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

Optimal. Leaf size=235 \[ \frac{405405 \sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{64 b^{15/2}}+\frac{405405 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{64 b^{15/2}}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}-\frac{405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac{135135 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

[Out]

(135135*Sqrt[x]*Cos[a + b*x^(1/3)])/(32*b^6) - (3861*x^(7/6)*Cos[a + b*x^(1/3)])/(8*b^4) + (39*x^(11/6)*Cos[a
+ b*x^(1/3)])/(2*b^2) + (405405*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/(64*b^(15/2)) + (40540
5*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/(64*b^(15/2)) - (405405*x^(1/6)*Sin[a + b*x^(1/3)])/
(64*b^7) + (27027*x^(5/6)*Sin[a + b*x^(1/3)])/(16*b^5) - (429*x^(3/2)*Sin[a + b*x^(1/3)])/(4*b^3) + (3*x^(13/6
)*Sin[a + b*x^(1/3)])/b

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Rubi [A]  time = 0.348002, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3416, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac{405405 \sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{64 b^{15/2}}+\frac{405405 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{64 b^{15/2}}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}-\frac{405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac{135135 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*Cos[a + b*x^(1/3)],x]

[Out]

(135135*Sqrt[x]*Cos[a + b*x^(1/3)])/(32*b^6) - (3861*x^(7/6)*Cos[a + b*x^(1/3)])/(8*b^4) + (39*x^(11/6)*Cos[a
+ b*x^(1/3)])/(2*b^2) + (405405*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/(64*b^(15/2)) + (40540
5*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/(64*b^(15/2)) - (405405*x^(1/6)*Sin[a + b*x^(1/3)])/
(64*b^7) + (27027*x^(5/6)*Sin[a + b*x^(1/3)])/(16*b^5) - (429*x^(3/2)*Sin[a + b*x^(1/3)])/(4*b^3) + (3*x^(13/6
)*Sin[a + b*x^(1/3)])/b

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 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 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 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \, dx &=3 \operatorname{Subst}\left (\int x^{13/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{39 \operatorname{Subst}\left (\int x^{11/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{429 \operatorname{Subst}\left (\int x^{9/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{4 b^2}\\ &=\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{3861 \operatorname{Subst}\left (\int x^{7/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{8 b^3}\\ &=-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{27027 \operatorname{Subst}\left (\int x^{5/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{16 b^4}\\ &=-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{135135 \operatorname{Subst}\left (\int x^{3/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{32 b^5}\\ &=\frac{135135 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{405405 \operatorname{Subst}\left (\int \sqrt{x} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{64 b^6}\\ &=\frac{135135 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{405405 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{128 b^7}\\ &=\frac{135135 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{(405405 \cos (a)) \operatorname{Subst}\left (\int \frac{\sin (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{128 b^7}+\frac{(405405 \sin (a)) \operatorname{Subst}\left (\int \frac{\cos (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{128 b^7}\\ &=\frac{135135 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac{405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac{(405405 \cos (a)) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{64 b^7}+\frac{(405405 \sin (a)) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{64 b^7}\\ &=\frac{135135 \sqrt{x} \cos \left (a+b \sqrt [3]{x}\right )}{32 b^6}-\frac{3861 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac{39 x^{11/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac{405405 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{64 b^{15/2}}+\frac{405405 \sqrt{\frac{\pi }{2}} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{64 b^{15/2}}-\frac{405405 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{64 b^7}+\frac{27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right )}{16 b^5}-\frac{429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac{3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.568362, size = 165, normalized size = 0.7 \[ \frac{6 \sqrt{b} \sqrt [6]{x} \left (\left (64 b^6 x^2-2288 b^4 x^{4/3}+36036 b^2 x^{2/3}-135135\right ) \sin \left (a+b \sqrt [3]{x}\right )+26 \left (16 b^5 x^{5/3}-396 b^3 x+3465 b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )\right )+405405 \sqrt{2 \pi } \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )+405405 \sqrt{2 \pi } \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{128 b^{15/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(405405*Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] + 405405*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]
*x^(1/6)]*Sin[a] + 6*Sqrt[b]*x^(1/6)*(26*(3465*b*x^(1/3) - 396*b^3*x + 16*b^5*x^(5/3))*Cos[a + b*x^(1/3)] + (-
135135 + 36036*b^2*x^(2/3) - 2288*b^4*x^(4/3) + 64*b^6*x^2)*Sin[a + b*x^(1/3)]))/(128*b^(15/2))

