3.1.63 \(\int x^{3/2} \cos (x) \, dx\) [63]

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

[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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Rubi [A]
time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {3377, 3385, 3433} \begin {gather*} -\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) \end {gather*}

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 3377

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 3385

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 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], 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} \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.01, size = 55, normalized size = 1.12 \begin {gather*} \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}} \end {gather*}

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 = 0.04, size = 34, normalized size = 0.69

method	result	size
derivativedivides	\(x^{\frac {3}{2}} \sin \left (x \right )-\frac {3 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{4}+\frac {3 \sqrt {x}\, \cos \left (x \right )}{2}\)	\(34\)
default	\(x^{\frac {3}{2}} \sin \left (x \right )-\frac {3 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{4}+\frac {3 \sqrt {x}\, \cos \left (x \right )}{2}\)	\(34\)
meijerg	\(2 \sqrt {2}\, \sqrt {\pi }\, \left (\frac {3 \sqrt {x}\, \sqrt {2}\, \cos \left (x \right )}{8 \sqrt {\pi }}+\frac {x^{\frac {3}{2}} \sqrt {2}\, \sin \left (x \right )}{4 \sqrt {\pi }}-\frac {3 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )}{8}\right )\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*cos(x),x,method=_RETURNVERBOSE)

[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] Result contains complex when optimal does not.
time = 0.52, size = 74, normalized size = 1.51 \begin {gather*} x^{\frac {3}{2}} \sin \left (x\right ) - \frac {3}{32} \, \sqrt {\pi } {\left (-\left (i - 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) - \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) + \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (\sqrt {-i} \sqrt {x}\right ) - \left (i - 1\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 {gather*}

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

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Fricas [A]
time = 0.38, size = 35, normalized size = 0.71 \begin {gather*} -\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 {gather*}

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 = 3.17, size = 83, normalized size = 1.69 \begin {gather*} \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 {gather*}

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] Result contains complex when optimal does not.
time = 0.44, size = 69, normalized size = 1.41 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^{3/2}\,\cos \left (x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^(3/2)*cos(x), x)

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