Integrand size = 18, antiderivative size = 310 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {405405 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {405405 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{32768 b^{15/2}}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7} \]
3861/256*x^(7/6)/b^4-39/16*x^(11/6)/b^2+1/5*x^(5/2)-3861/128*x^(7/6)*cos(a +b*x^(1/3))^2/b^4+39/8*x^(11/6)*cos(a+b*x^(1/3))^2/b^2+27027/512*x^(5/6)*c os(a+b*x^(1/3))*sin(a+b*x^(1/3))/b^5-429/32*x^(3/2)*cos(a+b*x^(1/3))*sin(a +b*x^(1/3))/b^3+3/2*x^(13/6)*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/b-405405/16 384*x^(1/6)*sin(2*a+2*b*x^(1/3))/b^7+405405/32768*cos(2*a)*FresnelS(2*x^(1 /6)*b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(15/2)+405405/32768*FresnelC(2*x^(1/6)*b^ (1/2)/Pi^(1/2))*sin(2*a)*Pi^(1/2)/b^(15/2)-135135/4096*x^(1/2)/b^6+135135/ 2048*cos(a+b*x^(1/3))^2*x^(1/2)/b^6
Time = 0.73 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.56 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\frac {2027025 \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+2027025 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+2 \sqrt {b} \sqrt [6]{x} \left (16384 b^7 x^{7/3}+780 \left (3465 b \sqrt [3]{x}-1584 b^3 x+256 b^5 x^{5/3}\right ) \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+15 \left (-135135+144144 b^2 x^{2/3}-36608 b^4 x^{4/3}+4096 b^6 x^2\right ) \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )\right )}{163840 b^{15/2}} \]
(2027025*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] + 202702 5*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + 2*Sqrt[b]*x^( 1/6)*(16384*b^7*x^(7/3) + 780*(3465*b*x^(1/3) - 1584*b^3*x + 256*b^5*x^(5/ 3))*Cos[2*(a + b*x^(1/3))] + 15*(-135135 + 144144*b^2*x^(2/3) - 36608*b^4* x^(4/3) + 4096*b^6*x^2)*Sin[2*(a + b*x^(1/3))]))/(163840*b^(15/2))
Time = 0.85 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {3897, 3042, 3792, 15, 3042, 3792, 15, 3042, 3792, 15, 3042, 3793, 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.
| \(\displaystyle \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx\) |
| \(\Big \downarrow \) 3897 |
| \(\displaystyle 3 \int x^{13/6} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \int x^{13/6} \sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle 3 \left (-\frac {143 \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}}{16 b^2}+\frac {1}{2} \int x^{13/6}d\sqrt [3]{x}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {143 \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {143 \int x^{3/2} \sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2d\sqrt [3]{x}}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle 3 \left (-\frac {143 \left (-\frac {63 \int x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}}{16 b^2}+\frac {1}{2} \int x^{3/2}d\sqrt [3]{x}+\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}\right )}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {143 \left (-\frac {63 \int x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}}{16 b^2}+\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{11/6}}{11}\right )}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {143 \left (-\frac {63 \int x^{5/6} \sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2d\sqrt [3]{x}}{16 b^2}+\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{11/6}}{11}\right )}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle 3 \left (-\frac {143 \left (-\frac {63 \left (-\frac {15 \int \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}}{16 b^2}+\frac {1}{2} \int x^{5/6}d\sqrt [3]{x}+\frac {5 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}\right )}{16 b^2}+\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{11/6}}{11}\right )}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {143 \left (-\frac {63 \left (-\frac {15 \int \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )d\sqrt [3]{x}}{16 b^2}+\frac {5 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{7/6}}{7}\right )}{16 b^2}+\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{11/6}}{11}\right )}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {143 \left (-\frac {63 \left (-\frac {15 \int \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2d\sqrt [3]{x}}{16 b^2}+\frac {5 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{7/6}}{7}\right )}{16 b^2}+\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{11/6}}{11}\right )}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle 3 \left (-\frac {143 \left (-\frac {63 \left (-\frac {15 \int \left (\frac {1}{2} \sqrt [6]{x} \cos \left (2 a+2 b \sqrt [3]{x}\right )+\frac {\sqrt [6]{x}}{2}\right )d\sqrt [3]{x}}{16 b^2}+\frac {5 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{7/6}}{7}\right )}{16 b^2}+\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{11/6}}{11}\right )}{16 b^2}+\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {13 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {143 \left (\frac {9 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {63 \left (\frac {5 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {15 \left (-\frac {\sqrt {\pi } \sin (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{8 b^{3/2}}+\frac {\sqrt [6]{x} \sin \left (2 a+2 b \sqrt [3]{x}\right )}{4 b}+\frac {\sqrt {x}}{3}\right )}{16 b^2}+\frac {x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{7/6}}{7}\right )}{16 b^2}+\frac {x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{11/6}}{11}\right )}{16 b^2}+\frac {x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {x^{5/2}}{15}\right )\) |
3*(x^(5/2)/15 + (13*x^(11/6)*Cos[a + b*x^(1/3)]^2)/(8*b^2) + (x^(13/6)*Cos [a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(2*b) - (143*(x^(11/6)/11 + (9*x^(7/6) *Cos[a + b*x^(1/3)]^2)/(8*b^2) + (x^(3/2)*Cos[a + b*x^(1/3)]*Sin[a + b*x^( 1/3)])/(2*b) - (63*(x^(7/6)/7 + (5*Sqrt[x]*Cos[a + b*x^(1/3)]^2)/(8*b^2) + (x^(5/6)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(2*b) - (15*(Sqrt[x]/3 - (Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(8*b^(3/2)) - ( Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/(8*b^(3/2)) + (x ^(1/6)*Sin[2*a + 2*b*x^(1/3)])/(4*b)))/(16*b^2)))/(16*b^2)))/(16*b^2))
3.1.55.3.1 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[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^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] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
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 = 0.58 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(\frac {x^{\frac {5}{2}}}{5}+\frac {3 x^{\frac {13}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {39 \left (-\frac {x^{\frac {11}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {11 x^{\frac {3}{2}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {99 \left (-\frac {x^{\frac {7}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {7 x^{\frac {5}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {35 \left (-\frac {\sqrt {x}\, \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{32 b^{\frac {3}{2}}}}{b}\right )}{16 b}}{b}\right )}{16 b}}{b}\right )}{4 b}\) | \(219\) |
| default | \(\frac {x^{\frac {5}{2}}}{5}+\frac {3 x^{\frac {13}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {39 \left (-\frac {x^{\frac {11}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {11 x^{\frac {3}{2}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {99 \left (-\frac {x^{\frac {7}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {7 x^{\frac {5}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {35 \left (-\frac {\sqrt {x}\, \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{32 b^{\frac {3}{2}}}}{b}\right )}{16 b}}{b}\right )}{16 b}}{b}\right )}{4 b}\) | \(219\) |
1/5*x^(5/2)+3/4/b*x^(13/6)*sin(2*a+2*b*x^(1/3))-39/4/b*(-1/4/b*x^(11/6)*co s(2*a+2*b*x^(1/3))+11/4/b*(1/4/b*x^(3/2)*sin(2*a+2*b*x^(1/3))-9/4/b*(-1/4/ b*x^(7/6)*cos(2*a+2*b*x^(1/3))+7/4/b*(1/4/b*x^(5/6)*sin(2*a+2*b*x^(1/3))-5 /4/b*(-1/4/b*x^(1/2)*cos(2*a+2*b*x^(1/3))+3/4/b*(1/4*x^(1/6)*sin(2*a+2*b*x ^(1/3))/b-1/8/b^(3/2)*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 = 184, normalized size of antiderivative = 0.59 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=-\frac {399360 \, b^{6} x^{\frac {11}{6}} - 2471040 \, b^{4} x^{\frac {7}{6}} - 2027025 \, \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 2027025 \, \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 3120 \, {\left (256 \, b^{6} x^{\frac {11}{6}} - 1584 \, b^{4} x^{\frac {7}{6}} + 3465 \, b^{2} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 60 \, {\left (36608 \, b^{5} x^{\frac {3}{2}} - 144144 \, b^{3} x^{\frac {5}{6}} - {\left (4096 \, b^{7} x^{2} - 135135 \, b\right )} x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) - 8 \, {\left (4096 \, b^{8} x^{2} - 675675 \, b^{2}\right )} \sqrt {x}}{163840 \, b^{8}} \]
