3.59 \(\int \frac{\cos ^2(a+b \sqrt [3]{x})}{x^{5/2}} \, dx\)

Optimal. Leaf size=228 \[ -\frac{512}{315} \sqrt{\pi } b^{9/2} \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-\frac{512}{315} \sqrt{\pi } b^{9/2} \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac{128 b^3 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{16 b^2}{105 x^{5/6}}+\frac{256 b^4}{315 \sqrt [6]{x}} \]

[Out]

(-16*b^2)/(105*x^(5/6)) + (256*b^4)/(315*x^(1/6)) - (2*Cos[a + b*x^(1/3)]^2)/(3*x^(3/2)) + (32*b^2*Cos[a + b*x
^(1/3)]^2)/(105*x^(5/6)) - (512*b^4*Cos[a + b*x^(1/3)]^2)/(315*x^(1/6)) - (512*b^(9/2)*Sqrt[Pi]*Cos[2*a]*Fresn
elS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/315 - (512*b^(9/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a]
)/315 + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(21*x^(7/6)) - (128*b^3*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/
3)])/(315*Sqrt[x])

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Rubi [A]  time = 0.250116, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {3416, 3314, 30, 3313, 12, 3306, 3305, 3351, 3304, 3352} \[ -\frac{512}{315} \sqrt{\pi } b^{9/2} \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-\frac{512}{315} \sqrt{\pi } b^{9/2} \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac{128 b^3 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{16 b^2}{105 x^{5/6}}+\frac{256 b^4}{315 \sqrt [6]{x}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^(1/3)]^2/x^(5/2),x]

[Out]

(-16*b^2)/(105*x^(5/6)) + (256*b^4)/(315*x^(1/6)) - (2*Cos[a + b*x^(1/3)]^2)/(3*x^(3/2)) + (32*b^2*Cos[a + b*x
^(1/3)]^2)/(105*x^(5/6)) - (512*b^4*Cos[a + b*x^(1/3)]^2)/(315*x^(1/6)) - (512*b^(9/2)*Sqrt[Pi]*Cos[2*a]*Fresn
elS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/315 - (512*b^(9/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a]
)/315 + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(21*x^(7/6)) - (128*b^3*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/
3)])/(315*Sqrt[x])

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 3313

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^{5/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{11/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}+\frac{1}{21} \left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^{7/2}} \, dx,x,\sqrt [3]{x}\right )-\frac{1}{21} \left (16 b^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{7/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{16 b^2}{105 x^{5/6}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}-\frac{1}{315} \left (128 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{315} \left (256 b^4\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(a+b x)}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{16 b^2}{105 x^{5/6}}+\frac{256 b^4}{315 \sqrt [6]{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}+\frac{1}{315} \left (1024 b^5\right ) \operatorname{Subst}\left (\int -\frac{\sin (2 a+2 b x)}{2 \sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{16 b^2}{105 x^{5/6}}+\frac{256 b^4}{315 \sqrt [6]{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}-\frac{1}{315} \left (512 b^5\right ) \operatorname{Subst}\left (\int \frac{\sin (2 a+2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{16 b^2}{105 x^{5/6}}+\frac{256 b^4}{315 \sqrt [6]{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}-\frac{1}{315} \left (512 b^5 \cos (2 a)\right ) \operatorname{Subst}\left (\int \frac{\sin (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )-\frac{1}{315} \left (512 b^5 \sin (2 a)\right ) \operatorname{Subst}\left (\int \frac{\cos (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{16 b^2}{105 x^{5/6}}+\frac{256 b^4}{315 \sqrt [6]{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}-\frac{1}{315} \left (1024 b^5 \cos (2 a)\right ) \operatorname{Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )-\frac{1}{315} \left (1024 b^5 \sin (2 a)\right ) \operatorname{Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=-\frac{16 b^2}{105 x^{5/6}}+\frac{256 b^4}{315 \sqrt [6]{x}}-\frac{2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac{32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac{512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac{512}{315} b^{9/2} \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-\frac{512}{315} b^{9/2} \sqrt{\pi } C\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right ) \sin (2 a)+\frac{8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac{128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.268309, size = 185, normalized size = 0.81 \[ \frac{-512 \sqrt{\pi } b^{9/2} x^{3/2} \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-512 \sqrt{\pi } b^{9/2} x^{3/2} \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-256 b^4 x^{4/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+48 b^2 x^{2/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-64 b^3 x \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+60 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-105 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-105}{315 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^(1/3)]^2/x^(5/2),x]

[Out]

(-105 - 105*Cos[2*(a + b*x^(1/3))] + 48*b^2*x^(2/3)*Cos[2*(a + b*x^(1/3))] - 256*b^4*x^(4/3)*Cos[2*(a + b*x^(1
/3))] - 512*b^(9/2)*Sqrt[Pi]*x^(3/2)*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] - 512*b^(9/2)*Sqrt[Pi]*x^
(3/2)*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + 60*b*x^(1/3)*Sin[2*(a + b*x^(1/3))] - 64*b^3*x*Sin[2*(
a + b*x^(1/3))])/(315*x^(3/2))

