3.60 \(\int \frac{\cos ^2(a+b \sqrt [3]{x})}{x^{7/2}} \, dx\)

Optimal. Leaf size=328 \[ \frac{32768 \sqrt{\pi } b^{15/2} \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{675675}-\frac{32768 \sqrt{\pi } b^{15/2} \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{675675}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac{2048 b^5 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{128 b^3 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}-\frac{32768 b^7 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{16 b^2}{715 x^{11/6}}-\frac{4096 b^6}{675675 \sqrt{x}} \]

[Out]

(-16*b^2)/(715*x^(11/6)) + (256*b^4)/(45045*x^(7/6)) - (4096*b^6)/(675675*Sqrt[x]) - (2*Cos[a + b*x^(1/3)]^2)/
(5*x^(5/2)) + (32*b^2*Cos[a + b*x^(1/3)]^2)/(715*x^(11/6)) - (512*b^4*Cos[a + b*x^(1/3)]^2)/(45045*x^(7/6)) +
(8192*b^6*Cos[a + b*x^(1/3)]^2)/(675675*Sqrt[x]) + (32768*b^(15/2)*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/
6))/Sqrt[Pi]])/675675 - (32768*b^(15/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/675675 + (8*
b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(65*x^(13/6)) - (128*b^3*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(6435
*x^(3/2)) + (2048*b^5*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(225225*x^(5/6)) - (32768*b^7*Cos[a + b*x^(1/3)]*
Sin[a + b*x^(1/3)])/(675675*x^(1/6))

________________________________________________________________________________________

Rubi [A]  time = 0.350694, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3416, 3314, 30, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac{32768 \sqrt{\pi } b^{15/2} \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{675675}-\frac{32768 \sqrt{\pi } b^{15/2} \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{675675}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac{2048 b^5 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{128 b^3 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}-\frac{32768 b^7 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{16 b^2}{715 x^{11/6}}-\frac{4096 b^6}{675675 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^(1/3)]^2/x^(7/2),x]

[Out]

(-16*b^2)/(715*x^(11/6)) + (256*b^4)/(45045*x^(7/6)) - (4096*b^6)/(675675*Sqrt[x]) - (2*Cos[a + b*x^(1/3)]^2)/
(5*x^(5/2)) + (32*b^2*Cos[a + b*x^(1/3)]^2)/(715*x^(11/6)) - (512*b^4*Cos[a + b*x^(1/3)]^2)/(45045*x^(7/6)) +
(8192*b^6*Cos[a + b*x^(1/3)]^2)/(675675*Sqrt[x]) + (32768*b^(15/2)*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/
6))/Sqrt[Pi]])/675675 - (32768*b^(15/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/675675 + (8*
b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(65*x^(13/6)) - (128*b^3*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(6435
*x^(3/2)) + (2048*b^5*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(225225*x^(5/6)) - (32768*b^7*Cos[a + b*x^(1/3)]*
Sin[a + b*x^(1/3)])/(675675*x^(1/6))

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 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

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 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 \frac{\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{17/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}+\frac{1}{65} \left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^{13/2}} \, dx,x,\sqrt [3]{x}\right )-\frac{1}{65} \left (16 b^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{13/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{16 b^2}{715 x^{11/6}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}-\frac{\left (128 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^{9/2}} \, dx,x,\sqrt [3]{x}\right )}{6435}+\frac{\left (256 b^4\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{9/2}} \, dx,x,\sqrt [3]{x}\right )}{6435}\\ &=-\frac{16 b^2}{715 x^{11/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}+\frac{\left (2048 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )}{225225}-\frac{\left (4096 b^6\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )}{225225}\\ &=-\frac{16 b^2}{715 x^{11/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{4096 b^6}{675675 \sqrt{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}-\frac{\left (32768 b^8\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}+\frac{\left (65536 b^8\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac{16 b^2}{715 x^{11/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{4096 b^6}{675675 \sqrt{x}}-\frac{65536 b^8 \sqrt [6]{x}}{675675}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{\left (65536 b^8\right ) \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 )}{675675}\\ &=-\frac{16 b^2}{715 x^{11/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{4096 b^6}{675675 \sqrt{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{\left (32768 b^8\right ) \operatorname{Subst}\left (\int \frac{\cos (2 a+2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac{16 b^2}{715 x^{11/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{4096 b^6}{675675 \sqrt{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{\left (32768 b^8 \cos (2 a)\right ) \operatorname{Subst}\left (\int \frac{\cos (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}-\frac{\left (32768 b^8 \sin (2 a)\right ) \operatorname{Subst}\left (\int \frac{\sin (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac{16 b^2}{715 x^{11/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{4096 b^6}{675675 \sqrt{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac{\left (65536 b^8 \cos (2 a)\right ) \operatorname{Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{675675}-\frac{\left (65536 b^8 \sin (2 a)\right ) \operatorname{Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{675675}\\ &=-\frac{16 b^2}{715 x^{11/6}}+\frac{256 b^4}{45045 x^{7/6}}-\frac{4096 b^6}{675675 \sqrt{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac{8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt{x}}+\frac{32768 b^{15/2} \sqrt{\pi } \cos (2 a) C\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{675675}-\frac{32768 b^{15/2} \sqrt{\pi } S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right ) \sin (2 a)}{675675}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac{2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac{32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}\\ \end{align*}

