Integrand size = 10, antiderivative size = 132 \[ \int \left (a \cos ^4(x)\right )^{5/2} \, dx=\frac {63}{256} a^2 x \sqrt {a \cos ^4(x)} \sec ^2(x)+\frac {21}{128} a^2 \cos (x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {21}{160} a^2 \cos ^3(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {9}{80} a^2 \cos ^5(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{10} a^2 \cos ^7(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {63}{256} a^2 \sqrt {a \cos ^4(x)} \tan (x) \]
63/256*a^2*x*sec(x)^2*(a*cos(x)^4)^(1/2)+21/128*a^2*cos(x)*sin(x)*(a*cos(x )^4)^(1/2)+21/160*a^2*cos(x)^3*sin(x)*(a*cos(x)^4)^(1/2)+9/80*a^2*cos(x)^5 *sin(x)*(a*cos(x)^4)^(1/2)+1/10*a^2*cos(x)^7*sin(x)*(a*cos(x)^4)^(1/2)+63/ 256*a^2*(a*cos(x)^4)^(1/2)*tan(x)
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.40 \[ \int \left (a \cos ^4(x)\right )^{5/2} \, dx=\frac {a \left (a \cos ^4(x)\right )^{3/2} \sec ^6(x) (2520 x+2100 \sin (2 x)+600 \sin (4 x)+150 \sin (6 x)+25 \sin (8 x)+2 \sin (10 x))}{10240} \]
(a*(a*Cos[x]^4)^(3/2)*Sec[x]^6*(2520*x + 2100*Sin[2*x] + 600*Sin[4*x] + 15 0*Sin[6*x] + 25*Sin[8*x] + 2*Sin[10*x]))/10240
Time = 0.48 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \left (a \cos ^4(x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (x+\frac {\pi }{2}\right )^4\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \int \cos ^{10}(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \int \sin \left (x+\frac {\pi }{2}\right )^{10}dx\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \int \cos ^8(x)dx+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \int \sin \left (x+\frac {\pi }{2}\right )^8dx+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \int \cos ^6(x)dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \int \sin \left (x+\frac {\pi }{2}\right )^6dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \cos ^4(x)dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^2 \sec ^2(x) \sqrt {a \cos ^4(x)} \left (\frac {1}{10} \sin (x) \cos ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sin (x) \cos ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )\right )\right )\) |
a^2*Sqrt[a*Cos[x]^4]*Sec[x]^2*((Cos[x]^9*Sin[x])/10 + (9*((Cos[x]^7*Sin[x] )/8 + (7*((Cos[x]^5*Sin[x])/6 + (5*((Cos[x]^3*Sin[x])/4 + (3*(x/2 + (Cos[x ]*Sin[x])/2))/4))/6))/8))/10)
3.1.51.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 13.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.42
| method | result | size |
| default | \(\frac {a^{2} \sqrt {a \left (\cos ^{4}\left (x \right )\right )}\, \left (128 \left (\cos ^{7}\left (x \right )\right ) \sin \left (x \right )+144 \left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )+168 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )+210 \cos \left (x \right ) \sin \left (x \right )+315 \tan \left (x \right )+315 \left (\sec ^{2}\left (x \right )\right ) x \right )}{1280}\) | \(56\) |
| risch | \(\frac {63 a^{2} {\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}\, x}{256 \left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {i a^{2} {\mathrm e}^{12 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}{10240 \left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {5 i a^{2} {\mathrm e}^{10 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}{4096 \left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {105 i a^{2} {\mathrm e}^{4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}{1024 \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {105 i a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}{1024 \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {15 i a^{2} {\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}{512 \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {15 i a^{2} {\mathrm e}^{-4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}{2048 \left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {37 i a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}\, \cos \left (8 x \right )}{5120 \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {19 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}\, \sin \left (8 x \right )}{2560 \left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {115 i a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}\, \cos \left (6 x \right )}{4096 \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {125 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}\, \sin \left (6 x \right )}{4096 \left ({\mathrm e}^{2 i x}+1\right )^{2}}\) | \(409\) |
1/1280*a^2*(a*cos(x)^4)^(1/2)*(128*cos(x)^7*sin(x)+144*cos(x)^5*sin(x)+168 *cos(x)^3*sin(x)+210*cos(x)*sin(x)+315*tan(x)+315*sec(x)^2*x)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.52 \[ \int \left (a \cos ^4(x)\right )^{5/2} \, dx=\frac {\sqrt {a \cos \left (x\right )^{4}} {\left (315 \, a^{2} x + {\left (128 \, a^{2} \cos \left (x\right )^{9} + 144 \, a^{2} \cos \left (x\right )^{7} + 168 \, a^{2} \cos \left (x\right )^{5} + 210 \, a^{2} \cos \left (x\right )^{3} + 315 \, a^{2} \cos \left (x\right )\right )} \sin \left (x\right )\right )}}{1280 \, \cos \left (x\right )^{2}} \]
1/1280*sqrt(a*cos(x)^4)*(315*a^2*x + (128*a^2*cos(x)^9 + 144*a^2*cos(x)^7 + 168*a^2*cos(x)^5 + 210*a^2*cos(x)^3 + 315*a^2*cos(x))*sin(x))/cos(x)^2
Timed out. \[ \int \left (a \cos ^4(x)\right )^{5/2} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.64 \[ \int \left (a \cos ^4(x)\right )^{5/2} \, dx=\frac {63}{256} \, a^{\frac {5}{2}} x + \frac {315 \, a^{\frac {5}{2}} \tan \left (x\right )^{9} + 1470 \, a^{\frac {5}{2}} \tan \left (x\right )^{7} + 2688 \, a^{\frac {5}{2}} \tan \left (x\right )^{5} + 2370 \, a^{\frac {5}{2}} \tan \left (x\right )^{3} + 965 \, a^{\frac {5}{2}} \tan \left (x\right )}{1280 \, {\left (\tan \left (x\right )^{10} + 5 \, \tan \left (x\right )^{8} + 10 \, \tan \left (x\right )^{6} + 10 \, \tan \left (x\right )^{4} + 5 \, \tan \left (x\right )^{2} + 1\right )}} \]
63/256*a^(5/2)*x + 1/1280*(315*a^(5/2)*tan(x)^9 + 1470*a^(5/2)*tan(x)^7 + 2688*a^(5/2)*tan(x)^5 + 2370*a^(5/2)*tan(x)^3 + 965*a^(5/2)*tan(x))/(tan(x )^10 + 5*tan(x)^8 + 10*tan(x)^6 + 10*tan(x)^4 + 5*tan(x)^2 + 1)
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.43 \[ \int \left (a \cos ^4(x)\right )^{5/2} \, dx=\frac {1}{10240} \, {\left (2520 \, a^{2} x + 2 \, a^{2} \sin \left (10 \, x\right ) + 25 \, a^{2} \sin \left (8 \, x\right ) + 150 \, a^{2} \sin \left (6 \, x\right ) + 600 \, a^{2} \sin \left (4 \, x\right ) + 2100 \, a^{2} \sin \left (2 \, x\right )\right )} \sqrt {a} \]
1/10240*(2520*a^2*x + 2*a^2*sin(10*x) + 25*a^2*sin(8*x) + 150*a^2*sin(6*x) + 600*a^2*sin(4*x) + 2100*a^2*sin(2*x))*sqrt(a)
Timed out. \[ \int \left (a \cos ^4(x)\right )^{5/2} \, dx=\int {\left (a\,{\cos \left (x\right )}^4\right )}^{5/2} \,d x \]