\(\int \frac {\cos ^2(a+b \sqrt [3]{x})}{x^{7/2}} \, dx\) [60]

Optimal result

Integrand size = 18, antiderivative size = 328 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=-\frac {16 b^2}{715 x^{11/6}}+\frac {256 b^4}{45045 x^{7/6}}-\frac {4096 b^6}{675675 \sqrt {x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {32768 b^{15/2} \sqrt {\pi } \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{675675}-\frac {32768 b^{15/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{675675}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3497, 3395, 30, 3393, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {32768 \sqrt {\pi } b^{15/2} \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{675675}-\frac {32768 \sqrt {\pi } b^{15/2} \sin (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{675675}-\frac {32768 b^7 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac {8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {2048 b^5 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}-\frac {128 b^3 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {4096 b^6}{675675 \sqrt {x}}+\frac {256 b^4}{45045 x^{7/6}}-\frac {16 b^2}{715 x^{11/6}} \]

[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 30

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

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 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 3395

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 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 \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 ) \text {Subst}\left (\int \frac {1}{x^{13/2}} \, dx,x,\sqrt [3]{x}\right )-\frac {1}{65} \left (16 b^2\right ) \text {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 ) \text {Subst}\left (\int \frac {1}{x^{9/2}} \, dx,x,\sqrt [3]{x}\right )}{6435}+\frac {\left (256 b^4\right ) \text {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 ) \text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )}{225225}-\frac {\left (4096 b^6\right ) \text {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 ) \text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{675675}+\frac {\left (65536 b^8\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{675675}-\frac {32768 b^{15/2} \sqrt {\pi } \operatorname {FresnelS}\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] (verified)

Time = 0.43 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {-135135-135135 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+15120 b^2 x^{2/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-3840 b^4 x^{4/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+4096 b^6 x^2 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+32768 b^{15/2} \sqrt {\pi } x^{5/2} \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-32768 b^{15/2} \sqrt {\pi } x^{5/2} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+41580 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-6720 b^3 x \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+3072 b^5 x^{5/3} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-16384 b^7 x^{7/3} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{675675 x^{5/2}} \]

[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))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(-\frac {1}{5 x^{\frac {5}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5}\)	\(207\)
default	\(-\frac {1}{5 x^{\frac {5}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5}\)	\(207\)

[In]

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

[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))))))))))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {2 \, {\left (16384 \, \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 16384 \, \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 2048 \, b^{6} x^{\frac {5}{2}} + 1920 \, b^{4} x^{\frac {11}{6}} - 7560 \, b^{2} x^{\frac {7}{6}} - {\left (3840 \, b^{4} x^{\frac {11}{6}} - 15120 \, b^{2} x^{\frac {7}{6}} - {\left (4096 \, b^{6} x^{2} - 135135\right )} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 4 \, {\left (768 \, b^{5} x^{\frac {13}{6}} - 1680 \, b^{3} x^{\frac {3}{2}} - {\left (4096 \, b^{7} x^{2} - 10395 \, b\right )} x^{\frac {5}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{675675 \, x^{3}} \]

[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

Sympy [F]

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

[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)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.27 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=-\frac {240 \, \sqrt {2} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, 2 i \, b x^{\frac {1}{3}}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (2 \, a\right ) + {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, 2 i \, b x^{\frac {1}{3}}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (2 \, a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b^{7} x^{\frac {7}{3}} + 1}{5 \, x^{\frac {5}{2}}} \]

[In]

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

[Out]

-1/5*(240*sqrt(2)*((-(I - 1)*sqrt(2)*gamma(-15/2, 2*I*b*x^(1/3)) + (I + 1)*sqrt(2)*gamma(-15/2, -2*I*b*x^(1/3)
))*cos(2*a) + (-(I + 1)*sqrt(2)*gamma(-15/2, 2*I*b*x^(1/3)) + (I - 1)*sqrt(2)*gamma(-15/2, -2*I*b*x^(1/3)))*si
n(2*a))*sqrt(b*x^(1/3))*b^7*x^(7/3) + 1)/x^(5/2)

Giac [F]

\[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int { \frac {\cos \left (b x^{\frac {1}{3}} + a\right )^{2}}{x^{\frac {7}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{x^{7/2}} \,d x \]

[In]

int(cos(a + b*x^(1/3))^2/x^(7/2),x)

[Out]

int(cos(a + b*x^(1/3))^2/x^(7/2), x)