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Maple [A]  time = 0.04, size = 146, normalized size = 0.6 \begin{align*} -{\frac{1}{3}{x}^{-{\frac{3}{2}}}}-{\frac{1}{3}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{3}{2}}}}-{\frac{4\,b}{3} \left ( -{\frac{1}{7}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{7}{6}}}}+{\frac{4\,b}{7} \left ( -{\frac{1}{5}\cos \left ( 2\,a+2\,b\sqrt [3]{x} \right ){x}^{-{\frac{5}{6}}}}-{\frac{4\,b}{5} \left ( -{\frac{1}{3}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ){\frac{1}{\sqrt{x}}}}+{\frac{4\,b}{3} \left ( -{\cos \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 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 ) \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^(5/2),x)

[Out]

-1/3/x^(3/2)-1/3/x^(3/2)*cos(2*a+2*b*x^(1/3))-4/3*b*(-1/7/x^(7/6)*sin(2*a+2*b*x^(1/3))+4/7*b*(-1/5/x^(5/6)*cos
(2*a+2*b*x^(1/3))-4/5*b*(-1/3/x^(1/2)*sin(2*a+2*b*x^(1/3))+4/3*b*(-1/x^(1/6)*cos(2*a+2*b*x^(1/3))-2*b^(1/2)*Pi
^(1/2)*(cos(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 = 1.57209, size = 381, normalized size = 1.67 \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^(5/2),x, algorithm="maxima")

[Out]

-1/3*(sqrt(2)*((18*(gamma(-9/2, 2*I*b*x^(1/3)) + gamma(-9/2, -2*I*b*x^(1/3)))*cos(9/4*pi + 9/2*arctan2(0, b))
+ 18*(gamma(-9/2, 2*I*b*x^(1/3)) + gamma(-9/2, -2*I*b*x^(1/3)))*cos(-9/4*pi + 9/2*arctan2(0, b)) + (18*I*gamma
(-9/2, 2*I*b*x^(1/3)) - 18*I*gamma(-9/2, -2*I*b*x^(1/3)))*sin(9/4*pi + 9/2*arctan2(0, b)) + (-18*I*gamma(-9/2,
2*I*b*x^(1/3)) + 18*I*gamma(-9/2, -2*I*b*x^(1/3)))*sin(-9/4*pi + 9/2*arctan2(0, b)))*cos(2*a) + ((-18*I*gamma
(-9/2, 2*I*b*x^(1/3)) + 18*I*gamma(-9/2, -2*I*b*x^(1/3)))*cos(9/4*pi + 9/2*arctan2(0, b)) + (-18*I*gamma(-9/2,
2*I*b*x^(1/3)) + 18*I*gamma(-9/2, -2*I*b*x^(1/3)))*cos(-9/4*pi + 9/2*arctan2(0, b)) + 18*(gamma(-9/2, 2*I*b*x
^(1/3)) + gamma(-9/2, -2*I*b*x^(1/3)))*sin(9/4*pi + 9/2*arctan2(0, b)) - 18*(gamma(-9/2, 2*I*b*x^(1/3)) + gamm
a(-9/2, -2*I*b*x^(1/3)))*sin(-9/4*pi + 9/2*arctan2(0, b)))*sin(2*a))*sqrt(x^(1/3)*abs(b))*b^4*x^(4/3) + 1)/x^(
3/2)

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Fricas [A]  time = 2.18392, size = 455, normalized size = 2. \begin{align*} -\frac{2 \,{\left (256 \, \pi b^{4} x^{2} \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{S}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) + 256 \, \pi b^{4} x^{2} \sqrt{\frac{b}{\pi }} \operatorname{C}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (2 \, a\right ) - 128 \, b^{4} x^{\frac{11}{6}} + 24 \, b^{2} x^{\frac{7}{6}} +{\left (256 \, b^{4} x^{\frac{11}{6}} - 48 \, b^{2} x^{\frac{7}{6}} + 105 \, \sqrt{x}\right )} \cos \left (b x^{\frac{1}{3}} + a\right )^{2} + 4 \,{\left (16 \, b^{3} x^{\frac{3}{2}} - 15 \, b x^{\frac{5}{6}}\right )} \cos \left (b x^{\frac{1}{3}} + a\right ) \sin \left (b x^{\frac{1}{3}} + a\right )\right )}}{315 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^(1/3))^2/x^(5/2),x, algorithm="fricas")

[Out]

-2/315*(256*pi*b^4*x^2*sqrt(b/pi)*cos(2*a)*fresnel_sin(2*x^(1/6)*sqrt(b/pi)) + 256*pi*b^4*x^2*sqrt(b/pi)*fresn
el_cos(2*x^(1/6)*sqrt(b/pi))*sin(2*a) - 128*b^4*x^(11/6) + 24*b^2*x^(7/6) + (256*b^4*x^(11/6) - 48*b^2*x^(7/6)
+ 105*sqrt(x))*cos(b*x^(1/3) + a)^2 + 4*(16*b^3*x^(3/2) - 15*b*x^(5/6))*cos(b*x^(1/3) + a)*sin(b*x^(1/3) + a)
)/x^2

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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{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x**(1/3))**2/x**(5/2),x)

[Out]

Integral(cos(a + b*x**(1/3))**2/x**(5/2), 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{5}{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^(5/2),x, algorithm="giac")

[Out]

integrate(cos(b*x^(1/3) + a)^2/x^(5/2), x)