Integrand size = 14, antiderivative size = 111 \[ \int x^{5/2} \cos \left (a+b x^2\right ) \, dx=-\frac {3 i e^{i a} x^{3/2} \Gamma \left (\frac {3}{4},-i b x^2\right )}{16 b \left (-i b x^2\right )^{3/4}}+\frac {3 i e^{-i a} x^{3/2} \Gamma \left (\frac {3}{4},i b x^2\right )}{16 b \left (i b x^2\right )^{3/4}}+\frac {x^{3/2} \sin \left (a+b x^2\right )}{2 b} \]
-3/16*I*exp(I*a)*x^(3/2)*GAMMA(3/4,-I*b*x^2)/b/(-I*b*x^2)^(3/4)+3/16*I*x^( 3/2)*GAMMA(3/4,I*b*x^2)/b/exp(I*a)/(I*b*x^2)^(3/4)+1/2*x^(3/2)*sin(b*x^2+a )/b
Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int x^{5/2} \cos \left (a+b x^2\right ) \, dx=\frac {b x^{11/2} \left (3 \left (i b x^2\right )^{3/4} \Gamma \left (\frac {3}{4},-i b x^2\right ) (-i \cos (a)+\sin (a))+3 \left (-i b x^2\right )^{3/4} \Gamma \left (\frac {3}{4},i b x^2\right ) (i \cos (a)+\sin (a))+8 \left (b^2 x^4\right )^{3/4} \sin \left (a+b x^2\right )\right )}{16 \left (b^2 x^4\right )^{7/4}} \]
(b*x^(11/2)*(3*(I*b*x^2)^(3/4)*Gamma[3/4, (-I)*b*x^2]*((-I)*Cos[a] + Sin[a ]) + 3*((-I)*b*x^2)^(3/4)*Gamma[3/4, I*b*x^2]*(I*Cos[a] + Sin[a]) + 8*(b^2 *x^4)^(3/4)*Sin[a + b*x^2]))/(16*(b^2*x^4)^(7/4))
Time = 0.30 (sec) , antiderivative size = 113, 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 = {3867, 3870, 2648}
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.
\(\displaystyle \int x^{5/2} \cos \left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {x^{3/2} \sin \left (a+b x^2\right )}{2 b}-\frac {3 \int \sqrt {x} \sin \left (b x^2+a\right )dx}{4 b}\) |
\(\Big \downarrow \) 3870 |
\(\displaystyle \frac {x^{3/2} \sin \left (a+b x^2\right )}{2 b}-\frac {3 \left (\frac {1}{2} i \int e^{-i b x^2-i a} \sqrt {x}dx-\frac {1}{2} i \int e^{i b x^2+i a} \sqrt {x}dx\right )}{4 b}\) |
\(\Big \downarrow \) 2648 |
\(\displaystyle \frac {x^{3/2} \sin \left (a+b x^2\right )}{2 b}-\frac {3 \left (\frac {i e^{i a} x^{3/2} \Gamma \left (\frac {3}{4},-i b x^2\right )}{4 \left (-i b x^2\right )^{3/4}}-\frac {i e^{-i a} x^{3/2} \Gamma \left (\frac {3}{4},i b x^2\right )}{4 \left (i b x^2\right )^{3/4}}\right )}{4 b}\) |
(-3*(((I/4)*E^(I*a)*x^(3/2)*Gamma[3/4, (-I)*b*x^2])/((-I)*b*x^2)^(3/4) - ( (I/4)*x^(3/2)*Gamma[3/4, I*b*x^2])/(E^(I*a)*(I*b*x^2)^(3/4))))/(4*b) + (x^ (3/2)*Sin[a + b*x^2])/(2*b)
3.1.23.3.1 Defintions of rubi rules used
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] /; FreeQ[{F , a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
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]
Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[I/2 Int[(e*x)^m*E^((-c)*I - d*I*x^n), x], x] - Simp[I/2 Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.57 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.06
method | result | size |
meijerg | \(\frac {2^{\frac {3}{4}} \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {2 x^{\frac {3}{2}} 2^{\frac {1}{4}} \left (b^{2}\right )^{\frac {7}{8}} \sin \left (b \,x^{2}\right )}{7 \sqrt {\pi }\, b}+\frac {3 x^{\frac {7}{2}} \left (b^{2}\right )^{\frac {7}{8}} 2^{\frac {1}{4}} \sin \left (b \,x^{2}\right ) s_{\frac {5}{4},\frac {3}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{14 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {5}{4}}}+\frac {3 x^{\frac {7}{2}} \left (b^{2}\right )^{\frac {7}{8}} 2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) s_{\frac {1}{4},\frac {1}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{8 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {9}{4}}}\right )}{2 \left (b^{2}\right )^{\frac {7}{8}}}-\frac {2^{\frac {3}{4}} \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {x^{\frac {7}{2}} b^{\frac {7}{4}} 2^{\frac {1}{4}} \sin \left (b \,x^{2}\right ) s_{\frac {1}{4},\frac {3}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{8 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {5}{4}}}-\frac {x^{\frac {7}{2}} b^{\frac {7}{4}} 2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) s_{\frac {5}{4},\frac {1}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{2 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {9}{4}}}\right )}{2 b^{\frac {7}{4}}}\) | \(229\) |
1/2*2^(3/4)/(b^2)^(7/8)*cos(a)*Pi^(1/2)*(2/7/Pi^(1/2)*x^(3/2)*2^(1/4)*(b^2 )^(7/8)/b*sin(b*x^2)+3/14/Pi^(1/2)*x^(7/2)*(b^2)^(7/8)*2^(1/4)/(b*x^2)^(5/ 4)*sin(b*x^2)*LommelS1(5/4,3/2,b*x^2)+3/8/Pi^(1/2)*x^(7/2)*(b^2)^(7/8)*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))- 1/2*2^(3/4)/b^(7/4)*sin(a)*Pi^(1/2)*(-1/8/Pi^(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)-1/2/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))
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.58 \[ \int x^{5/2} \cos \left (a+b x^2\right ) \, dx=\frac {8 \, b x^{\frac {3}{2}} \sin \left (b x^{2} + a\right ) + 3 \, \left (i \, b\right )^{\frac {1}{4}} {\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \Gamma \left (\frac {3}{4}, i \, b x^{2}\right ) + 3 \, \left (-i \, b\right )^{\frac {1}{4}} {\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \Gamma \left (\frac {3}{4}, -i \, b x^{2}\right )}{16 \, b^{2}} \]
1/16*(8*b*x^(3/2)*sin(b*x^2 + a) + 3*(I*b)^(1/4)*(cos(a) - I*sin(a))*gamma (3/4, I*b*x^2) + 3*(-I*b)^(1/4)*(cos(a) + I*sin(a))*gamma(3/4, -I*b*x^2))/ b^2
\[ \int x^{5/2} \cos \left (a+b x^2\right ) \, dx=\int x^{\frac {5}{2}} \cos {\left (a + b x^{2} \right )}\, dx \]
Exception generated. \[ \int x^{5/2} \cos \left (a+b x^2\right ) \, dx=\text {Exception raised: RuntimeError} \]
\[ \int x^{5/2} \cos \left (a+b x^2\right ) \, dx=\int { x^{\frac {5}{2}} \cos \left (b x^{2} + a\right ) \,d x } \]
Timed out. \[ \int x^{5/2} \cos \left (a+b x^2\right ) \, dx=\int x^{5/2}\,\cos \left (b\,x^2+a\right ) \,d x \]