3.1.27 \(\int \frac {\cos (a+b x^2)}{x^{3/2}} \, dx\) [27]

Optimal. Leaf size=98 \[ -\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-\frac {i b e^{i a} x^{3/2} \text {Gamma}\left (\frac {3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac {i b e^{-i a} x^{3/2} \text {Gamma}\left (\frac {3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}} \]

[Out]

-I*b*exp(I*a)*x^(3/2)*GAMMA(3/4,-I*b*x^2)/(-I*b*x^2)^(3/4)+I*b*x^(3/2)*GAMMA(3/4,I*b*x^2)/exp(I*a)/(I*b*x^2)^(
3/4)-2*cos(b*x^2+a)/x^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3469, 3470, 2250} \begin {gather*} -\frac {i e^{i a} b x^{3/2} \text {Gamma}\left (\frac {3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac {i e^{-i a} b x^{3/2} \text {Gamma}\left (\frac {3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}}-\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^2]/x^(3/2),x]

[Out]

(-2*Cos[a + b*x^2])/Sqrt[x] - (I*b*E^(I*a)*x^(3/2)*Gamma[3/4, (-I)*b*x^2])/((-I)*b*x^2)^(3/4) + (I*b*x^(3/2)*G
amma[3/4, I*b*x^2])/(E^(I*a)*(I*b*x^2)^(3/4))

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 3469

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*x)^(m + 1)*(Cos[c + d*x^n]/(e*(m + 1)
)), x] + Dist[d*(n/(e^n*(m + 1))), Int[(e*x)^(m + n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n,
0] && LtQ[m, -1]

Rule 3470

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[I/2, Int[(e*x)^m*E^((-c)*I - d*I*x^n),
x], x] - Dist[I/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx &=-\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-(4 b) \int \sqrt {x} \sin \left (a+b x^2\right ) \, dx\\ &=-\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-(2 i b) \int e^{-i a-i b x^2} \sqrt {x} \, dx+(2 i b) \int e^{i a+i b x^2} \sqrt {x} \, dx\\ &=-\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-\frac {i b e^{i a} x^{3/2} \Gamma \left (\frac {3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac {i b e^{-i a} x^{3/2} \Gamma \left (\frac {3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 114, normalized size = 1.16 \begin {gather*} \frac {-2 \left (b^2 x^4\right )^{3/4} \cos \left (a+b x^2\right )+b x^2 \left (i b x^2\right )^{3/4} \text {Gamma}\left (\frac {3}{4},-i b x^2\right ) (-i \cos (a)+\sin (a))+i \left (-i b x^2\right )^{7/4} \text {Gamma}\left (\frac {3}{4},i b x^2\right ) (i \cos (a)+\sin (a))}{\sqrt {x} \left (b^2 x^4\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^2]/x^(3/2),x]

[Out]

(-2*(b^2*x^4)^(3/4)*Cos[a + b*x^2] + b*x^2*(I*b*x^2)^(3/4)*Gamma[3/4, (-I)*b*x^2]*((-I)*Cos[a] + Sin[a]) + I*(
(-I)*b*x^2)^(7/4)*Gamma[3/4, I*b*x^2]*(I*Cos[a] + Sin[a]))/(Sqrt[x]*(b^2*x^4)^(3/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.05, size = 338, normalized size = 3.45

