 2027025*cos(-1/4*pi + 1/2*arctan2(0, b)) -
2027025*I*sin(1/4*pi + 1/2*arctan2(0, b)) + 2027025*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(2*a))*erf(sqrt(2*I
*b)*x^(1/6)) + ((2027025*I*cos(1/4*pi + 1/2*arctan2(0, b)) + 2027025*I*cos(-1/4*pi + 1/2*arctan2(0, b)) - 2027
025*sin(1/4*pi + 1/2*arctan2(0, b)) + 2027025*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(2*a) - (2027025*cos(1/4*pi
+ 1/2*arctan2(0, b)) + 2027025*cos(-1/4*pi + 1/2*arctan2(0, b)) + 2027025*I*sin(1/4*pi + 1/2*arctan2(0, b)) -
2027025*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(2*a))*erf(sqrt(-2*I*b)*x^(1/6)))*sqrt(abs(b)) + 12480*(256*b^
5*x^(11/6)*abs(b) - 1584*b^3*x^(7/6)*abs(b) + 3465*b*sqrt(x)*abs(b))*cos(2*b*x^(1/3) + 2*a) + 240*(4096*b^6*x^
(13/6)*abs(b) - 36608*b^4*x^(3/2)*abs(b) + 144144*b^2*x^(5/6)*abs(b) - 135135*x^(1/6)*abs(b))*sin(2*b*x^(1/3)
+ 2*a))/(b^7*abs(b))

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Fricas [A]  time = 2.25505, size = 591, normalized size = 1.91 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{3}{2}} \cos ^{2}{\left (a + b \sqrt [3]{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [C]  time = 1.18605, size = 302, normalized size = 0.97 \begin{align*} \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 i \, \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 )}}{65536 \, b^{\frac{15}{2}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}} - \frac{405405 i \, \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 )}}{65536 \, b^{\frac{15}{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^(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*I*sqrt(pi)*erf(-sqrt(b)*x^(1/6)*(-I*b/a
bs(b) + 1))*e^(2*I*a)/(b^(15/2)*(-I*b/abs(b) + 1)) - 405405/65536*I*sqrt(pi)*erf(-sqrt(b)*x^(1/6)*(I*b/abs(b)
+ 1))*e^(-2*I*a)/(b^(15/2)*(I*b/abs(b) + 1))