Integrand size = 12, antiderivative size = 91 \[ \int x^2 \cos \left (a+b x^2\right ) \, dx=-\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{2 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{2 b^{3/2}}+\frac {x \sin \left (a+b x^2\right )}{2 b} \] Output:
-1/4*2^(1/2)*Pi^(1/2)*cos(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*x)/b^(3/2)- 1/4*2^(1/2)*Pi^(1/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*x)*sin(a)/b^(3/2)+1 /2*x*sin(b*x^2+a)/b
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int x^2 \cos \left (a+b x^2\right ) \, dx=\frac {-\sqrt {2 \pi } \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)+2 \sqrt {b} x \sin \left (a+b x^2\right )}{4 b^{3/2}} \] Input:
Integrate[x^2*Cos[a + b*x^2],x]
Output:
(-(Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]) - Sqrt[2*Pi]*FresnelC [Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a] + 2*Sqrt[b]*x*Sin[a + b*x^2])/(4*b^(3/2))
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3867, 3834, 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.
\(\displaystyle \int x^2 \cos \left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {x \sin \left (a+b x^2\right )}{2 b}-\frac {\int \sin \left (b x^2+a\right )dx}{2 b}\) |
\(\Big \downarrow \) 3834 |
\(\displaystyle \frac {x \sin \left (a+b x^2\right )}{2 b}-\frac {\sin (a) \int \cos \left (b x^2\right )dx+\cos (a) \int \sin \left (b x^2\right )dx}{2 b}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {x \sin \left (a+b x^2\right )}{2 b}-\frac {\sin (a) \int \cos \left (b x^2\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}}{2 b}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {x \sin \left (a+b x^2\right )}{2 b}-\frac {\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}}{2 b}\) |
Input:
Int[x^2*Cos[a + b*x^2],x]
Output:
-1/2*((Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x])/Sqrt[b] + (Sqrt[P i/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/Sqrt[b])/b + (x*Sin[a + b*x^2] )/(2*b)
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]
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]
Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[Sin[c] In t[Cos[d*(e + f*x)^2], x], x] + Simp[Cos[c] Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sin[c + d*x^n]/(d*n)), x] - Simp[e^n*((m - n + 1)/ (d*n)) Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
Time = 1.36 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {x \sin \left (b \,x^{2}+a \right )}{2 b}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {FresnelS}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )+\sin \left (a \right ) \operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}\) | \(58\) |
risch | \(-\frac {i {\mathrm e}^{-i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i b}\, x \right )}{8 b \sqrt {i b}}+\frac {i {\mathrm e}^{i a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i b}\, x \right )}{8 b \sqrt {-i b}}+\frac {x \sin \left (b \,x^{2}+a \right )}{2 b}\) | \(74\) |
meijerg | \(\frac {\cos \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (\frac {x \sqrt {2}\, \left (b^{2}\right )^{\frac {3}{4}} \sin \left (b \,x^{2}\right )}{2 \sqrt {\pi }\, b}-\frac {\left (b^{2}\right )^{\frac {3}{4}} \operatorname {FresnelS}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )}{2 b^{\frac {3}{2}}}\right )}{2 \left (b^{2}\right )^{\frac {3}{4}}}-\frac {\sin \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {x \sqrt {2}\, \sqrt {b}\, \cos \left (b \,x^{2}\right )}{2 \sqrt {\pi }}+\frac {\operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )}{2}\right )}{2 b^{\frac {3}{2}}}\) | \(109\) |
parts | \(\frac {\sqrt {2}\, \sqrt {\pi }\, x^{2} \cos \left (a \right ) \operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )}{2 \sqrt {b}}-\frac {\sqrt {2}\, \sqrt {\pi }\, x^{2} \sin \left (a \right ) \operatorname {FresnelS}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )}{2 \sqrt {b}}-\frac {\sqrt {2}\, \pi ^{\frac {3}{2}} \left (\cos \left (a \right ) \left (\frac {\operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right ) b \,x^{2}}{\pi }-\frac {\sqrt {b}\, \sqrt {2}\, x \sin \left (b \,x^{2}\right )}{2 \pi ^{\frac {3}{2}}}+\frac {\operatorname {FresnelS}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )}{2 \pi }\right )-\sin \left (a \right ) \left (\frac {\operatorname {FresnelS}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right ) b \,x^{2}}{\pi }+\frac {\cos \left (b \,x^{2}\right ) \sqrt {b}\, \sqrt {2}\, x}{2 \pi ^{\frac {3}{2}}}-\frac {\operatorname {FresnelC}\left (\frac {\sqrt {b}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )}{2 \pi }\right )\right )}{2 b^{\frac {3}{2}}}\) | \(189\) |
Input:
int(x^2*cos(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2*x*sin(b*x^2+a)/b-1/4/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(b^(1/2) *2^(1/2)/Pi^(1/2)*x)+sin(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*x))
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.79 \[ \int x^2 \cos \left (a+b x^2\right ) \, dx=-\frac {\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - 2 \, b x \sin \left (b x^{2} + a\right )}{4 \, b^{2}} \] Input:
integrate(x^2*cos(b*x^2+a),x, algorithm="fricas")
Output:
-1/4*(sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_sin(sqrt(2)*x*sqrt(b/pi)) + sqr t(2)*pi*sqrt(b/pi)*fresnel_cos(sqrt(2)*x*sqrt(b/pi))*sin(a) - 2*b*x*sin(b* x^2 + a))/b^2
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (90) = 180\).
