Integrand size = 10, antiderivative size = 71 \[ \int x^2 \sin ^3\left (x^2\right ) \, dx=-\frac {1}{2} x \cos \left (x^2\right )+\frac {1}{6} x \cos ^3\left (x^2\right )+\frac {3}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} x\right )-\frac {1}{24} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} x\right ) \] Output:
-1/2*x*cos(x^2)+1/6*x*cos(x^2)^3+3/16*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi ^(1/2)*x)-1/144*6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*x)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int x^2 \sin ^3\left (x^2\right ) \, dx=\frac {1}{144} \left (6 x \left (-9 \cos \left (x^2\right )+\cos \left (3 x^2\right )\right )+27 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} x\right )-\sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} x\right )\right ) \] Input:
Integrate[x^2*Sin[x^2]^3,x]
Output:
(6*x*(-9*Cos[x^2] + Cos[3*x^2]) + 27*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*x] - S qrt[6*Pi]*FresnelC[Sqrt[6/Pi]*x])/144
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3884, 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.
\(\displaystyle \int x^2 \sin ^3\left (x^2\right ) \, dx\) |
\(\Big \downarrow \) 3884 |
\(\displaystyle \int \left (\frac {3}{4} x^2 \sin \left (x^2\right )-\frac {1}{4} x^2 \sin \left (3 x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} x\right )-\frac {1}{24} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} x\right )-\frac {3}{8} x \cos \left (x^2\right )+\frac {1}{24} x \cos \left (3 x^2\right )\) |
Input:
Int[x^2*Sin[x^2]^3,x]
Output:
(-3*x*Cos[x^2])/8 + (x*Cos[3*x^2])/24 + (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]* x])/8 - (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*x])/24
Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x _Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
Time = 1.52 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {3 x \cos \left (x^{2}\right )}{8}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}\right )}{16}+\frac {x \cos \left (3 x^{2}\right )}{24}-\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, x}{\sqrt {\pi }}\right )}{144}\) | \(58\) |
risch | \(-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 i}\, x \right )}{96 \sqrt {-3 i}}+\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, \left (-1\right )^{\frac {1}{4}} x \right )}{288}-\frac {3 \left (-1\right )^{\frac {3}{4}} \sqrt {\pi }\, \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} x \right )}{32}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i}\, x \right )}{32 \sqrt {-i}}-\frac {3 x \cos \left (x^{2}\right )}{8}+\frac {x \cos \left (3 x^{2}\right )}{24}\) | \(84\) |
Input:
int(x^2*sin(x^2)^3,x,method=_RETURNVERBOSE)
Output:
-3/8*x*cos(x^2)+3/16*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*x)+1/24*x* cos(3*x^2)-1/144*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2 )*x)
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72 \[ \int x^2 \sin ^3\left (x^2\right ) \, dx=\frac {1}{6} \, x \cos \left (x^{2}\right )^{3} - \frac {1}{2} \, x \cos \left (x^{2}\right ) - \frac {1}{144} \, \sqrt {6} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {6} x}{\sqrt {\pi }}\right ) + \frac {3}{16} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} x}{\sqrt {\pi }}\right ) \] Input:
integrate(x^2*sin(x^2)^3,x, algorithm="fricas")
Output:
1/6*x*cos(x^2)^3 - 1/2*x*cos(x^2) - 1/144*sqrt(6)*sqrt(pi)*fresnel_cos(sqr t(6)*x/sqrt(pi)) + 3/16*sqrt(2)*sqrt(pi)*fresnel_cos(sqrt(2)*x/sqrt(pi))