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Maple [A]  time = 0.033, size = 196, normalized size = 0.8 \begin{align*} 3\,{\frac{\sin \left ( a+b\sqrt [3]{x} \right ) }{b}{x}^{{\frac{13}{6}}}}-39\,{\frac{1}{b} \left ( -1/2\,{\frac{\cos \left ( a+b\sqrt [3]{x} \right ) }{b}{x}^{{\frac{11}{6}}}}+11/2\,{\frac{1}{b} \left ( 1/2\,{\frac{{x}^{3/2}\sin \left ( a+b\sqrt [3]{x} \right ) }{b}}-9/2\,{\frac{1}{b} \left ( -1/2\,{\frac{{x}^{7/6}\cos \left ( a+b\sqrt [3]{x} \right ) }{b}}+7/2\,{\frac{1}{b} \left ( 1/2\,{\frac{{x}^{5/6}\sin \left ( a+b\sqrt [3]{x} \right ) }{b}}-5/2\,{\frac{1}{b} \left ( -1/2\,{\frac{\sqrt{x}\cos \left ( a+b\sqrt [3]{x} \right ) }{b}}+3/2\,{\frac{1}{b} \left ( 1/2\,{\frac{\sqrt [6]{x}\sin \left ( a+b\sqrt [3]{x} \right ) }{b}}-1/4\,{\frac{\sqrt{2}\sqrt{\pi }}{{b}^{3/2}} \left ( \cos \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt [6]{x}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) +\sin \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt [6]{x}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(13/6)*sin(a+b*x^(1/3))/b-39/b*(-1/2/b*x^(11/6)*cos(a+b*x^(1/3))+11/2/b*(1/2/b*x^(3/2)*sin(a+b*x^(1/3))-9/
2/b*(-1/2/b*x^(7/6)*cos(a+b*x^(1/3))+7/2/b*(1/2/b*x^(5/6)*sin(a+b*x^(1/3))-5/2/b*(-1/2/b*x^(1/2)*cos(a+b*x^(1/
3))+3/2/b*(1/2*x^(1/6)*sin(a+b*x^(1/3))/b-1/4/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2
)/Pi^(1/2))+sin(a)*FresnelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2)))))))))

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Maxima [C]  time = 2.76905, size = 459, normalized size = 1.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/512*(sqrt(pi)*(((-135135*I*cos(1/4*pi + 1/2*arctan2(0, b)) - 135135*I*cos(-1/4*pi + 1/2*arctan2(0, b)) - 13
5135*sin(1/4*pi + 1/2*arctan2(0, b)) + 135135*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (135135*cos(1/4*pi +
1/2*arctan2(0, b)) + 135135*cos(-1/4*pi + 1/2*arctan2(0, b)) - 135135*I*sin(1/4*pi + 1/2*arctan2(0, b)) + 1351
35*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf(sqrt(I*b)*x^(1/6)) + ((135135*I*cos(1/4*pi + 1/2*arctan2(0,
b)) + 135135*I*cos(-1/4*pi + 1/2*arctan2(0, b)) - 135135*sin(1/4*pi + 1/2*arctan2(0, b)) + 135135*sin(-1/4*pi
+ 1/2*arctan2(0, b)))*cos(a) - (135135*cos(1/4*pi + 1/2*arctan2(0, b)) + 135135*cos(-1/4*pi + 1/2*arctan2(0,
b)) + 135135*I*sin(1/4*pi + 1/2*arctan2(0, b)) - 135135*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf(sqrt(-
I*b)*x^(1/6)))*sqrt(abs(b)) - 208*(16*b^5*x^(11/6)*abs(b) - 396*b^3*x^(7/6)*abs(b) + 3465*b*sqrt(x)*abs(b))*co
s(b*x^(1/3) + a) - 8*(64*b^6*x^(13/6)*abs(b) - 2288*b^4*x^(3/2)*abs(b) + 36036*b^2*x^(5/6)*abs(b) - 135135*x^(
1/6)*abs(b))*sin(b*x^(1/3) + a))/(b^7*abs(b))

