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

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

[Out]

-16/105*b^2/x^(5/6)+256/315*b^4/x^(1/6)-2/3*cos(a+b*x^(1/3))^2/x^(3/2)+32/105*b^2*cos(a+b*x^(1/3))^2/x^(5/6)-5
12/315*b^4*cos(a+b*x^(1/3))^2/x^(1/6)+8/21*b*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/x^(7/6)-512/315*b^(9/2)*cos(2*a
)*FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))*Pi^(1/2)-512/315*b^(9/2)*FresnelC(2*x^(1/6)*b^(1/2)/Pi^(1/2))*sin(2*a)*
Pi^(1/2)-128/315*b^3*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/x^(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 228, normalized size of antiderivative = 1.00, 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 12

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

Rule 30

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

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 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 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 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 0.27, size = 185, normalized size = 0.81 \[ \frac {-512 \sqrt {\pi } b^{9/2} x^{3/2} \sin (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-512 \sqrt {\pi } b^{9/2} x^{3/2} \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-256 b^4 x^{4/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-64 b^3 x \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+48 b^2 x^{2/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+60 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-105 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-105}{315 x^{3/2}} \]

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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fricas [A]  time = 0.86, size = 154, normalized size = 0.68 \[ -\frac {2 \, {\left (256 \, \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + 256 \, \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 128 \, b^{4} x^{\frac {11}{6}} + 24 \, b^{2} x^{\frac {7}{6}} + {\left (256 \, b^{4} x^{\frac {11}{6}} - 48 \, b^{2} x^{\frac {7}{6}} + 105 \, \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 4 \, {\left (16 \, b^{3} x^{\frac {3}{2}} - 15 \, b x^{\frac {5}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{315 \, x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x^{\frac {1}{3}} + a\right )^{2}}{x^{\frac {5}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.06, size = 146, normalized size = 0.64 \[ -\frac {1}{3 x^{\frac {3}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 x^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}-2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3} \]

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 = 1.71, size = 89, normalized size = 0.39 \[ \frac {\sqrt {2} {\left ({\left (-\left (18 i + 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, 2 i \, b x^{\frac {1}{3}}\right ) + \left (18 i - 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (2 \, a\right ) + {\left (\left (18 i - 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, 2 i \, b x^{\frac {1}{3}}\right ) - \left (18 i + 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (2 \, a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b^{4} x^{\frac {4}{3}} - 1}{3 \, x^{\frac {3}{2}}} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{x^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^(1/3))^2/x^(5/2),x)

[Out]

int(cos(a + b*x^(1/3))^2/x^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {5}{2}}}\, dx \]

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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