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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.336934, antiderivative size = 250, normalized size of antiderivative = 1., 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 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 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 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 \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*}

Mathematica [A]  time = 0.316166, size = 238, normalized size = 0.95 \[ \frac{2 \left (128 \sqrt{2 \pi } b^{15/2} x^{5/2} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )-128 \sqrt{2 \pi } b^{15/2} x^{5/2} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )-128 b^7 x^{7/3} \sin \left (a+b \sqrt [3]{x}\right )+96 b^5 x^{5/3} \sin \left (a+b \sqrt [3]{x}\right )+64 b^6 x^2 \cos \left (a+b \sqrt [3]{x}\right )-240 b^4 x^{4/3} \cos \left (a+b \sqrt [3]{x}\right )+3780 b^2 x^{2/3} \cos \left (a+b \sqrt [3]{x}\right )-840 b^3 x \sin \left (a+b \sqrt [3]{x}\right )+20790 b \sqrt [3]{x} \sin \left (a+b \sqrt [3]{x}\right )-135135 \cos \left (a+b \sqrt [3]{x}\right )\right )}{675675 x^{5/2}} \]

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))

________________________________________________________________________________________

Maple [A]  time = 0.032, size = 180, normalized size = 0.7 \begin{align*} -{\frac{2}{5}\cos \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{5}{2}}}}-{\frac{4\,b}{5} \left ( -{\frac{1}{13}\sin \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{13}{6}}}}+{\frac{2\,b}{13} \left ( -{\frac{1}{11}\cos \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{11}{6}}}}-{\frac{2\,b}{11} \left ( -{\frac{1}{9}\sin \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{3}{2}}}}+{\frac{2\,b}{9} \left ( -{\frac{1}{7}\cos \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{7}{6}}}}-{\frac{2\,b}{7} \left ( -{\frac{1}{5}\sin \left ( a+b\sqrt [3]{x} \right ){x}^{-{\frac{5}{6}}}}+{\frac{2\,b}{5} \left ( -{\frac{1}{3}\cos \left ( a+b\sqrt [3]{x} \right ){\frac{1}{\sqrt{x}}}}-{\frac{2\,b}{3} \left ( -{\sin \left ( a+b\sqrt [3]{x} \right ){\frac{1}{\sqrt [6]{x}}}}+\sqrt{b}\sqrt{2}\sqrt{\pi } \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \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))/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))))))))))

________________________________________________________________________________________

Maxima [C]  time = 1.43609, size = 362, normalized size = 1.45 \begin{align*} -\frac{3 \,{\left ({\left ({\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) - i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) +{\left ({\left (-i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + i \, \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \cos \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (-\frac{15}{2}, i \, b x^{\frac{1}{3}}\right ) + \Gamma \left (-\frac{15}{2}, -i \, b x^{\frac{1}{3}}\right )\right )} \sin \left (-\frac{15}{4} \, \pi + \frac{15}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \sqrt{x^{\frac{1}{3}}{\left | b \right |}} b^{6}{\left | b \right |}}{4 \, x^{\frac{1}{6}}} \end{align*}

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*(((gamma(-15/2, I*b*x^(1/3)) + gamma(-15/2, -I*b*x^(1/3)))*cos(15/4*pi + 15/2*arctan2(0, b)) + (gamma(-15
/2, I*b*x^(1/3)) + gamma(-15/2, -I*b*x^(1/3)))*cos(-15/4*pi + 15/2*arctan2(0, b)) + (I*gamma(-15/2, I*b*x^(1/3
)) - I*gamma(-15/2, -I*b*x^(1/3)))*sin(15/4*pi + 15/2*arctan2(0, b)) + (-I*gamma(-15/2, I*b*x^(1/3)) + I*gamma
(-15/2, -I*b*x^(1/3)))*sin(-15/4*pi + 15/2*arctan2(0, b)))*cos(a) + ((-I*gamma(-15/2, I*b*x^(1/3)) + I*gamma(-
15/2, -I*b*x^(1/3)))*cos(15/4*pi + 15/2*arctan2(0, b)) + (-I*gamma(-15/2, I*b*x^(1/3)) + I*gamma(-15/2, -I*b*x
^(1/3)))*cos(-15/4*pi + 15/2*arctan2(0, b)) + (gamma(-15/2, I*b*x^(1/3)) + gamma(-15/2, -I*b*x^(1/3)))*sin(15/
4*pi + 15/2*arctan2(0, b)) - (gamma(-15/2, I*b*x^(1/3)) + gamma(-15/2, -I*b*x^(1/3)))*sin(-15/4*pi + 15/2*arct
an2(0, b)))*sin(a))*sqrt(x^(1/3)*abs(b))*b^6*abs(b)/x^(1/6)

________________________________________________________________________________________

Fricas [A]  time = 2.13131, size = 490, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (128 \, \sqrt{2} \pi b^{7} x^{3} \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \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 \left (a\right ) -{\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}} \end{align*}

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\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))/x**(7/2),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

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