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

Optimal. Leaf size=250 \[ \frac {256 \sqrt {2 \pi } b^{15/2} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{675675}-\frac {256 \sqrt {2 \pi } b^{15/2} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{675675}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}} \]

[Out]

-2/5*cos(a+b*x^(1/3))/x^(5/2)+8/715*b^2*cos(a+b*x^(1/3))/x^(11/6)-32/45045*b^4*cos(a+b*x^(1/3))/x^(7/6)+4/65*b
*sin(a+b*x^(1/3))/x^(13/6)-16/6435*b^3*sin(a+b*x^(1/3))/x^(3/2)+64/225225*b^5*sin(a+b*x^(1/3))/x^(5/6)-256/675
675*b^7*sin(a+b*x^(1/3))/x^(1/6)+256/675675*b^(15/2)*cos(a)*FresnelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)
*Pi^(1/2)-256/675675*b^(15/2)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)+128/675675*b^
6*cos(a+b*x^(1/3))/x^(1/2)

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Rubi [A]  time = 0.34, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3416, 3297, 3306, 3305, 3351, 3304, 3352} \[ \frac {256 \sqrt {2 \pi } b^{15/2} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [6]{x}\right )}{675675}-\frac {256 \sqrt {2 \pi } b^{15/2} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{675675}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^(1/3)]/x^(7/2),x]

[Out]

(-2*Cos[a + b*x^(1/3)])/(5*x^(5/2)) + (8*b^2*Cos[a + b*x^(1/3)])/(715*x^(11/6)) - (32*b^4*Cos[a + b*x^(1/3)])/
(45045*x^(7/6)) + (128*b^6*Cos[a + b*x^(1/3)])/(675675*Sqrt[x]) + (256*b^(15/2)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqr
t[b]*Sqrt[2/Pi]*x^(1/6)])/675675 - (256*b^(15/2)*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/67567
5 + (4*b*Sin[a + b*x^(1/3)])/(65*x^(13/6)) - (16*b^3*Sin[a + b*x^(1/3)])/(6435*x^(3/2)) + (64*b^5*Sin[a + b*x^
(1/3)])/(225225*x^(5/6)) - (256*b^7*Sin[a + b*x^(1/3)])/(675675*x^(1/6))

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
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 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 \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^{17/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}-\frac {1}{5} (2 b) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^{15/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {1}{65} \left (4 b^2\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^{13/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}+\frac {1}{715} \left (8 b^3\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^{11/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {\left (16 b^4\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^{9/2}} \, dx,x,\sqrt [3]{x}\right )}{6435}\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}-\frac {\left (32 b^5\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^{7/2}} \, dx,x,\sqrt [3]{x}\right )}{45045}\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {\left (64 b^6\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )}{225225}\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}+\frac {\left (128 b^7\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac {\left (256 b^8\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac {\left (256 b^8 \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{675675}-\frac {\left (256 b^8 \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{675675}\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}+\frac {\left (512 b^8 \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{675675}-\frac {\left (512 b^8 \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{675675}\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {256 b^{15/2} \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{675675}-\frac {256 b^{15/2} \sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{675675}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 238, normalized size = 0.95 \[ \frac {2 \left (128 \sqrt {2 \pi } b^{15/2} x^{5/2} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-128 \sqrt {2 \pi } b^{15/2} x^{5/2} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-128 b^7 x^{7/3} \sin \left (a+b \sqrt [3]{x}\right )+64 b^6 x^2 \cos \left (a+b \sqrt [3]{x}\right )+96 b^5 x^{5/3} \sin \left (a+b \sqrt [3]{x}\right )-240 b^4 x^{4/3} \cos \left (a+b \sqrt [3]{x}\right )-840 b^3 x \sin \left (a+b \sqrt [3]{x}\right )+3780 b^2 x^{2/3} \cos \left (a+b \sqrt [3]{x}\right )+20790 b \sqrt [3]{x} \sin \left (a+b \sqrt [3]{x}\right )-135135 \cos \left (a+b \sqrt [3]{x}\right )\right )}{675675 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-135135*Cos[a + b*x^(1/3)] + 3780*b^2*x^(2/3)*Cos[a + b*x^(1/3)] - 240*b^4*x^(4/3)*Cos[a + b*x^(1/3)] + 64
*b^6*x^2*Cos[a + b*x^(1/3)] + 128*b^(15/2)*Sqrt[2*Pi]*x^(5/2)*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] - 12
8*b^(15/2)*Sqrt[2*Pi]*x^(5/2)*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a] + 20790*b*x^(1/3)*Sin[a + b*x^(1/3)]
- 840*b^3*x*Sin[a + b*x^(1/3)] + 96*b^5*x^(5/3)*Sin[a + b*x^(1/3)] - 128*b^7*x^(7/3)*Sin[a + b*x^(1/3)]))/(67
5675*x^(5/2))