Time = 2.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.63 \[ \int x^2 \sin ^3\left (x^2\right ) \, dx=- \frac {15 x \cos {\left (x^{2} \right )} \Gamma \left (\frac {5}{4}\right )}{32 \Gamma \left (\frac {9}{4}\right )} + \frac {5 x \cos {\left (3 x^{2} \right )} \Gamma \left (\frac {5}{4}\right )}{96 \Gamma \left (\frac {9}{4}\right )} + \frac {15 \sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} x}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{64 \Gamma \left (\frac {9}{4}\right )} - \frac {5 \sqrt {6} \sqrt {\pi } C\left (\frac {\sqrt {6} x}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{576 \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate(x**2*sin(x**2)**3,x)
Output:
-15*x*cos(x**2)*gamma(5/4)/(32*gamma(9/4)) + 5*x*cos(3*x**2)*gamma(5/4)/(9 6*gamma(9/4)) + 15*sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)*x/sqrt(pi))*gamma(5/4 )/(64*gamma(9/4)) - 5*sqrt(6)*sqrt(pi)*fresnelc(sqrt(6)*x/sqrt(pi))*gamma( 5/4)/(576*gamma(9/4))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.37 \[ \int x^2 \sin ^3\left (x^2\right ) \, dx=\frac {1}{24} \, x \cos \left (3 \, x^{2}\right ) - \frac {3}{8} \, x \cos \left (x^{2}\right ) + \frac {1}{1152} \, \sqrt {\pi } {\left (\left (2 i - 2\right ) \, \sqrt {3} \sqrt {2} \operatorname {erf}\left (\sqrt {3 i} x\right ) - \left (2 i + 2\right ) \, \sqrt {3} \sqrt {2} \operatorname {erf}\left (\sqrt {-3 i} x\right ) - \left (27 i - 27\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} x\right ) - \left (27 i + 27\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} x\right ) + \left (27 i + 27\right ) \, \sqrt {2} \operatorname {erf}\left (\sqrt {-i} x\right ) - \left (27 i - 27\right ) \, \sqrt {2} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} x\right )\right )} \] Input:
integrate(x^2*sin(x^2)^3,x, algorithm="maxima")
Output:
1/24*x*cos(3*x^2) - 3/8*x*cos(x^2) + 1/1152*sqrt(pi)*((2*I - 2)*sqrt(3)*sq rt(2)*erf(sqrt(3*I)*x) - (2*I + 2)*sqrt(3)*sqrt(2)*erf(sqrt(-3*I)*x) - (27 *I - 27)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) - (27*I + 27)*sqrt(2)*erf((1 /2*I - 1/2)*sqrt(2)*x) + (27*I + 27)*sqrt(2)*erf(sqrt(-I)*x) - (27*I - 27) *sqrt(2)*erf((-1)^(1/4)*x))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.37 \[ \int x^2 \sin ^3\left (x^2\right ) \, dx=\left (\frac {1}{576} i + \frac {1}{576}\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} x\right ) - \left (\frac {1}{576} i - \frac {1}{576}\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} x\right ) - \left (\frac {3}{64} i + \frac {3}{64}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} x\right ) + \left (\frac {3}{64} i - \frac {3}{64}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} x\right ) + \frac {1}{48} \, x e^{\left (3 i \, x^{2}\right )} - \frac {3}{16} \, x e^{\left (i \, x^{2}\right )} - \frac {3}{16} \, x e^{\left (-i \, x^{2}\right )} + \frac {1}{48} \, x e^{\left (-3 i \, x^{2}\right )} \] Input:
integrate(x^2*sin(x^2)^3,x, algorithm="giac")
Output:
(1/576*I + 1/576)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*x) - (1/576*I - 1/576)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*x) - (3/64*I + 3/64) *sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*x) + (3/64*I - 3/64)*sqrt(2)*s qrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*x) + 1/48*x*e^(3*I*x^2) - 3/16*x*e^(I*x ^2) - 3/16*x*e^(-I*x^2) + 1/48*x*e^(-3*I*x^2)
Timed out. \[ \int x^2 \sin ^3\left (x^2\right ) \, dx=\int x^2\,{\sin \left (x^2\right )}^3 \,d x \] Input:
int(x^2*sin(x^2)^3,x)
Output:
int(x^2*sin(x^2)^3, x)
\[ \int x^2 \sin ^3\left (x^2\right ) \, dx=\int \sin \left (x^{2}\right )^{3} x^{2}d x \] Input:
int(x^2*sin(x^2)^3,x)
Output:
int(sin(x**2)**3*x**2,x)