method	result	size
meijerg	\(\frac {\cos \left (a \right ) \sqrt {\pi }\, 2^{\frac {3}{4}} \left (b^{2}\right )^{\frac {1}{8}} \left (-\frac {12 \,2^{\frac {1}{4}} \left (\frac {8 x^{4} b^{2}}{21}+\frac {2}{3}\right ) \sin \left (b \,x^{2}\right )}{\sqrt {\pi }\, x^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{8}} b}-\frac {8 \,2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{\sqrt {\pi }\, x^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{8}} b}+\frac {32 x^{\frac {7}{2}} b^{2} 2^{\frac {1}{4}} \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {5}{4}, \frac {3}{2}, b \,x^{2}\right )}{7 \sqrt {\pi }\, \left (b^{2}\right )^{\frac {1}{8}} \left (b \,x^{2}\right )^{\frac {5}{4}}}+\frac {8 x^{\frac {7}{2}} b^{2} 2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {1}{4}, \frac {1}{2}, b \,x^{2}\right )}{\sqrt {\pi }\, \left (b^{2}\right )^{\frac {1}{8}} \left (b \,x^{2}\right )^{\frac {9}{4}}}\right )}{8}-\frac {\sin \left (a \right ) \sqrt {\pi }\, 2^{\frac {3}{4}} b^{\frac {1}{4}} \left (\frac {8 \,2^{\frac {1}{4}} \sin \left (b \,x^{2}\right )}{3 \sqrt {\pi }\, \sqrt {x}\, b^{\frac {1}{4}}}+\frac {32 \,2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{3 \sqrt {\pi }\, \sqrt {x}\, b^{\frac {1}{4}}}-\frac {8 x^{\frac {7}{2}} b^{\frac {7}{4}} 2^{\frac {1}{4}} \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {1}{4}, \frac {3}{2}, b \,x^{2}\right )}{3 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {5}{4}}}-\frac {32 x^{\frac {7}{2}} b^{\frac {7}{4}} 2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {5}{4}, \frac {1}{2}, b \,x^{2}\right )}{3 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {9}{4}}}\right )}{8}\)	\(338\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*cos(a)*Pi^(1/2)*2^(3/4)*(b^2)^(1/8)*(-12/Pi^(1/2)/x^(5/2)*2^(1/4)/(b^2)^(1/8)*(8/21*x^4*b^2+2/3)/b*sin(b*x
^2)-8/Pi^(1/2)/x^(5/2)*2^(1/4)/(b^2)^(1/8)/b*(cos(b*x^2)*x^2*b-sin(b*x^2))+32/7/Pi^(1/2)*x^(7/2)/(b^2)^(1/8)*b
^2*2^(1/4)/(b*x^2)^(5/4)*sin(b*x^2)*LommelS1(5/4,3/2,b*x^2)+8/Pi^(1/2)*x^(7/2)/(b^2)^(1/8)*b^2*2^(1/4)/(b*x^2)
^(9/4)*(cos(b*x^2)*x^2*b-sin(b*x^2))*LommelS1(1/4,1/2,b*x^2))-1/8*sin(a)*Pi^(1/2)*2^(3/4)*b^(1/4)*(8/3/Pi^(1/2
)/x^(1/2)*2^(1/4)/b^(1/4)*sin(b*x^2)+32/3/Pi^(1/2)/x^(1/2)*2^(1/4)/b^(1/4)*(cos(b*x^2)*x^2*b-sin(b*x^2))-8/3/P
i^(1/2)*x^(7/2)*b^(7/4)*2^(1/4)/(b*x^2)^(5/4)*sin(b*x^2)*LommelS1(1/4,3/2,b*x^2)-32/3/Pi^(1/2)*x^(7/2)*b^(7/4)
*2^(1/4)/(b*x^2)^(9/4)*(cos(b*x^2)*x^2*b-sin(b*x^2))*LommelS1(5/4,1/2,b*x^2))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (66) = 132\).
time = 0.12, size = 133, normalized size = 1.36 \begin {gather*} -\frac {\left (b x^{2}\right )^{\frac {1}{4}} {\left ({\left (\sqrt {\sqrt {2} + 2} {\left (\Gamma \left (-\frac {1}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac {1}{4}, -i \, b x^{2}\right )\right )} + \sqrt {-\sqrt {2} + 2} {\left (i \, \Gamma \left (-\frac {1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (-\frac {1}{4}, -i \, b x^{2}\right )\right )}\right )} \cos \left (a\right ) + {\left (\sqrt {-\sqrt {2} + 2} {\left (\Gamma \left (-\frac {1}{4}, i \, b x^{2}\right ) + \Gamma \left (-\frac {1}{4}, -i \, b x^{2}\right )\right )} + \sqrt {\sqrt {2} + 2} {\left (-i \, \Gamma \left (-\frac {1}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac {1}{4}, -i \, b x^{2}\right )\right )}\right )} \sin \left (a\right )\right )}}{8 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(b*x^2)^(1/4)*((sqrt(sqrt(2) + 2)*(gamma(-1/4, I*b*x^2) + gamma(-1/4, -I*b*x^2)) + sqrt(-sqrt(2) + 2)*(I*
gamma(-1/4, I*b*x^2) - I*gamma(-1/4, -I*b*x^2)))*cos(a) + (sqrt(-sqrt(2) + 2)*(gamma(-1/4, I*b*x^2) + gamma(-1
/4, -I*b*x^2)) + sqrt(sqrt(2) + 2)*(-I*gamma(-1/4, I*b*x^2) + I*gamma(-1/4, -I*b*x^2)))*sin(a))/sqrt(x)

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Fricas [A]
time = 0.12, size = 56, normalized size = 0.57 \begin {gather*} \frac {\left (i \, b\right )^{\frac {1}{4}} x e^{\left (-i \, a\right )} \Gamma \left (\frac {3}{4}, i \, b x^{2}\right ) + \left (-i \, b\right )^{\frac {1}{4}} x e^{\left (i \, a\right )} \Gamma \left (\frac {3}{4}, -i \, b x^{2}\right ) - 2 \, \sqrt {x} \cos \left (b x^{2} + a\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((I*b)^(1/4)*x*e^(-I*a)*gamma(3/4, I*b*x^2) + (-I*b)^(1/4)*x*e^(I*a)*gamma(3/4, -I*b*x^2) - 2*sqrt(x)*cos(b*x^
2 + a))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (a + b x^{2} \right )}}{x^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b*x**2)/x**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(b*x^2 + a)/x^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\cos \left (b\,x^2+a\right )}{x^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^2)/x^(3/2),x)

[Out]

int(cos(a + b*x^2)/x^(3/2), x)

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