Integrand size = 18, antiderivative size = 228 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx=-\frac {16 b^2}{105 x^{5/6}}+\frac {256 b^4}{315 \sqrt [6]{x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac {512}{315} b^{9/2} \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-\frac {512}{315} b^{9/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}} \]
-16/105*b^2/x^(5/6)+256/315*b^4/x^(1/6)-2/3*cos(a+b*x^(1/3))^2/x^(3/2)+32/ 105*b^2*cos(a+b*x^(1/3))^2/x^(5/6)-512/315*b^4*cos(a+b*x^(1/3))^2/x^(1/6)+ 8/21*b*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/x^(7/6)-512/315*b^(9/2)*cos(2*a)* FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))*Pi^(1/2)-512/315*b^(9/2)*FresnelC(2*x ^(1/6)*b^(1/2)/Pi^(1/2))*sin(2*a)*Pi^(1/2)-128/315*b^3*cos(a+b*x^(1/3))*si n(a+b*x^(1/3))/x^(1/2)
Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx=\frac {-105-105 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+48 b^2 x^{2/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-256 b^4 x^{4/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-512 b^{9/2} \sqrt {\pi } x^{3/2} \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-512 b^{9/2} \sqrt {\pi } x^{3/2} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+60 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-64 b^3 x \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{315 x^{3/2}} \]
(-105 - 105*Cos[2*(a + b*x^(1/3))] + 48*b^2*x^(2/3)*Cos[2*(a + b*x^(1/3))] - 256*b^4*x^(4/3)*Cos[2*(a + b*x^(1/3))] - 512*b^(9/2)*Sqrt[Pi]*x^(3/2)*C os[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] - 512*b^(9/2)*Sqrt[Pi]*x^(3 /2)*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + 60*b*x^(1/3)*Sin[2*( a + b*x^(1/3))] - 64*b^3*x*Sin[2*(a + b*x^(1/3))])/(315*x^(3/2))
Time = 0.98 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.944, Rules used = {3897, 3042, 3795, 15, 3042, 3795, 15, 3042, 3794, 27, 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.
| \(\displaystyle \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3897 |
| \(\displaystyle 3 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{11/6}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{x^{11/6}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/6}}d\sqrt [3]{x}+\frac {8}{63} b^2 \int \frac {1}{x^{7/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{x^{7/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}+\frac {8}{15} b^2 \int \frac {1}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (4 b \int -\frac {\sin \left (2 a+2 b \sqrt [3]{x}\right )}{2 \sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \int \frac {\sin \left (2 a+2 b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \int \frac {\sin \left (2 a+2 b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \left (\sin (2 a) \int \frac {\cos \left (2 b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}+\cos (2 a) \int \frac {\sin \left (2 b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \left (\sin (2 a) \int \frac {\sin \left (2 \sqrt [3]{x} b+\frac {\pi }{2}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}+\cos (2 a) \int \frac {\sin \left (2 b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \left (2 \sin (2 a) \int \cos \left (2 b x^{2/3}\right )d\sqrt [6]{x}+\cos (2 a) \int \frac {\sin \left (2 b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \left (2 \sin (2 a) \int \cos \left (2 b x^{2/3}\right )d\sqrt [6]{x}+2 \cos (2 a) \int \sin \left (2 b x^{2/3}\right )d\sqrt [6]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \left (2 \sin (2 a) \int \cos \left (2 b x^{2/3}\right )d\sqrt [6]{x}+\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{\sqrt {b}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle 3 \left (-\frac {16}{63} b^2 \left (-\frac {16}{15} b^2 \left (-2 b \left (\frac {\sqrt {\pi } \sin (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{\sqrt {b}}+\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{\sqrt {b}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{15 \sqrt {x}}-\frac {16 b^2}{15 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{63 x^{7/6}}-\frac {16 b^2}{315 x^{5/6}}\right )\) |