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Fricas [A]  time = 1.78436, size = 462, normalized size = 1.97 \begin{align*} \frac{3 \,{\left (135135 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{S}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) + 135135 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) + 52 \,{\left (16 \, b^{6} x^{\frac{11}{6}} - 396 \, b^{4} x^{\frac{7}{6}} + 3465 \, b^{2} \sqrt{x}\right )} \cos \left (b x^{\frac{1}{3}} + a\right ) - 2 \,{\left (2288 \, b^{5} x^{\frac{3}{2}} - 36036 \, b^{3} x^{\frac{5}{6}} -{\left (64 \, b^{7} x^{2} - 135135 \, b\right )} x^{\frac{1}{6}}\right )} \sin \left (b x^{\frac{1}{3}} + a\right )\right )}}{128 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*(135135*sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_sin(sqrt(2)*x^(1/6)*sqrt(b/pi)) + 135135*sqrt(2)*pi*sqrt(b/
pi)*fresnel_cos(sqrt(2)*x^(1/6)*sqrt(b/pi))*sin(a) + 52*(16*b^6*x^(11/6) - 396*b^4*x^(7/6) + 3465*b^2*sqrt(x))
*cos(b*x^(1/3) + a) - 2*(2288*b^5*x^(3/2) - 36036*b^3*x^(5/6) - (64*b^7*x^2 - 135135*b)*x^(1/6))*sin(b*x^(1/3)
+ a))/b^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{3}{2}} \cos{\left (a + b \sqrt [3]{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.15408, size = 325, normalized size = 1.38 \begin{align*} -\frac{3 \,{\left (64 i \, b^{6} x^{\frac{13}{6}} - 416 \, b^{5} x^{\frac{11}{6}} - 2288 i \, b^{4} x^{\frac{3}{2}} + 10296 \, b^{3} x^{\frac{7}{6}} + 36036 i \, b^{2} x^{\frac{5}{6}} - 90090 \, b \sqrt{x} - 135135 i \, x^{\frac{1}{6}}\right )} e^{\left (i \, b x^{\frac{1}{3}} + i \, a\right )}}{128 \, b^{7}} - \frac{3 \,{\left (-64 i \, b^{6} x^{\frac{13}{6}} - 416 \, b^{5} x^{\frac{11}{6}} + 2288 i \, b^{4} x^{\frac{3}{2}} + 10296 \, b^{3} x^{\frac{7}{6}} - 36036 i \, b^{2} x^{\frac{5}{6}} - 90090 \, b \sqrt{x} + 135135 i \, x^{\frac{1}{6}}\right )} e^{\left (-i \, b x^{\frac{1}{3}} - i \, a\right )}}{128 \, b^{7}} + \frac{405405 i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x^{\frac{1}{6}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{256 \, b^{7}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} - \frac{405405 i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x^{\frac{1}{6}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{256 \, b^{7}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/128*(64*I*b^6*x^(13/6) - 416*b^5*x^(11/6) - 2288*I*b^4*x^(3/2) + 10296*b^3*x^(7/6) + 36036*I*b^2*x^(5/6) -
90090*b*sqrt(x) - 135135*I*x^(1/6))*e^(I*b*x^(1/3) + I*a)/b^7 - 3/128*(-64*I*b^6*x^(13/6) - 416*b^5*x^(11/6) +
2288*I*b^4*x^(3/2) + 10296*b^3*x^(7/6) - 36036*I*b^2*x^(5/6) - 90090*b*sqrt(x) + 135135*I*x^(1/6))*e^(-I*b*x^
(1/3) - I*a)/b^7 + 405405/256*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x^(1/6)*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I
*a)/(b^7*(-I*b/abs(b) + 1)*sqrt(abs(b))) - 405405/256*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x^(1/6)*(I*b/abs(b)
+ 1)*sqrt(abs(b)))*e^(-I*a)/(b^7*(I*b/abs(b) + 1)*sqrt(abs(b)))