Integrand size = 10, antiderivative size = 132 \[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\frac {63 x \sec ^2(x)}{256 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt {a \sec ^4(x)}}+\frac {9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {\cos ^7(x) \sin (x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {63 \tan (x)}{256 a^2 \sqrt {a \sec ^4(x)}} \] Output:
63/256*x*sec(x)^2/a^2/(a*sec(x)^4)^(1/2)+21/128*cos(x)*sin(x)/a^2/(a*sec(x )^4)^(1/2)+21/160*cos(x)^3*sin(x)/a^2/(a*sec(x)^4)^(1/2)+9/80*cos(x)^5*sin (x)/a^2/(a*sec(x)^4)^(1/2)+1/10*cos(x)^7*sin(x)/a^2/(a*sec(x)^4)^(1/2)+63/ 256*tan(x)/a^2/(a*sec(x)^4)^(1/2)
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\frac {\cos ^2(x) \sqrt {a \sec ^4(x)} (2520 x+2100 \sin (2 x)+600 \sin (4 x)+150 \sin (6 x)+25 \sin (8 x)+2 \sin (10 x))}{10240 a^3} \] Input:
Integrate[(a*Sec[x]^4)^(-5/2),x]
Output:
(Cos[x]^2*Sqrt[a*Sec[x]^4]*(2520*x + 2100*Sin[2*x] + 600*Sin[4*x] + 150*Si n[6*x] + 25*Sin[8*x] + 2*Sin[10*x]))/(10240*a^3)
Time = 0.49 (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, 4611, 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.
| \(\displaystyle \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sec (x)^4\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle \frac {\sec ^2(x) \int \cos ^{10}(x)dx}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^2(x) \int \sin \left (x+\frac {\pi }{2}\right )^{10}dx}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \int \cos ^8(x)dx+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \int \sin \left (x+\frac {\pi }{2}\right )^8dx+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \int \cos ^6(x)dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \int \sin \left (x+\frac {\pi }{2}\right )^6dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \cos ^4(x)dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )+\frac {1}{10} \sin (x) \cos ^9(x)\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\sec ^2(x) \left (\frac {1}{10} \sin (x) \cos ^9(x)+\frac {9}{10} \left (\frac {1}{8} \sin (x) \cos ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )\right )\right )}{a^2 \sqrt {a \sec ^4(x)}}\) |
Input:
Int[(a*Sec[x]^4)^(-5/2),x]
Output:
(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))/(a^2*Sqrt[a*Sec[x]^4])
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[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart [p])) Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
Time = 0.66 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.39
| method | result | size |
| default | \(\frac {\frac {\left (128 \cos \left (x \right )^{8}+144 \cos \left (x \right )^{6}+168 \cos \left (x \right )^{4}+210 \cos \left (x \right )^{2}+315\right ) \tan \left (x \right )}{1280}+\frac {63 x \sec \left (x \right )^{2}}{256}}{\sqrt {a \sec \left (x \right )^{4}}\, a^{2}}\) | \(51\) |
| risch | \(\frac {63 \,{\mathrm e}^{2 i x} x}{256 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}-\frac {i {\mathrm e}^{12 i x}}{10240 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}-\frac {5 i {\mathrm e}^{10 i x}}{4096 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}-\frac {105 i {\mathrm e}^{4 i x}}{1024 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}+\frac {105 i}{1024 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}+\frac {15 i {\mathrm e}^{-2 i x}}{512 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}+\frac {15 i {\mathrm e}^{-4 i x}}{2048 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}-\frac {37 i \cos \left (8 x \right )}{5120 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}+\frac {19 \sin \left (8 x \right )}{2560 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}-\frac {115 i \cos \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}+\frac {125 \sin \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}}\) | \(409\) |
Input:
int(1/(a*sec(x)^4)^(5/2),x,method=_RETURNVERBOSE)
Output:
(1/1280*(128*cos(x)^8+144*cos(x)^6+168*cos(x)^4+210*cos(x)^2+315)*tan(x)+6 3/256*x*sec(x)^2)/(a*sec(x)^4)^(1/2)/a^2
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\frac {{\left (315 \, x \cos \left (x\right )^{2} + {\left (128 \, \cos \left (x\right )^{11} + 144 \, \cos \left (x\right )^{9} + 168 \, \cos \left (x\right )^{7} + 210 \, \cos \left (x\right )^{5} + 315 \, \cos \left (x\right )^{3}\right )} \sin \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4}}}}{1280 \, a^{3}} \] Input:
integrate(1/(a*sec(x)^4)^(5/2),x, algorithm="fricas")
Output:
1/1280*(315*x*cos(x)^2 + (128*cos(x)^11 + 144*cos(x)^9 + 168*cos(x)^7 + 21 0*cos(x)^5 + 315*cos(x)^3)*sin(x))*sqrt(a/cos(x)^4)/a^3
\[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \sec ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a*sec(x)**4)**(5/2),x)
Output:
Integral((a*sec(x)**4)**(-5/2), x)
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\frac {315 \, \tan \left (x\right )^{9} + 1470 \, \tan \left (x\right )^{7} + 2688 \, \tan \left (x\right )^{5} + 2370 \, \tan \left (x\right )^{3} + 965 \, \tan \left (x\right )}{1280 \, {\left (a^{\frac {5}{2}} \tan \left (x\right )^{10} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{8} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{6} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{4} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{2} + a^{\frac {5}{2}}\right )}} + \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \] Input:
integrate(1/(a*sec(x)^4)^(5/2),x, algorithm="maxima")
Output:
1/1280*(315*tan(x)^9 + 1470*tan(x)^7 + 2688*tan(x)^5 + 2370*tan(x)^3 + 965 *tan(x))/(a^(5/2)*tan(x)^10 + 5*a^(5/2)*tan(x)^8 + 10*a^(5/2)*tan(x)^6 + 1 0*a^(5/2)*tan(x)^4 + 5*a^(5/2)*tan(x)^2 + a^(5/2)) + 63/256*x/a^(5/2)
Exception generated. \[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a*sec(x)^4)^(5/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\cos \left (x\right )}^4}\right )}^{5/2}} \,d x \] Input:
int(1/(a/cos(x)^4)^(5/2),x)
Output:
int(1/(a/cos(x)^4)^(5/2), x)
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (128 \cos \left (x \right ) \sin \left (x \right )^{9}-656 \cos \left (x \right ) \sin \left (x \right )^{7}+1368 \cos \left (x \right ) \sin \left (x \right )^{5}-1490 \cos \left (x \right ) \sin \left (x \right )^{3}+965 \cos \left (x \right ) \sin \left (x \right )+315 x \right )}{1280 a^{3}} \] Input:
int(1/(a*sec(x)^4)^(5/2),x)
Output:
(sqrt(a)*(128*cos(x)*sin(x)**9 - 656*cos(x)*sin(x)**7 + 1368*cos(x)*sin(x) **5 - 1490*cos(x)*sin(x)**3 + 965*cos(x)*sin(x) + 315*x))/(1280*a**3)