Integrand size = 31, antiderivative size = 288 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=-\frac {2 d^3 (4 c+3 d) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^3 (c-d)^{9/2} (c+d)^{3/2} f}+\frac {d \left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \tan (e+f x)}{15 a^3 (c-d)^4 (c+d) f (c+d \sec (e+f x))}+\frac {\tan (e+f x)}{5 (c-d) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac {(2 c-9 d) \tan (e+f x)}{15 a (c-d)^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}+\frac {\left (2 c^2-12 c d+45 d^2\right ) \tan (e+f x)}{15 (c-d)^3 f \left (a^3+a^3 \sec (e+f x)\right ) (c+d \sec (e+f x))} \]
-2*d^3*(4*c+3*d)*arctanh((c-d)^(1/2)*tan(1/2*f*x+1/2*e)/(c+d)^(1/2))/a^3/( c-d)^(9/2)/(c+d)^(3/2)/f+1/15*d*(2*c^3-12*c^2*d+43*c*d^2+72*d^3)*tan(f*x+e )/a^3/(c-d)^4/(c+d)/f/(c+d*sec(f*x+e))+1/5*tan(f*x+e)/(c-d)/f/(a+a*sec(f*x +e))^3/(c+d*sec(f*x+e))+1/15*(2*c-9*d)*tan(f*x+e)/a/(c-d)^2/f/(a+a*sec(f*x +e))^2/(c+d*sec(f*x+e))+1/15*(2*c^2-12*c*d+45*d^2)*tan(f*x+e)/(c-d)^3/f/(a ^3+a^3*sec(f*x+e))/(c+d*sec(f*x+e))
Result contains complex when optimal does not.
Time = 7.96 (sec) , antiderivative size = 1772, normalized size of antiderivative = 6.15 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx =\text {Too large to display} \]
((4*c + 3*d)*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])^2*Sec[e + f*x]^5*(( (16*I)*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I* Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((- I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt [Cos[2*e] - I*Sin[2*e]]) + (16*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[ 2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Sin[e]) /(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((-c + d)^4*(c + d)*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2) + (Cos[e/2 + (f*x)/2]*(d + c*C os[e + f*x])*Sec[e/2]*Sec[e]*Sec[e + f*x]^5*(-55*c^5*Sin[(f*x)/2] + 135*c^ 4*d*Sin[(f*x)/2] - 20*c^3*d^2*Sin[(f*x)/2] - 810*c^2*d^3*Sin[(f*x)/2] - 45 0*c*d^4*Sin[(f*x)/2] + 150*d^5*Sin[(f*x)/2] + 47*c^5*Sin[(3*f*x)/2] - 137* c^4*d*Sin[(3*f*x)/2] + 88*c^3*d^2*Sin[(3*f*x)/2] + 812*c^2*d^3*Sin[(3*f*x) /2] + 690*c*d^4*Sin[(3*f*x)/2] + 75*d^5*Sin[(3*f*x)/2] - 50*c^5*Sin[e - (f *x)/2] + 130*c^4*d*Sin[e - (f*x)/2] - 10*c^3*d^2*Sin[e - (f*x)/2] - 1030*c ^2*d^3*Sin[e - (f*x)/2] - 990*c*d^4*Sin[e - (f*x)/2] - 150*d^5*Sin[e - (f* x)/2] + 50*c^5*Sin[e + (f*x)/2] - 130*c^4*d*Sin[e + (f*x)/2] + 10*c^3*d^2* Sin[e + (f*x)/2] + 1030*c^2*d^3*Sin[e + (f*x)/2] + 765*c*d^4*Sin[e + (f*x) /2] - 150*d^5*Sin[e + (f*x)/2] - 55*c^5*Sin[2*e + (f*x)/2] + 135*c^4*d*Sin [2*e + (f*x)/2] - 20*c^3*d^2*Sin[2*e + (f*x)/2] - 810*c^2*d^3*Sin[2*e +...
