Integrand size = 14, antiderivative size = 168 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx=-\frac {\cos ^3\left (a+b x^2\right )}{x}-\frac {3}{2} \sqrt {b} \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\frac {1}{2} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \cos (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )-\frac {3}{2} \sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)-\frac {1}{2} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a) \]
-cos(b*x^2+a)^3/x-3/4*cos(a)*FresnelS(x*b^(1/2)*2^(1/2)/Pi^(1/2))*b^(1/2)* 2^(1/2)*Pi^(1/2)-3/4*FresnelC(x*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*b^(1/2)*2 ^(1/2)*Pi^(1/2)-1/4*cos(3*a)*FresnelS(x*b^(1/2)*6^(1/2)/Pi^(1/2))*b^(1/2)* 6^(1/2)*Pi^(1/2)-1/4*FresnelC(x*b^(1/2)*6^(1/2)/Pi^(1/2))*sin(3*a)*b^(1/2) *6^(1/2)*Pi^(1/2)
Time = 0.72 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx=-\frac {3 \cos \left (a+b x^2\right )+\cos \left (3 \left (a+b x^2\right )\right )+3 \sqrt {b} \sqrt {2 \pi } x \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\sqrt {b} \sqrt {6 \pi } x \cos (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )+3 \sqrt {b} \sqrt {2 \pi } x \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)+\sqrt {b} \sqrt {6 \pi } x \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{4 x} \]
-1/4*(3*Cos[a + b*x^2] + Cos[3*(a + b*x^2)] + 3*Sqrt[b]*Sqrt[2*Pi]*x*Cos[a ]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x] + Sqrt[b]*Sqrt[6*Pi]*x*Cos[3*a]*FresnelS[ Sqrt[b]*Sqrt[6/Pi]*x] + 3*Sqrt[b]*Sqrt[2*Pi]*x*FresnelC[Sqrt[b]*Sqrt[2/Pi] *x]*Sin[a] + Sqrt[b]*Sqrt[6*Pi]*x*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a]) /x
Time = 0.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3875, 5085, 2009}
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.
| \(\displaystyle \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 3875 |
| \(\displaystyle -6 b \int \cos ^2\left (b x^2+a\right ) \sin \left (b x^2+a\right )dx-\frac {\cos ^3\left (a+b x^2\right )}{x}\) |
| \(\Big \downarrow \) 5085 |
| \(\displaystyle -6 b \int \left (\frac {1}{4} \sin \left (b x^2+a\right )+\frac {1}{4} \sin \left (3 b x^2+3 a\right )\right )dx-\frac {\cos ^3\left (a+b x^2\right )}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 b \left (\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}\right )-\frac {\cos ^3\left (a+b x^2\right )}{x}\) |
-(Cos[a + b*x^2]^3/x) - 6*b*((Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi ]*x])/(4*Sqrt[b]) + (Sqrt[Pi/6]*Cos[3*a]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x])/( 4*Sqrt[b]) + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(4*Sqrt[b] ) + (Sqrt[Pi/6]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(4*Sqrt[b]))
3.1.20.3.1 Defintions of rubi rules used
Int[Cos[(a_.) + (b_.)*(x_)^(n_)]^(p_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Cos[a + b*x^n]^p/(m + 1)), x] + Simp[b*n*(p/(m + 1)) Int[Cos[a + b*x^ n]^(p - 1)*Sin[a + b*x^n], x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 1] && EqQ[ m + n, 0] && NeQ[n, 1] && IntegerQ[n]
Int[Cos[w_]^(q_.)*Sin[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p *Cos[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && Pol ynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]))