-1/163840*(399360*b^6*x^(11/6) - 2471040*b^4*x^(7/6) - 2027025*pi*sqrt(b/p i)*cos(2*a)*fresnel_sin(2*x^(1/6)*sqrt(b/pi)) - 2027025*pi*sqrt(b/pi)*fres nel_cos(2*x^(1/6)*sqrt(b/pi))*sin(2*a) - 3120*(256*b^6*x^(11/6) - 1584*b^4 *x^(7/6) + 3465*b^2*sqrt(x))*cos(b*x^(1/3) + a)^2 + 60*(36608*b^5*x^(3/2) - 144144*b^3*x^(5/6) - (4096*b^7*x^2 - 135135*b)*x^(1/6))*cos(b*x^(1/3) + a)*sin(b*x^(1/3) + a) - 8*(4096*b^8*x^2 - 675675*b^2)*sqrt(x))/b^8
\[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\int x^{\frac {3}{2}} \cos ^{2}{\left (a + b \sqrt [3]{x} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.52 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\frac {262144 \, b^{9} x^{\frac {5}{2}} + 2027025 \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \cos \left (2 \, a\right ) - \left (i - 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (2 \, a\right ) + \left (i + 1\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 12480 \, {\left (256 \, b^{7} x^{\frac {11}{6}} - 1584 \, b^{5} x^{\frac {7}{6}} + 3465 \, b^{3} \sqrt {x}\right )} \cos \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right ) + 240 \, {\left (4096 \, b^{8} x^{\frac {13}{6}} - 36608 \, b^{6} x^{\frac {3}{2}} + 144144 \, b^{4} x^{\frac {5}{6}} - 135135 \, b^{2} x^{\frac {1}{6}}\right )} \sin \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right )}{1310720 \, b^{9}} \]
1/1310720*(262144*b^9*x^(5/2) + 2027025*4^(1/4)*sqrt(2)*sqrt(pi)*(((I + 1) *cos(2*a) - (I - 1)*sin(2*a))*erf(sqrt(2*I*b)*x^(1/6)) + (-(I - 1)*cos(2*a ) + (I + 1)*sin(2*a))*erf(sqrt(-2*I*b)*x^(1/6)))*b^(3/2) + 12480*(256*b^7* x^(11/6) - 1584*b^5*x^(7/6) + 3465*b^3*sqrt(x))*cos(2*b*x^(1/3) + 2*a) + 2 40*(4096*b^8*x^(13/6) - 36608*b^6*x^(3/2) + 144144*b^4*x^(5/6) - 135135*b^ 2*x^(1/6))*sin(2*b*x^(1/3) + 2*a))/b^9
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.72 \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\frac {1}{5} \, x^{\frac {5}{2}} - \frac {3 \, {\left (4096 i \, b^{6} x^{\frac {13}{6}} - 13312 \, b^{5} x^{\frac {11}{6}} - 36608 i \, b^{4} x^{\frac {3}{2}} + 82368 \, b^{3} x^{\frac {7}{6}} + 144144 i \, b^{2} x^{\frac {5}{6}} - 180180 \, b \sqrt {x} - 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (2 i \, b x^{\frac {1}{3}} + 2 i \, a\right )}}{32768 \, b^{7}} - \frac {3 \, {\left (-4096 i \, b^{6} x^{\frac {13}{6}} - 13312 \, b^{5} x^{\frac {11}{6}} + 36608 i \, b^{4} x^{\frac {3}{2}} + 82368 \, b^{3} x^{\frac {7}{6}} - 144144 i \, b^{2} x^{\frac {5}{6}} - 180180 \, b \sqrt {x} + 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (-2 i \, b x^{\frac {1}{3}} - 2 i \, a\right )}}{32768 \, b^{7}} + \frac {405405 \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {b} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{65536 \, b^{\frac {15}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} + \frac {405405 \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {b} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{65536 \, b^{\frac {15}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
1/5*x^(5/2) - 3/32768*(4096*I*b^6*x^(13/6) - 13312*b^5*x^(11/6) - 36608*I* b^4*x^(3/2) + 82368*b^3*x^(7/6) + 144144*I*b^2*x^(5/6) - 180180*b*sqrt(x) - 135135*I*x^(1/6))*e^(2*I*b*x^(1/3) + 2*I*a)/b^7 - 3/32768*(-4096*I*b^6*x ^(13/6) - 13312*b^5*x^(11/6) + 36608*I*b^4*x^(3/2) + 82368*b^3*x^(7/6) - 1 44144*I*b^2*x^(5/6) - 180180*b*sqrt(x) + 135135*I*x^(1/6))*e^(-2*I*b*x^(1/ 3) - 2*I*a)/b^7 + 405405/65536*sqrt(pi)*erf(-I*sqrt(b)*x^(1/6)*(I*b/abs(b) + 1))*e^(2*I*a)/(b^(15/2)*(I*b/abs(b) + 1)) + 405405/65536*sqrt(pi)*erf(I *sqrt(b)*x^(1/6)*(-I*b/abs(b) + 1))*e^(-2*I*a)/(b^(15/2)*(-I*b/abs(b) + 1) )
Timed out. \[ \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx=\int x^{3/2}\,{\cos \left (a+b\,x^{1/3}\right )}^2 \,d x \]