Time = 0.59 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.36, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {3042, 4475, 114, 27, 169, 25, 27, 169, 25, 27, 169, 27, 104, 218}
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.
| \(\displaystyle \int \frac {\sec (e+f x)}{(a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4475 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {1}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))^2}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {a^2 (c+3 d-3 d \sec (e+f x))}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))}d\sec (e+f x)}{a^2 \left (c^2-d^2\right )}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\int \frac {c+3 d-3 d \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{7/2} (c+d \sec (e+f x))}d\sec (e+f x)}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {-\frac {\int -\frac {a^2 \left (2 c^2-8 d c-15 d^2+2 d (c+6 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{5 a^3 (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {a^2 \left (2 c^2-8 d c-15 d^2+2 d (c+6 d) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{5 a^3 (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\int \frac {2 c^2-8 d c-15 d^2+2 d (c+6 d) \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{5/2} (c+d \sec (e+f x))}d\sec (e+f x)}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {-\frac {\int -\frac {a^2 \left ((c+d) \left (2 c^2-12 d c+45 d^2\right )+d \left (2 c^2-10 d c-27 d^2\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a^3 (c-d)}-\frac {\left (2 c^2-10 c d-27 d^2\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\int \frac {a^2 \left ((c+d) \left (2 c^2-12 d c+45 d^2\right )+d \left (2 c^2-10 d c-27 d^2\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a^3 (c-d)}-\frac {\left (2 c^2-10 c d-27 d^2\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {\int \frac {(c+d) \left (2 c^2-12 d c+45 d^2\right )+d \left (2 c^2-10 d c-27 d^2\right ) \sec (e+f x)}{\sqrt {a-a \sec (e+f x)} (\sec (e+f x) a+a)^{3/2} (c+d \sec (e+f x))}d\sec (e+f x)}{3 a (c-d)}-\frac {\left (2 c^2-10 c d-27 d^2\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {-\frac {\int \frac {15 a^2 d^3 (4 c+3 d)}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}d\sec (e+f x)}{a^3 (c-d)}-\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (2 c^2-10 c d-27 d^2\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {-\frac {15 d^3 (4 c+3 d) \int \frac {1}{\sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}d\sec (e+f x)}{a (c-d)}-\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (2 c^2-10 c d-27 d^2\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {-\frac {30 d^3 (4 c+3 d) \int \frac {1}{a (c-d)+\frac {a (c+d) (\sec (e+f x) a+a)}{a-a \sec (e+f x)}}d\frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {a-a \sec (e+f x)}}}{a (c-d)}-\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (2 c^2-10 c d-27 d^2\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \left (\frac {\frac {\frac {-\frac {30 d^3 (4 c+3 d) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{a^2 (c-d)^{3/2} \sqrt {c+d}}-\frac {\left (2 c^3-12 c^2 d+43 c d^2+72 d^3\right ) \sqrt {a-a \sec (e+f x)}}{a^2 (c-d) \sqrt {a \sec (e+f x)+a}}}{3 a (c-d)}-\frac {\left (2 c^2-10 c d-27 d^2\right ) \sqrt {a-a \sec (e+f x)}}{3 a^2 (c-d) (a \sec (e+f x)+a)^{3/2}}}{5 a (c-d)}-\frac {(c+6 d) \sqrt {a-a \sec (e+f x)}}{5 a^2 (c-d) (a \sec (e+f x)+a)^{5/2}}}{c^2-d^2}+\frac {d \sqrt {a-a \sec (e+f x)}}{a^2 \left (c^2-d^2\right ) (a \sec (e+f x)+a)^{5/2} (c+d \sec (e+f x))}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^2*((d*Sqrt[a - a*Sec[e + f*x]])/(a^2*(c^2 - d^2)*(a + a*Sec[e + f*x]) ^(5/2)*(c + d*Sec[e + f*x])) + (-1/5*((c + 6*d)*Sqrt[a - a*Sec[e + f*x]])/ (a^2*(c - d)*(a + a*Sec[e + f*x])^(5/2)) + (-1/3*((2*c^2 - 10*c*d - 27*d^2 )*Sqrt[a - a*Sec[e + f*x]])/(a^2*(c - d)*(a + a*Sec[e + f*x])^(3/2)) + ((- 30*d^3*(4*c + 3*d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])])/(a^2*(c - d)^(3/2)*Sqrt[c + d]) - ((2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*Sqrt[a - a*Sec[e + f*x]])/(a^2*(c - d)*Sqr t[a + a*Sec[e + f*x]]))/(3*a*(c - d)))/(5*a*(c - d)))/(c^2 - d^2))*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))