Mathematica [A]  time = 0.375443, size = 249, normalized size = 0.76 \[ \frac{32768 \sqrt{\pi } b^{15/2} x^{5/2} \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-32768 \sqrt{\pi } b^{15/2} x^{5/2} \sin (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-16384 b^7 x^{7/3} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+3072 b^5 x^{5/3} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+4096 b^6 x^2 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-3840 b^4 x^{4/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+15120 b^2 x^{2/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-6720 b^3 x \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+41580 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-135135 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-135135}{675675 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^(1/3)]^2/x^(7/2),x]

[Out]

(-135135 - 135135*Cos[2*(a + b*x^(1/3))] + 15120*b^2*x^(2/3)*Cos[2*(a + b*x^(1/3))] - 3840*b^4*x^(4/3)*Cos[2*(
a + b*x^(1/3))] + 4096*b^6*x^2*Cos[2*(a + b*x^(1/3))] + 32768*b^(15/2)*Sqrt[Pi]*x^(5/2)*Cos[2*a]*FresnelC[(2*S
qrt[b]*x^(1/6))/Sqrt[Pi]] - 32768*b^(15/2)*Sqrt[Pi]*x^(5/2)*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] +
41580*b*x^(1/3)*Sin[2*(a + b*x^(1/3))] - 6720*b^3*x*Sin[2*(a + b*x^(1/3))] + 3072*b^5*x^(5/3)*Sin[2*(a + b*x^(
1/3))] - 16384*b^7*x^(7/3)*Sin[2*(a + b*x^(1/3))])/(675675*x^(5/2))

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Maple [A]  time = 0.039, size = 207, normalized size = 0.6 \begin{align*} -{\frac{1}{5}{x}^{-{\frac{5}{2}}}}-{\frac{1}{5}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{5}{2}}}}-{\frac{4\,b}{5} \left ( -{\frac{1}{13}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{13}{6}}}}+{\frac{4\,b}{13} \left ( -{\frac{1}{11}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{11}{6}}}}-{\frac{4\,b}{11} \left ( -{\frac{1}{9}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{3}{2}}}}+{\frac{4\,b}{9} \left ( -{\frac{1}{7}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{7}{6}}}}-{\frac{4\,b}{7} \left ( -{\frac{1}{5}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{5}{6}}}}+{\frac{4\,b}{5} \left ( -{\frac{1}{3}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ){\frac{1}{\sqrt{x}}}}-{\frac{4\,b}{3} \left ( -{\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ){\frac{1}{\sqrt [6]{x}}}}+2\,\sqrt{b}\sqrt{\pi } \left ( \cos \left ( 2\,a \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\sqrt{\pi }}} \right ) -\sin \left ( 2\,a \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\sqrt{\pi }}} \right ) \right ) \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b*x^(1/3))^2/x^(7/2),x)

[Out]

-1/5/x^(5/2)-1/5/x^(5/2)*cos(2*a+2*b*x^(1/3))-4/5*b*(-1/13/x^(13/6)*sin(2*a+2*b*x^(1/3))+4/13*b*(-1/11/x^(11/6
)*cos(2*a+2*b*x^(1/3))-4/11*b*(-1/9/x^(3/2)*sin(2*a+2*b*x^(1/3))+4/9*b*(-1/7/x^(7/6)*cos(2*a+2*b*x^(1/3))-4/7*
b*(-1/5/x^(5/6)*sin(2*a+2*b*x^(1/3))+4/5*b*(-1/3/x^(1/2)*cos(2*a+2*b*x^(1/3))-4/3*b*(-1/x^(1/6)*sin(2*a+2*b*x^
(1/3))+2*b^(1/2)*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)/P
i^(1/2))))))))))