Time = 0.82 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.30 \[ \int x^2 \cos \left (a+b x^2\right ) \, dx=\frac {b^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}} \sin {\left (a \right )} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {9}{4} \end {matrix}\middle | {- \frac {b^{2} x^{4}}{4}} \right )}}{8 \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right )} - \frac {\sqrt {b} x^{3} \sqrt {\frac {1}{b}} \cos {\left (a \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {7}{4} \end {matrix}\middle | {- \frac {b^{2} x^{4}}{4}} \right )}}{8 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right )} - \frac {\sqrt {2} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \sin {\left (a \right )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )}{2} + \frac {\sqrt {2} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \cos {\left (a \right )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )}{2} \] Input:
integrate(x**2*cos(b*x**2+a),x)
Output:
b**(3/2)*x**5*sqrt(1/b)*sin(a)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/ 2, 7/4, 9/4), -b**2*x**4/4)/(8*gamma(7/4)*gamma(9/4)) - sqrt(b)*x**3*sqrt( 1/b)*cos(a)*gamma(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -b**2 *x**4/4)/(8*gamma(5/4)*gamma(7/4)) - sqrt(2)*sqrt(pi)*x**2*sqrt(1/b)*sin(a )*fresnels(sqrt(2)*sqrt(b)*x/sqrt(pi))/2 + sqrt(2)*sqrt(pi)*x**2*sqrt(1/b) *cos(a)*fresnelc(sqrt(2)*sqrt(b)*x/sqrt(pi))/2
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int x^2 \cos \left (a+b x^2\right ) \, dx=\frac {8 \, b^{2} x \sin \left (b x^{2} + a\right ) + \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (\left (i - 1\right ) \, \cos \left (a\right ) - \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}}}{16 \, b^{3}} \] Input:
integrate(x^2*cos(b*x^2+a),x, algorithm="maxima")
Output:
1/16*(8*b^2*x*sin(b*x^2 + a) + sqrt(2)*sqrt(pi)*((-(I + 1)*cos(a) + (I - 1 )*sin(a))*erf(sqrt(I*b)*x) + ((I - 1)*cos(a) - (I + 1)*sin(a))*erf(sqrt(-I *b)*x))*b^(3/2))/b^3
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.48 \[ \int x^2 \cos \left (a+b x^2\right ) \, dx=-\frac {i \, x e^{\left (i \, b x^{2} + i \, a\right )}}{4 \, b} + \frac {i \, x e^{\left (-i \, b x^{2} - i \, a\right )}}{4 \, b} - \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{8 \, b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{8 \, b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} \] Input:
integrate(x^2*cos(b*x^2+a),x, algorithm="giac")
Output:
-1/4*I*x*e^(I*b*x^2 + I*a)/b + 1/4*I*x*e^(-I*b*x^2 - I*a)/b - 1/8*I*sqrt(2 )*sqrt(pi)*erf(-1/2*sqrt(2)*x*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(b*( -I*b/abs(b) + 1)*sqrt(abs(b))) + 1/8*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x *(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(-I*a)/(b*(I*b/abs(b) + 1)*sqrt(abs(b)))
Timed out. \[ \int x^2 \cos \left (a+b x^2\right ) \, dx=\int x^2\,\cos \left (b\,x^2+a\right ) \,d x \] Input:
int(x^2*cos(a + b*x^2),x)
Output:
int(x^2*cos(a + b*x^2), x)
\[ \int x^2 \cos \left (a+b x^2\right ) \, dx=\int \cos \left (b \,x^{2}+a \right ) x^{2}d x \] Input:
int(x^2*cos(b*x^2+a),x)
Output:
int(cos(a + b*x**2)*x**2,x)