3.3.32.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[a ^2*g*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*((c + d*x)^n/Sqrt[a - b*x]), x ], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[ b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p, 1] || In tegerQ[m - 1/2])
Time = 0.89 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c^{2}}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c d}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} d^{2}}{5}-\frac {2 c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{2}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +17 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )^{2}}+\frac {16 d^{3} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (4 c +3 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{4}}}{4 f \,a^{3}}\) | \(284\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c^{2}}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c d}{5}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} d^{2}}{5}-\frac {2 c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{2}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c d +17 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{2}}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )^{2}}+\frac {16 d^{3} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (4 c +3 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{4}}}{4 f \,a^{3}}\) | \(284\) |
| risch | \(\frac {2 i \left (7 c^{5}+10 c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-137 c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+106 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}-76 c^{4} d \,{\mathrm e}^{i \left (f x +e \right )}+195 c \,d^{4} {\mathrm e}^{5 i \left (f x +e \right )}+990 c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+60 c^{3} d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+360 c^{2} d^{3} {\mathrm e}^{5 i \left (f x +e \right )}+20 c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+810 c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+88 c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+812 c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+1030 c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-130 c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}-45 c^{4} d \,{\mathrm e}^{6 i \left (f x +e \right )}-135 c^{4} d \,{\mathrm e}^{4 i \left (f x +e \right )}+72 c^{2} d^{3}+15 c \,d^{4}+38 c^{3} d^{2}+47 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 d^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 d^{5} {\mathrm e}^{5 i \left (f x +e \right )}+15 d^{5} {\mathrm e}^{6 i \left (f x +e \right )}-27 c^{4} d +219 c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}-90 c^{4} d \,{\mathrm e}^{5 i \left (f x +e \right )}+346 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}+675 c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+690 c \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+90 c^{2} d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+30 c^{3} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+50 c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+150 d^{5} {\mathrm e}^{4 i \left (f x +e \right )}+15 c^{5} {\mathrm e}^{6 i \left (f x +e \right )}+30 c^{5} {\mathrm e}^{5 i \left (f x +e \right )}+55 c^{5} {\mathrm e}^{4 i \left (f x +e \right )}+15 d^{5} {\mathrm e}^{i \left (f x +e \right )}+150 d^{5} {\mathrm e}^{3 i \left (f x +e \right )}+20 c^{5} {\mathrm e}^{i \left (f x +e \right )}\right )}{15 \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right ) \left (-c^{2}+d^{2}\right ) c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left (-c +d \right )^{3} a^{3} f}+\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}+\frac {3 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}-\frac {3 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{4} f \,a^{3}}\) | \(993\) |
1/4/f/a^3*(1/(c^2-2*c*d+d^2)/(c-d)^2*(1/5*tan(1/2*f*x+1/2*e)^5*c^2-2/5*tan (1/2*f*x+1/2*e)^5*c*d+1/5*tan(1/2*f*x+1/2*e)^5*d^2-2/3*c^2*tan(1/2*f*x+1/2 *e)^3+8/3*tan(1/2*f*x+1/2*e)^3*c*d-2*tan(1/2*f*x+1/2*e)^3*d^2+tan(1/2*f*x+ 1/2*e)*c^2-6*tan(1/2*f*x+1/2*e)*c*d+17*tan(1/2*f*x+1/2*e)*d^2)+16*d^3/(c-d )^4*(-1/2*d/(c+d)*tan(1/2*f*x+1/2*e)/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1 /2*e)^2*d-c-d)-1/2*(4*c+3*d)/(c+d)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1 /2*f*x+1/2*e)/((c+d)*(c-d))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (271) = 542\).