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Maxima [C]  time = 2.03386, size = 383, normalized size = 1.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^(1/3))^2/x^(7/2),x, algorithm="maxima")

[Out]

-1/5*(sqrt(2)*((240*(gamma(-15/2, 2*I*b*x^(1/3)) + gamma(-15/2, -2*I*b*x^(1/3)))*cos(15/4*pi + 15/2*arctan2(0,
b)) + 240*(gamma(-15/2, 2*I*b*x^(1/3)) + gamma(-15/2, -2*I*b*x^(1/3)))*cos(-15/4*pi + 15/2*arctan2(0, b)) + (
240*I*gamma(-15/2, 2*I*b*x^(1/3)) - 240*I*gamma(-15/2, -2*I*b*x^(1/3)))*sin(15/4*pi + 15/2*arctan2(0, b)) + (-
240*I*gamma(-15/2, 2*I*b*x^(1/3)) + 240*I*gamma(-15/2, -2*I*b*x^(1/3)))*sin(-15/4*pi + 15/2*arctan2(0, b)))*co
s(2*a) + ((-240*I*gamma(-15/2, 2*I*b*x^(1/3)) + 240*I*gamma(-15/2, -2*I*b*x^(1/3)))*cos(15/4*pi + 15/2*arctan2
(0, b)) + (-240*I*gamma(-15/2, 2*I*b*x^(1/3)) + 240*I*gamma(-15/2, -2*I*b*x^(1/3)))*cos(-15/4*pi + 15/2*arctan
2(0, b)) + 240*(gamma(-15/2, 2*I*b*x^(1/3)) + gamma(-15/2, -2*I*b*x^(1/3)))*sin(15/4*pi + 15/2*arctan2(0, b))
- 240*(gamma(-15/2, 2*I*b*x^(1/3)) + gamma(-15/2, -2*I*b*x^(1/3)))*sin(-15/4*pi + 15/2*arctan2(0, b)))*sin(2*a
))*sqrt(x^(1/3)*abs(b))*b^6*x^(7/3)*abs(b) + 1)/x^(5/2)

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Fricas [A]  time = 2.26457, size = 581, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (16384 \, \pi b^{7} x^{3} \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{C}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) - 16384 \, \pi b^{7} x^{3} \sqrt{\frac{b}{\pi }} \operatorname{S}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (2 \, a\right ) - 2048 \, b^{6} x^{\frac{5}{2}} + 1920 \, b^{4} x^{\frac{11}{6}} - 7560 \, b^{2} x^{\frac{7}{6}} -{\left (3840 \, b^{4} x^{\frac{11}{6}} - 15120 \, b^{2} x^{\frac{7}{6}} -{\left (4096 \, b^{6} x^{2} - 135135\right )} \sqrt{x}\right )} \cos \left (b x^{\frac{1}{3}} + a\right )^{2} + 4 \,{\left (768 \, b^{5} x^{\frac{13}{6}} - 1680 \, b^{3} x^{\frac{3}{2}} -{\left (4096 \, b^{7} x^{2} - 10395 \, b\right )} x^{\frac{5}{6}}\right )} \cos \left (b x^{\frac{1}{3}} + a\right ) \sin \left (b x^{\frac{1}{3}} + a\right )\right )}}{675675 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^(1/3))^2/x^(7/2),x, algorithm="fricas")

[Out]

2/675675*(16384*pi*b^7*x^3*sqrt(b/pi)*cos(2*a)*fresnel_cos(2*x^(1/6)*sqrt(b/pi)) - 16384*pi*b^7*x^3*sqrt(b/pi)
*fresnel_sin(2*x^(1/6)*sqrt(b/pi))*sin(2*a) - 2048*b^6*x^(5/2) + 1920*b^4*x^(11/6) - 7560*b^2*x^(7/6) - (3840*
b^4*x^(11/6) - 15120*b^2*x^(7/6) - (4096*b^6*x^2 - 135135)*sqrt(x))*cos(b*x^(1/3) + a)^2 + 4*(768*b^5*x^(13/6)
- 1680*b^3*x^(3/2) - (4096*b^7*x^2 - 10395*b)*x^(5/6))*cos(b*x^(1/3) + a)*sin(b*x^(1/3) + a))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x**(1/3))**2/x**(7/2),x)

[Out]

Integral(cos(a + b*x**(1/3))**2/x**(7/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x^{\frac{1}{3}} + a\right )^{2}}{x^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^(1/3))^2/x^(7/2),x, algorithm="giac")

[Out]

integrate(cos(b*x^(1/3) + a)^2/x^(7/2), x)