3*((-16*b^2)/(315*x^(5/6)) - (2*Cos[a + b*x^(1/3)]^2)/(9*x^(3/2)) + (8*b*C os[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(63*x^(7/6)) - (16*b^2*((-16*b^2)/(1 5*x^(1/6)) - (2*Cos[a + b*x^(1/3)]^2)/(5*x^(5/6)) - (16*b^2*((-2*Cos[a + b *x^(1/3)]^2)/x^(1/6) - 2*b*((Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6) )/Sqrt[Pi]])/Sqrt[b] + (Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Si n[2*a])/Sqrt[b])))/15 + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(15*Sq rt[x])))/63)
3.1.59.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
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[((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]
Time = 0.66 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.64
| method | result | size |
| derivativedivides | \(-\frac {1}{3 x^{\frac {3}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 x^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}-2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3}\) | \(146\) |
| default | \(-\frac {1}{3 x^{\frac {3}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 x^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}-2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3}\) | \(146\) |
-1/3/x^(3/2)-1/3/x^(3/2)*cos(2*a+2*b*x^(1/3))-4/3*b*(-1/7/x^(7/6)*sin(2*a+ 2*b*x^(1/3))+4/7*b*(-1/5/x^(5/6)*cos(2*a+2*b*x^(1/3))-4/5*b*(-1/3/x^(1/2)* sin(2*a+2*b*x^(1/3))+4/3*b*(-1/x^(1/6)*cos(2*a+2*b*x^(1/3))-2*b^(1/2)*Pi^( 1/2)*(cos(2*a)*FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))+sin(2*a)*FresnelC(2*x^ (1/6)*b^(1/2)/Pi^(1/2)))))))
Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx=-\frac {2 \, {\left (256 \, \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + 256 \, \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 128 \, b^{4} x^{\frac {11}{6}} + 24 \, b^{2} x^{\frac {7}{6}} + {\left (256 \, b^{4} x^{\frac {11}{6}} - 48 \, b^{2} x^{\frac {7}{6}} + 105 \, \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 4 \, {\left (16 \, b^{3} x^{\frac {3}{2}} - 15 \, b x^{\frac {5}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{315 \, x^{2}} \]
-2/315*(256*pi*b^4*x^2*sqrt(b/pi)*cos(2*a)*fresnel_sin(2*x^(1/6)*sqrt(b/pi )) + 256*pi*b^4*x^2*sqrt(b/pi)*fresnel_cos(2*x^(1/6)*sqrt(b/pi))*sin(2*a) - 128*b^4*x^(11/6) + 24*b^2*x^(7/6) + (256*b^4*x^(11/6) - 48*b^2*x^(7/6) + 105*sqrt(x))*cos(b*x^(1/3) + a)^2 + 4*(16*b^3*x^(3/2) - 15*b*x^(5/6))*cos (b*x^(1/3) + a)*sin(b*x^(1/3) + a))/x^2
\[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx=\int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {5}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.39 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx=-\frac {18 \, \sqrt {2} {\left ({\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, 2 i \, b x^{\frac {1}{3}}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (2 \, a\right ) + {\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, 2 i \, b x^{\frac {1}{3}}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (2 \, a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b^{4} x^{\frac {4}{3}} + 1}{3 \, x^{\frac {3}{2}}} \]
-1/3*(18*sqrt(2)*(((I + 1)*sqrt(2)*gamma(-9/2, 2*I*b*x^(1/3)) - (I - 1)*sq rt(2)*gamma(-9/2, -2*I*b*x^(1/3)))*cos(2*a) + (-(I - 1)*sqrt(2)*gamma(-9/2 , 2*I*b*x^(1/3)) + (I + 1)*sqrt(2)*gamma(-9/2, -2*I*b*x^(1/3)))*sin(2*a))* sqrt(b*x^(1/3))*b^4*x^(4/3) + 1)/x^(3/2)
\[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx=\int { \frac {\cos \left (b x^{\frac {1}{3}} + a\right )^{2}}{x^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx=\int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{x^{5/2}} \,d x \]