Time = 0.66 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(-\frac {3 \cos \left (b \,x^{2}+a \right )}{4 x}-\frac {3 \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \left (a \right ) \operatorname {C}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4}-\frac {\cos \left (3 b \,x^{2}+3 a \right )}{4 x}-\frac {\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (3 a \right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )+\sin \left (3 a \right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )\right )}{4}\) | \(128\) |
| risch | \(-\frac {i {\mathrm e}^{-3 i a} b \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, \sqrt {i b}\, x \right )}{8 \sqrt {i b}}-\frac {3 i {\mathrm e}^{-i a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i b}\, x \right )}{8 \sqrt {i b}}+\frac {3 i {\mathrm e}^{i a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i b}\, x \right )}{8 \sqrt {-i b}}+\frac {3 i {\mathrm e}^{3 i a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 i b}\, x \right )}{8 \sqrt {-3 i b}}-\frac {3 \cos \left (b \,x^{2}+a \right )}{4 x}-\frac {\cos \left (3 b \,x^{2}+3 a \right )}{4 x}\) | \(145\) |
-3/4*cos(b*x^2+a)/x-3/4*b^(1/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(x*b^(1/2 )*2^(1/2)/Pi^(1/2))+sin(a)*FresnelC(x*b^(1/2)*2^(1/2)/Pi^(1/2)))-1/4*cos(3 *b*x^2+3*a)/x-1/4*b^(1/2)*2^(1/2)*Pi^(1/2)*3^(1/2)*(cos(3*a)*FresnelS(2^(1 /2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x)+sin(3*a)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2) *b^(1/2)*x))
Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx=-\frac {\sqrt {6} \pi x \sqrt {\frac {b}{\pi }} \cos \left (3 \, a\right ) \operatorname {S}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) + 3 \, \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {6} \pi x \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (3 \, a\right ) + 3 \, \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) + 4 \, \cos \left (b x^{2} + a\right )^{3}}{4 \, x} \]
-1/4*(sqrt(6)*pi*x*sqrt(b/pi)*cos(3*a)*fresnel_sin(sqrt(6)*x*sqrt(b/pi)) + 3*sqrt(2)*pi*x*sqrt(b/pi)*cos(a)*fresnel_sin(sqrt(2)*x*sqrt(b/pi)) + sqrt (6)*pi*x*sqrt(b/pi)*fresnel_cos(sqrt(6)*x*sqrt(b/pi))*sin(3*a) + 3*sqrt(2) *pi*x*sqrt(b/pi)*fresnel_cos(sqrt(2)*x*sqrt(b/pi))*sin(a) + 4*cos(b*x^2 + a)^3)/x
\[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx=\int \frac {\cos ^{3}{\left (a + b x^{2} \right )}}{x^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx=\frac {\sqrt {3} \sqrt {b x^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, 3 i \, b x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -3 i \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, 3 i \, b x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -3 i \, b x^{2}\right )\right )} \sin \left (3 \, a\right )\right )} - 3 \, \sqrt {b x^{2}} {\left ({\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \cos \left (a\right ) + {\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \sin \left (a\right )\right )}}{32 \, x} \]
1/32*(sqrt(3)*sqrt(b*x^2)*((-(I + 1)*sqrt(2)*gamma(-1/2, 3*I*b*x^2) + (I - 1)*sqrt(2)*gamma(-1/2, -3*I*b*x^2))*cos(3*a) + ((I - 1)*sqrt(2)*gamma(-1/ 2, 3*I*b*x^2) - (I + 1)*sqrt(2)*gamma(-1/2, -3*I*b*x^2))*sin(3*a)) - 3*sqr t(b*x^2)*(((I + 1)*sqrt(2)*gamma(-1/2, I*b*x^2) - (I - 1)*sqrt(2)*gamma(-1 /2, -I*b*x^2))*cos(a) + (-(I - 1)*sqrt(2)*gamma(-1/2, I*b*x^2) + (I + 1)*s qrt(2)*gamma(-1/2, -I*b*x^2))*sin(a)))/x
\[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx=\int { \frac {\cos \left (b x^{2} + a\right )^{3}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^2} \, dx=\int \frac {{\cos \left (b\,x^2+a\right )}^3}{x^2} \,d x \]