3.1.52 \(\int \frac {\cos (a+b \sqrt [3]{x})}{x^{3/2}} \, dx\) [52]

3.1.52.1 Optimal result

Integrand size = 16, antiderivative size = 110 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}-4 b^{3/2} \sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )+4 b^{3/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}} \]

output

4*b*sin(a+b*x^(1/3))/x^(1/6)-4*b^(3/2)*cos(a)*FresnelC(x^(1/6)*b^(1/2)*2^( 
1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)+4*b^(3/2)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2) 
/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)-2*cos(a+b*x^(1/3))/x^(1/2)
 

3.1.52.2 Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}-4 b^{3/2} \sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )+4 b^{3/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}} \]

input

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

output

(-2*Cos[a + b*x^(1/3)])/Sqrt[x] - 4*b^(3/2)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqr 
t[b]*Sqrt[2/Pi]*x^(1/6)] + 4*b^(3/2)*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi 
]*x^(1/6)]*Sin[a] + (4*b*Sin[a + b*x^(1/3)])/x^(1/6)
 

3.1.52.3 Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {3897, 3042, 3778, 25, 3042, 3778, 3042, 3787, 3042, 3785, 3786, 3832, 3833}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3897

\(\displaystyle 3 \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{5/6}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )}{x^{5/6}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 3778

\(\displaystyle 3 \left (\frac {2}{3} b \int -\frac {\sin \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 3 \left (-\frac {2}{3} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 \left (-\frac {2}{3} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3778

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3787

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \left (\cos (a) \int \frac {\cos \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\sin (a) \int \frac {\sin \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \left (\cos (a) \int \frac {\sin \left (\sqrt [3]{x} b+\frac {\pi }{2}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\sin (a) \int \frac {\sin \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3785

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \left (2 \cos (a) \int \cos \left (b x^{2/3}\right )d\sqrt [6]{x}-\sin (a) \int \frac {\sin \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3786

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \left (2 \cos (a) \int \cos \left (b x^{2/3}\right )d\sqrt [6]{x}-2 \sin (a) \int \sin \left (b x^{2/3}\right )d\sqrt [6]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \left (2 \cos (a) \int \cos \left (b x^{2/3}\right )d\sqrt [6]{x}-\frac {\sqrt {2 \pi } \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{\sqrt {b}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

\(\Big \downarrow \) 3833

\(\displaystyle 3 \left (-\frac {2}{3} b \left (2 b \left (\frac {\sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{\sqrt {b}}-\frac {\sqrt {2 \pi } \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{\sqrt {b}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )\)

input

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

output

3*((-2*Cos[a + b*x^(1/3)])/(3*Sqrt[x]) - (2*b*(2*b*((Sqrt[2*Pi]*Cos[a]*Fre 
snelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/Sqrt[b] - (Sqrt[2*Pi]*FresnelS[Sqrt[b]* 
Sqrt[2/Pi]*x^(1/6)]*Sin[a])/Sqrt[b]) - (2*Sin[a + b*x^(1/3)])/x^(1/6)))/3)
 

3.1.52.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

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] - Simp[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 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[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 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3833

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]
 

rule 3897

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol 
] :> Module[{k = Denominator[n]}, Simp[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]
 

3.1.52.4 Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(-\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{\sqrt {x}}-4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )\)	\(78\)
default	\(-\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{\sqrt {x}}-4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )\)	\(78\)
meijerg	\(\frac {3 \cos \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (b^{2}\right )^{\frac {3}{4}} \left (-\frac {8 \sqrt {2}\, \cos \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, \sqrt {x}\, \left (b^{2}\right )^{\frac {3}{4}}}+\frac {16 \sqrt {2}\, b \sin \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, x^{\frac {1}{6}} \left (b^{2}\right )^{\frac {3}{4}}}-\frac {32 b^{\frac {3}{2}} \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{3 \left (b^{2}\right )^{\frac {3}{4}}}\right )}{8}-\frac {3 \sin \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, b^{\frac {3}{2}} \left (-\frac {16 \sqrt {2}\, \cos \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, x^{\frac {1}{6}} \sqrt {b}}-\frac {8 \sqrt {2}\, \sin \left (b \,x^{\frac {1}{3}}\right )}{3 \sqrt {\pi }\, \sqrt {x}\, b^{\frac {3}{2}}}-\frac {32 \,\operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{3}\right )}{8}\)	\(157\)

input

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

output

-2*cos(a+b*x^(1/3))/x^(1/2)-4*b*(-1/x^(1/6)*sin(a+b*x^(1/3))+b^(1/2)*2^(1/ 
2)*Pi^(1/2)*(cos(a)*FresnelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*Fres 
nelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))))
 

3.1.52.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.87 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} \pi b x \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 2 \, \sqrt {2} \pi b x \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - 2 \, b x^{\frac {5}{6}} \sin \left (b x^{\frac {1}{3}} + a\right ) + \sqrt {x} \cos \left (b x^{\frac {1}{3}} + a\right )\right )}}{x} \]

input

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

output

-2*(2*sqrt(2)*pi*b*x*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x^(1/6)*sqrt(b/ 
pi)) - 2*sqrt(2)*pi*b*x*sqrt(b/pi)*fresnel_sin(sqrt(2)*x^(1/6)*sqrt(b/pi)) 
*sin(a) - 2*b*x^(5/6)*sin(b*x^(1/3) + a) + sqrt(x)*cos(b*x^(1/3) + a))/x
 

3.1.52.6 Sympy [F]

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

input

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

output

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

3.1.52.7 Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx=-\frac {3 \, {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (a\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b}{4 \, x^{\frac {1}{6}}} \]

input

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

output

-3/4*(((I - 1)*sqrt(2)*gamma(-3/2, I*b*x^(1/3)) - (I + 1)*sqrt(2)*gamma(-3 
/2, -I*b*x^(1/3)))*cos(a) + ((I + 1)*sqrt(2)*gamma(-3/2, I*b*x^(1/3)) - (I 
 - 1)*sqrt(2)*gamma(-3/2, -I*b*x^(1/3)))*sin(a))*sqrt(b*x^(1/3))*b/x^(1/6)
 

3.1.52.8 Giac [F]

\[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx=\int { \frac {\cos \left (b x^{\frac {1}{3}} + a\right )}{x^{\frac {3}{2}}} \,d x } \]

input

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

output

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

3.1.52.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx=\int \frac {\cos \left (a+b\,x^{1/3}\right )}{x^{3/2}} \,d x \]

input

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

output

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