Time = 0.34 (sec) , antiderivative size = 1693, normalized size of antiderivative = 5.88 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\text {Too large to display} \]
[1/30*(15*(4*c*d^4 + 3*d^5 + (4*c^2*d^3 + 3*c*d^4)*cos(f*x + e)^4 + (12*c^ 2*d^3 + 13*c*d^4 + 3*d^5)*cos(f*x + e)^3 + 3*(4*c^2*d^3 + 7*c*d^4 + 3*d^5) *cos(f*x + e)^2 + (4*c^2*d^3 + 15*c*d^4 + 9*d^5)*cos(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) + 2*(2*c^5*d - 12*c^4*d^2 + 41*c^3*d^3 + 84* c^2*d^4 - 43*c*d^5 - 72*d^6 + (7*c^6 - 27*c^5*d + 31*c^4*d^2 + 99*c^3*d^3 - 23*c^2*d^4 - 72*c*d^5 - 15*d^6)*cos(f*x + e)^3 + (6*c^6 - 29*c^5*d + 51* c^4*d^2 + 193*c^3*d^3 + 60*c^2*d^4 - 164*c*d^5 - 117*d^6)*cos(f*x + e)^2 + (2*c^6 - 6*c^5*d + 5*c^4*d^2 + 147*c^3*d^3 + 164*c^2*d^4 - 141*c*d^5 - 17 1*d^6)*cos(f*x + e))*sin(f*x + e))/((a^3*c^8 - 3*a^3*c^7*d + a^3*c^6*d^2 + 5*a^3*c^5*d^3 - 5*a^3*c^4*d^4 - a^3*c^3*d^5 + 3*a^3*c^2*d^6 - a^3*c*d^7)* f*cos(f*x + e)^4 + (3*a^3*c^8 - 8*a^3*c^7*d + 16*a^3*c^5*d^3 - 10*a^3*c^4* d^4 - 8*a^3*c^3*d^5 + 8*a^3*c^2*d^6 - a^3*d^8)*f*cos(f*x + e)^3 + 3*(a^3*c ^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c ^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e)^2 + (a^3*c^8 - 8*a^3*c^6*d^ 2 + 8*a^3*c^5*d^3 + 10*a^3*c^4*d^4 - 16*a^3*c^3*d^5 + 8*a^3*c*d^7 - 3*a^3* d^8)*f*cos(f*x + e) + (a^3*c^7*d - 3*a^3*c^6*d^2 + a^3*c^5*d^3 + 5*a^3*c^4 *d^4 - 5*a^3*c^3*d^5 - a^3*c^2*d^6 + 3*a^3*c*d^7 - a^3*d^8)*f), -1/15*(15* (4*c*d^4 + 3*d^5 + (4*c^2*d^3 + 3*c*d^4)*cos(f*x + e)^4 + (12*c^2*d^3 +...
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sec ^{3}{\left (e + f x \right )} + 3 c^{2} \sec ^{2}{\left (e + f x \right )} + 3 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{4}{\left (e + f x \right )} + 6 c d \sec ^{3}{\left (e + f x \right )} + 6 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{5}{\left (e + f x \right )} + 3 d^{2} \sec ^{4}{\left (e + f x \right )} + 3 d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx}{a^{3}} \]
Integral(sec(e + f*x)/(c**2*sec(e + f*x)**3 + 3*c**2*sec(e + f*x)**2 + 3*c **2*sec(e + f*x) + c**2 + 2*c*d*sec(e + f*x)**4 + 6*c*d*sec(e + f*x)**3 + 6*c*d*sec(e + f*x)**2 + 2*c*d*sec(e + f*x) + d**2*sec(e + f*x)**5 + 3*d**2 *sec(e + f*x)**4 + 3*d**2*sec(e + f*x)**3 + d**2*sec(e + f*x)**2), x)/a**3
Exception generated. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (271) = 542\).