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fricas [A]  time = 0.65, size = 164, normalized size = 0.66 \[ \frac {2 \, {\left (128 \, \sqrt {2} \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 128 \, \sqrt {2} \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a) - {\left (240 \, b^{4} x^{\frac {11}{6}} - 3780 \, b^{2} x^{\frac {7}{6}} - {\left (64 \, b^{6} x^{2} - 135135\right )} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) + 2 \, {\left (48 \, b^{5} x^{\frac {13}{6}} - 420 \, b^{3} x^{\frac {3}{2}} - {\left (64 \, b^{7} x^{2} - 10395 \, b\right )} x^{\frac {5}{6}}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{675675 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/675675*(128*sqrt(2)*pi*b^7*x^3*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x^(1/6)*sqrt(b/pi)) - 128*sqrt(2)*pi*b^
7*x^3*sqrt(b/pi)*fresnel_sin(sqrt(2)*x^(1/6)*sqrt(b/pi))*sin(a) - (240*b^4*x^(11/6) - 3780*b^2*x^(7/6) - (64*b
^6*x^2 - 135135)*sqrt(x))*cos(b*x^(1/3) + a) + 2*(48*b^5*x^(13/6) - 420*b^3*x^(3/2) - (64*b^7*x^2 - 10395*b)*x
^(5/6))*sin(b*x^(1/3) + a))/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x^{\frac {1}{3}} + a\right )}{x^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 180, normalized size = 0.72 \[ -\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*cos(a+b*x^(1/3))/x^(5/2)-4/5*b*(-1/13/x^(13/6)*sin(a+b*x^(1/3))+2/13*b*(-1/11/x^(11/6)*cos(a+b*x^(1/3))-2
/11*b*(-1/9/x^(3/2)*sin(a+b*x^(1/3))+2/9*b*(-1/7/x^(7/6)*cos(a+b*x^(1/3))-2/7*b*(-1/5/x^(5/6)*sin(a+b*x^(1/3))
+2/5*b*(-1/3*cos(a+b*x^(1/3))/x^(1/2)-2/3*b*(-1/x^(1/6)*sin(a+b*x^(1/3))+b^(1/2)*2^(1/2)*Pi^(1/2)*(cos(a)*Fres
nelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))))))))))

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maxima [C]  time = 1.29, size = 76, normalized size = 0.30 \[ \frac {3 \, {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \cos \relax (a) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \sin \relax (a)\right )} \sqrt {b x^{\frac {1}{3}}} b^{7}}{4 \, x^{\frac {1}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(((I - 1)*sqrt(2)*gamma(-15/2, I*b*x^(1/3)) - (I + 1)*sqrt(2)*gamma(-15/2, -I*b*x^(1/3)))*cos(a) + ((I + 1
)*sqrt(2)*gamma(-15/2, I*b*x^(1/3)) - (I - 1)*sqrt(2)*gamma(-15/2, -I*b*x^(1/3)))*sin(a))*sqrt(b*x^(1/3))*b^7/
x^(1/6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\cos \left (a+b\,x^{1/3}\right )}{x^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^(1/3))/x^(7/2),x)

[Out]

int(cos(a + b*x^(1/3))/x^(7/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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