Integrand size = 10, antiderivative size = 117 \[ \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx=\frac {\cos (x) \sin (x)}{a^2 \sqrt {a \cos ^4(x)}}+\frac {4 \sin ^2(x) \tan (x)}{3 a^2 \sqrt {a \cos ^4(x)}}+\frac {6 \sin ^2(x) \tan ^3(x)}{5 a^2 \sqrt {a \cos ^4(x)}}+\frac {4 \sin ^2(x) \tan ^5(x)}{7 a^2 \sqrt {a \cos ^4(x)}}+\frac {\sin ^2(x) \tan ^7(x)}{9 a^2 \sqrt {a \cos ^4(x)}} \]
cos(x)*sin(x)/a^2/(a*cos(x)^4)^(1/2)+4/3*sin(x)^2*tan(x)/a^2/(a*cos(x)^4)^ (1/2)+6/5*sin(x)^2*tan(x)^3/a^2/(a*cos(x)^4)^(1/2)+4/7*sin(x)^2*tan(x)^5/a ^2/(a*cos(x)^4)^(1/2)+1/9*sin(x)^2*tan(x)^7/a^2/(a*cos(x)^4)^(1/2)
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx=\frac {(128+130 \cos (2 x)+46 \cos (4 x)+10 \cos (6 x)+\cos (8 x)) \sec ^6(x) \tan (x)}{315 a^2 \sqrt {a \cos ^4(x)}} \]
((128 + 130*Cos[2*x] + 46*Cos[4*x] + 10*Cos[6*x] + Cos[8*x])*Sec[x]^6*Tan[ x])/(315*a^2*Sqrt[a*Cos[x]^4])
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.48, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3686, 3042, 4254, 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.
\(\displaystyle \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sin \left (x+\frac {\pi }{2}\right )^4\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\cos ^2(x) \int \sec ^{10}(x)dx}{a^2 \sqrt {a \cos ^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos ^2(x) \int \csc \left (x+\frac {\pi }{2}\right )^{10}dx}{a^2 \sqrt {a \cos ^4(x)}}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\cos ^2(x) \int \left (\tan ^8(x)+4 \tan ^6(x)+6 \tan ^4(x)+4 \tan ^2(x)+1\right )d(-\tan (x))}{a^2 \sqrt {a \cos ^4(x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\cos ^2(x) \left (-\frac {1}{9} \tan ^9(x)-\frac {4 \tan ^7(x)}{7}-\frac {6 \tan ^5(x)}{5}-\frac {4 \tan ^3(x)}{3}-\tan (x)\right )}{a^2 \sqrt {a \cos ^4(x)}}\) |
-((Cos[x]^2*(-Tan[x] - (4*Tan[x]^3)/3 - (6*Tan[x]^5)/5 - (4*Tan[x]^7)/7 - Tan[x]^9/9))/(a^2*Sqrt[a*Cos[x]^4]))
3.1.56.3.1 Defintions of rubi rules used
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]])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 0.96 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.39
method | result | size |
default | \(\frac {\tan \left (x \right ) \left (\sec ^{6}\left (x \right )\right ) \left (128 \left (\cos ^{8}\left (x \right )\right )+64 \left (\cos ^{6}\left (x \right )\right )+48 \left (\cos ^{4}\left (x \right )\right )+40 \left (\cos ^{2}\left (x \right )\right )+35\right )}{315 a^{2} \sqrt {a \left (\cos ^{4}\left (x \right )\right )}}\) | \(46\) |
risch | \(\frac {256 i \left (126 \,{\mathrm e}^{6 i x}+84 \,{\mathrm e}^{4 i x}+9+37 \cos \left (2 x \right )+35 i \sin \left (2 x \right )\right )}{315 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{7} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}\) | \(63\) |
1/315*tan(x)*sec(x)^6*(128*cos(x)^8+64*cos(x)^6+48*cos(x)^4+40*cos(x)^2+35 )/a^2/(a*cos(x)^4)^(1/2)
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx=\frac {{\left (128 \, \cos \left (x\right )^{8} + 64 \, \cos \left (x\right )^{6} + 48 \, \cos \left (x\right )^{4} + 40 \, \cos \left (x\right )^{2} + 35\right )} \sqrt {a \cos \left (x\right )^{4}} \sin \left (x\right )}{315 \, a^{3} \cos \left (x\right )^{11}} \]
1/315*(128*cos(x)^8 + 64*cos(x)^6 + 48*cos(x)^4 + 40*cos(x)^2 + 35)*sqrt(a *cos(x)^4)*sin(x)/(a^3*cos(x)^11)
Timed out. \[ \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx=\frac {35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{\frac {5}{2}}} \]
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx=\frac {35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{\frac {5}{2}}} \]
Time = 18.03 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.62 \[ \int \frac {1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx=\frac {{\mathrm {e}}^{x\,4{}\mathrm {i}}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}}{2}\right )}^4}\,2048{}\mathrm {i}}{5\,a^3\,{\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )}^5\,\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+2\,{\mathrm {e}}^{x\,4{}\mathrm {i}}+{\mathrm {e}}^{x\,6{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{x\,4{}\mathrm {i}}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}}{2}\right )}^4}\,4096{}\mathrm {i}}{3\,a^3\,{\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )}^6\,\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+2\,{\mathrm {e}}^{x\,4{}\mathrm {i}}+{\mathrm {e}}^{x\,6{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{x\,4{}\mathrm {i}}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}}{2}\right )}^4}\,12288{}\mathrm {i}}{7\,a^3\,{\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )}^7\,\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+2\,{\mathrm {e}}^{x\,4{}\mathrm {i}}+{\mathrm {e}}^{x\,6{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{x\,4{}\mathrm {i}}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}}{2}\right )}^4}\,1024{}\mathrm {i}}{a^3\,{\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )}^8\,\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+2\,{\mathrm {e}}^{x\,4{}\mathrm {i}}+{\mathrm {e}}^{x\,6{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{x\,4{}\mathrm {i}}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}}{2}\right )}^4}\,2048{}\mathrm {i}}{9\,a^3\,{\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )}^9\,\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+2\,{\mathrm {e}}^{x\,4{}\mathrm {i}}+{\mathrm {e}}^{x\,6{}\mathrm {i}}\right )} \]
(exp(x*4i)*(a*(exp(-x*1i)/2 + exp(x*1i)/2)^4)^(1/2)*2048i)/(5*a^3*(exp(x*2 i) + 1)^5*(exp(x*2i) + 2*exp(x*4i) + exp(x*6i))) - (exp(x*4i)*(a*(exp(-x*1 i)/2 + exp(x*1i)/2)^4)^(1/2)*4096i)/(3*a^3*(exp(x*2i) + 1)^6*(exp(x*2i) + 2*exp(x*4i) + exp(x*6i))) + (exp(x*4i)*(a*(exp(-x*1i)/2 + exp(x*1i)/2)^4)^ (1/2)*12288i)/(7*a^3*(exp(x*2i) + 1)^7*(exp(x*2i) + 2*exp(x*4i) + exp(x*6i ))) - (exp(x*4i)*(a*(exp(-x*1i)/2 + exp(x*1i)/2)^4)^(1/2)*1024i)/(a^3*(exp (x*2i) + 1)^8*(exp(x*2i) + 2*exp(x*4i) + exp(x*6i))) + (exp(x*4i)*(a*(exp( -x*1i)/2 + exp(x*1i)/2)^4)^(1/2)*2048i)/(9*a^3*(exp(x*2i) + 1)^9*(exp(x*2i ) + 2*exp(x*4i) + exp(x*6i)))