Time = 0.39 (sec) , antiderivative size = 918, normalized size of antiderivative = 3.19 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\text {Too large to display} \]
-1/60*(120*d^4*tan(1/2*f*x + 1/2*e)/((a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^ 2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*(c*tan(1/2*f*x + 1/2*e)^2 - d*t an(1/2*f*x + 1/2*e)^2 - c - d)) + 120*(4*c*d^3 + 3*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + arctan(-(c*tan(1/2*f*x + 1/2*e) - d*tan( 1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d ^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*sqrt(-c^2 + d^2)) - (3*a^12*c^ 8*tan(1/2*f*x + 1/2*e)^5 - 24*a^12*c^7*d*tan(1/2*f*x + 1/2*e)^5 + 84*a^12* c^6*d^2*tan(1/2*f*x + 1/2*e)^5 - 168*a^12*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 + 210*a^12*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 - 168*a^12*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 + 84*a^12*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 24*a^12*c*d^7*tan(1/2* f*x + 1/2*e)^5 + 3*a^12*d^8*tan(1/2*f*x + 1/2*e)^5 - 10*a^12*c^8*tan(1/2*f *x + 1/2*e)^3 + 100*a^12*c^7*d*tan(1/2*f*x + 1/2*e)^3 - 420*a^12*c^6*d^2*t an(1/2*f*x + 1/2*e)^3 + 980*a^12*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 - 1400*a^1 2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 1260*a^12*c^3*d^5*tan(1/2*f*x + 1/2*e)^ 3 - 700*a^12*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 220*a^12*c*d^7*tan(1/2*f*x + 1/2*e)^3 - 30*a^12*d^8*tan(1/2*f*x + 1/2*e)^3 + 15*a^12*c^8*tan(1/2*f*x + 1/2*e) - 180*a^12*c^7*d*tan(1/2*f*x + 1/2*e) + 1020*a^12*c^6*d^2*tan(1/2* f*x + 1/2*e) - 3180*a^12*c^5*d^3*tan(1/2*f*x + 1/2*e) + 5850*a^12*c^4*d^4* tan(1/2*f*x + 1/2*e) - 6540*a^12*c^3*d^5*tan(1/2*f*x + 1/2*e) + 4380*a^12* c^2*d^6*tan(1/2*f*x + 1/2*e) - 1620*a^12*c*d^7*tan(1/2*f*x + 1/2*e) + 2...
Time = 13.88 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.61 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20\,a^3\,f\,{\left (c-d\right )}^2}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {2\,\left (c^2-d^2\right )\,\left (\frac {1}{a^3\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{2\,a^3\,{\left (c-d\right )}^4}\right )}{{\left (c-d\right )}^2}-\frac {3}{2\,a^3\,{\left (c-d\right )}^2}+\frac {{\left (c+d\right )}^2}{4\,a^3\,{\left (c-d\right )}^4}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {1}{3\,a^3\,{\left (c-d\right )}^2}-\frac {c^2-d^2}{6\,a^3\,{\left (c-d\right )}^4}\right )}{f}+\frac {2\,d^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left (a^3\,c^5-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^3\,c^5-5\,a^3\,c^4\,d+10\,a^3\,c^3\,d^2-10\,a^3\,c^2\,d^3+5\,a^3\,c\,d^4-a^3\,d^5\right )+a^3\,d^5-3\,a^3\,c\,d^4-3\,a^3\,c^4\,d+2\,a^3\,c^2\,d^3+2\,a^3\,c^3\,d^2\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^5-5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4\,d+10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d^2-10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^3+5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^4-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^5}{\sqrt {c+d}\,{\left (c-d\right )}^{9/2}}\right )\,\left (4\,c+3\,d\right )\,2{}\mathrm {i}}{a^3\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{9/2}} \]
tan(e/2 + (f*x)/2)^5/(20*a^3*f*(c - d)^2) - (tan(e/2 + (f*x)/2)*((2*(c^2 - d^2)*(1/(a^3*(c - d)^2) - (c^2 - d^2)/(2*a^3*(c - d)^4)))/(c - d)^2 - 3/( 2*a^3*(c - d)^2) + (c + d)^2/(4*a^3*(c - d)^4)))/f - (tan(e/2 + (f*x)/2)^3 *(1/(3*a^3*(c - d)^2) - (c^2 - d^2)/(6*a^3*(c - d)^4)))/f + (2*d^4*tan(e/2 + (f*x)/2))/(f*(c + d)*(a^3*c^5 - tan(e/2 + (f*x)/2)^2*(a^3*c^5 - a^3*d^5 + 5*a^3*c*d^4 - 5*a^3*c^4*d - 10*a^3*c^2*d^3 + 10*a^3*c^3*d^2) + a^3*d^5 - 3*a^3*c*d^4 - 3*a^3*c^4*d + 2*a^3*c^2*d^3 + 2*a^3*c^3*d^2)) + (d^3*atan( (c^5*tan(e/2 + (f*x)/2)*1i - d^5*tan(e/2 + (f*x)/2)*1i + c*d^4*tan(e/2 + ( f*x)/2)*5i - c^4*d*tan(e/2 + (f*x)/2)*5i - c^2*d^3*tan(e/2 + (f*x)/2)*10i + c^3*d^2*tan(e/2 + (f*x)/2)*10i)/((c + d)^(1/2)*(c - d)^(9/2)))*(4*c + 3* d)*2i)/(a^3*f*(c + d)^(3/2)*(c - d)^(9/2))