3.63 \(\int x^{3/2} \cos (x) \, dx\)

Optimal. Leaf size=49 \[ -\frac{3}{2} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )+x^{3/2} \sin (x)+\frac{3}{2} \sqrt{x} \cos (x) \]

[Out]

(3*Sqrt[x]*Cos[x])/2 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[x]])/2 + x^(3/2)*Sin[x]

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Rubi [A]  time = 0.0572119, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3296, 3304, 3352} \[ -\frac{3}{2} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )+x^{3/2} \sin (x)+\frac{3}{2} \sqrt{x} \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*Cos[x],x]

[Out]

(3*Sqrt[x]*Cos[x])/2 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[x]])/2 + x^(3/2)*Sin[x]

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 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 (x) \, dx &=x^{3/2} \sin (x)-\frac{3}{2} \int \sqrt{x} \sin (x) \, dx\\ &=\frac{3}{2} \sqrt{x} \cos (x)+x^{3/2} \sin (x)-\frac{3}{4} \int \frac{\cos (x)}{\sqrt{x}} \, dx\\ &=\frac{3}{2} \sqrt{x} \cos (x)+x^{3/2} \sin (x)-\frac{3}{2} \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{3}{2} \sqrt{x} \cos (x)-\frac{3}{2} \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )+x^{3/2} \sin (x)\\ \end{align*}

Mathematica [C]  time = 0.0133136, size = 55, normalized size = 1.12 \[ \frac{\sqrt{x} \text{Gamma}\left (\frac{5}{2},-i x\right )}{2 \sqrt{-i x}}+\frac{\sqrt{x} \text{Gamma}\left (\frac{5}{2},i x\right )}{2 \sqrt{i x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*Cos[x],x]

[Out]

(Sqrt[x]*Gamma[5/2, (-I)*x])/(2*Sqrt[(-I)*x]) + (Sqrt[x]*Gamma[5/2, I*x])/(2*Sqrt[I*x])

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Maple [A]  time = 0.027, size = 34, normalized size = 0.7 \begin{align*}{x}^{{\frac{3}{2}}}\sin \left ( x \right ) -{\frac{3\,\sqrt{2}\sqrt{\pi }}{4}{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{x}} \right ) }+{\frac{3\,\cos \left ( x \right ) }{2}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*cos(x),x)

[Out]

x^(3/2)*sin(x)-3/4*FresnelC(2^(1/2)/Pi^(1/2)*x^(1/2))*2^(1/2)*Pi^(1/2)+3/2*x^(1/2)*cos(x)

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Maxima [C]  time = 1.78026, size = 100, normalized size = 2.04 \begin{align*} x^{\frac{3}{2}} \sin \left (x\right ) + \frac{1}{32} \, \sqrt{\pi }{\left (\left (3 i - 3\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) + \left (3 i + 3\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \left (3 i + 3\right ) \, \sqrt{2} \operatorname{erf}\left (\sqrt{-i} \sqrt{x}\right ) + \left (3 i - 3\right ) \, \sqrt{2} \operatorname{erf}\left (\left (-1\right )^{\frac{1}{4}} \sqrt{x}\right )\right )} + \frac{3}{2} \, \sqrt{x} \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*cos(x),x, algorithm="maxima")

[Out]

x^(3/2)*sin(x) + 1/32*sqrt(pi)*((3*I - 3)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*sqrt(x)) + (3*I + 3)*sqrt(2)*erf((
1/2*I - 1/2)*sqrt(2)*sqrt(x)) - (3*I + 3)*sqrt(2)*erf(sqrt(-I)*sqrt(x)) + (3*I - 3)*sqrt(2)*erf((-1)^(1/4)*sqr
t(x))) + 3/2*sqrt(x)*cos(x)

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Fricas [A]  time = 1.5896, size = 134, normalized size = 2.73 \begin{align*} -\frac{3}{4} \, \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) + \frac{1}{2} \,{\left (2 \, x \sin \left (x\right ) + 3 \, \cos \left (x\right )\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*cos(x),x, algorithm="fricas")

[Out]

-3/4*sqrt(2)*sqrt(pi)*fresnel_cos(sqrt(2)*sqrt(x)/sqrt(pi)) + 1/2*(2*x*sin(x) + 3*cos(x))*sqrt(x)

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Sympy [A]  time = 16.2414, size = 83, normalized size = 1.69 \begin{align*} \frac{5 x^{\frac{3}{2}} \sin{\left (x \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{15 \sqrt{x} \cos{\left (x \right )} \Gamma \left (\frac{5}{4}\right )}{8 \Gamma \left (\frac{9}{4}\right )} - \frac{15 \sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{5}{4}\right )}{16 \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*cos(x),x)

[Out]

5*x**(3/2)*sin(x)*gamma(5/4)/(4*gamma(9/4)) + 15*sqrt(x)*cos(x)*gamma(5/4)/(8*gamma(9/4)) - 15*sqrt(2)*sqrt(pi
)*fresnelc(sqrt(2)*sqrt(x)/sqrt(pi))*gamma(5/4)/(16*gamma(9/4))

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Giac [C]  time = 1.14162, size = 93, normalized size = 1.9 \begin{align*} \left (\frac{3}{16} i + \frac{3}{16}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \left (\frac{3}{16} i - \frac{3}{16}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \frac{1}{4} \,{\left (2 i \, x^{\frac{3}{2}} - 3 \, \sqrt{x}\right )} e^{\left (i \, x\right )} - \frac{1}{4} \,{\left (-2 i \, x^{\frac{3}{2}} - 3 \, \sqrt{x}\right )} e^{\left (-i \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*cos(x),x, algorithm="giac")

[Out]

(3/16*I + 3/16)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(x)) - (3/16*I - 3/16)*sqrt(2)*sqrt(pi)*erf(-(1
/2*I + 1/2)*sqrt(2)*sqrt(x)) - 1/4*(2*I*x^(3/2) - 3*sqrt(x))*e^(I*x) - 1/4*(-2*I*x^(3/2) - 3*sqrt(x))*e^(-I*x)