Integrand size = 25, antiderivative size = 177 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {3 a (a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{8 (a+b)^{7/2} f}-\frac {5 a \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(13 a-2 b) b \sec (e+f x)}{8 (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}} \]
-3/8*a*(a-4*b)*arctanh(sec(f*x+e)*(a+b)^(1/2)/(a+b*sec(f*x+e)^2)^(1/2))/(a +b)^(7/2)/f-5/8*a*cot(f*x+e)*csc(f*x+e)/(a+b)^2/f/(a+b*sec(f*x+e)^2)^(1/2) -1/4*cot(f*x+e)^3*csc(f*x+e)/(a+b)/f/(a+b*sec(f*x+e)^2)^(1/2)-1/8*(13*a-2* b)*b*sec(f*x+e)/(a+b)^3/f/(a+b*sec(f*x+e)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.56 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(a+2 b+a \cos (2 (e+f x))) \left ((a+b)^2 \csc ^4(e+f x)-a (a-4 b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},1-\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sec ^3(e+f x)}{8 (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
-1/8*((a + 2*b + a*Cos[2*(e + f*x)])*((a + b)^2*Csc[e + f*x]^4 - a*(a - 4* b)*Hypergeometric2F1[-1/2, 2, 1/2, 1 - (a*Sin[e + f*x]^2)/(a + b)])*Sec[e + f*x]^3)/((a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2))
Time = 0.38 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 4622, 25, 372, 27, 402, 25, 402, 27, 291, 219}
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.
| \(\displaystyle \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^5 \left (a+b \sec (e+f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4622 |
| \(\displaystyle \frac {\int -\frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (4 \sec ^2(e+f x)+1\right )}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \int \frac {4 \sec ^2(e+f x)+1}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {a \left (\frac {\int -\frac {-10 b \sec ^2(e+f x)+3 a-2 b}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{2 (a+b)}+\frac {5 \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}\right )}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a \left (\frac {5 \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\int \frac {-10 b \sec ^2(e+f x)+3 a-2 b}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{2 (a+b)}\right )}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {a \left (\frac {5 \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\int -\frac {3 a (a-4 b)}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec (e+f x)}{a (a+b)}}{2 (a+b)}\right )}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \left (\frac {5 \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\frac {3 (a-4 b) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec (e+f x)}{a+b}+\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}\right )}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {a \left (\frac {5 \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\frac {3 (a-4 b) \int \frac {1}{1-\frac {(a+b) \sec ^2(e+f x)}{b \sec ^2(e+f x)+a}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a}}}{a+b}+\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}\right )}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {a \left (\frac {5 \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\frac {3 (a-4 b) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{(a+b)^{3/2}}+\frac {b (13 a-2 b) \sec (e+f x)}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}\right )}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{f}\) |
(-1/4*Sec[e + f*x]/((a + b)*(1 - Sec[e + f*x]^2)^2*Sqrt[a + b*Sec[e + f*x] ^2]) + (a*((5*Sec[e + f*x])/(2*(a + b)*(1 - Sec[e + f*x]^2)*Sqrt[a + b*Sec [e + f*x]^2]) - ((3*(a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(a + b)^(3/2) + ((13*a - 2*b)*b*Sec[e + f*x])/(a*(a + b)*Sqrt[a + b*Sec[e + f*x]^2]))/(2*(a + b))))/(4*(a + b)))/f
3.2.11.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*sin[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Si mp[1/(f*ff^m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p /x^(m + 1)), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4])
Leaf count of result is larger than twice the leaf count of optimal. \(3611\) vs. \(2(157)=314\).
Time = 1.23 (sec) , antiderivative size = 3612, normalized size of antiderivative = 20.41
1/64/f/(a+b)^(13/2)*(2*a*b*(a+b)^(7/2)+(a+b)^(7/2)*a^2+(a+b)^(7/2)*b^2+7*a ^2*(1-cos(f*x+e))^2*(a+b)^(7/2)*csc(f*x+e)^2-3*b^2*(1-cos(f*x+e))^2*(a+b)^ (7/2)*csc(f*x+e)^2+(a+b)^(7/2)*a^2*(1-cos(f*x+e))^10*csc(f*x+e)^10+(a+b)^( 7/2)*b^2*(1-cos(f*x+e))^10*csc(f*x+e)^10-12*ln(2/(1-cos(f*x+e))^2*(-a*(1-c os(f*x+e))^2+b*(1-cos(f*x+e))^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos( f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e) )^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2+b*sin( f*x+e)^2))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+e)^ 4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b) ^(1/2)*a^4*b*(1-cos(f*x+e))^4*csc(f*x+e)^4-108*ln(2/(1-cos(f*x+e))^2*(-a*( 1-cos(f*x+e))^2+b*(1-cos(f*x+e))^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-c os(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x +e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2+b*s in(f*x+e)^2))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*csc(f*x+ e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a +b)^(1/2)*a^3*b^2*(1-cos(f*x+e))^4*csc(f*x+e)^4-132*ln(2/(1-cos(f*x+e))^2* (-a*(1-cos(f*x+e))^2+b*(1-cos(f*x+e))^2+(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b *(1-cos(f*x+e))^4*csc(f*x+e)^4-2*a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(1-co s(f*x+e))^2*csc(f*x+e)^2+a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^ 2+b*sin(f*x+e)^2))*(a*(1-cos(f*x+e))^4*csc(f*x+e)^4+b*(1-cos(f*x+e))^4*...
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (157) = 314\).
Time = 0.42 (sec) , antiderivative size = 873, normalized size of antiderivative = 4.93 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (a^{3} - 4 \, a^{2} b\right )} \cos \left (f x + e\right )^{6} - {\left (2 \, a^{3} - 9 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{2} b - 4 \, a b^{2} + {\left (a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a + b} \log \left (\frac {2 \, {\left (a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a + b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) + a + 2 \, b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (3 \, {\left (a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (5 \, a^{3} - 16 \, a^{2} b - 17 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, a^{2} b + 11 \, a b^{2} - 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{16 \, {\left ({\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{4} + {\left (a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} f\right )}}, \frac {3 \, {\left ({\left (a^{3} - 4 \, a^{2} b\right )} \cos \left (f x + e\right )^{6} - {\left (2 \, a^{3} - 9 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{2} b - 4 \, a b^{2} + {\left (a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {-a - b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a + b}\right ) + {\left (3 \, {\left (a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (5 \, a^{3} - 16 \, a^{2} b - 17 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, a^{2} b + 11 \, a b^{2} - 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{8 \, {\left ({\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} f \cos \left (f x + e\right )^{6} - {\left (2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{4} + {\left (a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} f\right )}}\right ] \]
[-1/16*(3*((a^3 - 4*a^2*b)*cos(f*x + e)^6 - (2*a^3 - 9*a^2*b + 4*a*b^2)*co s(f*x + e)^4 + a^2*b - 4*a*b^2 + (a^3 - 6*a^2*b + 8*a*b^2)*cos(f*x + e)^2) *sqrt(a + b)*log(2*(a*cos(f*x + e)^2 + 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^ 2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*( 3*(a^3 - 3*a^2*b - 4*a*b^2)*cos(f*x + e)^5 - (5*a^3 - 16*a^2*b - 17*a*b^2 + 4*b^3)*cos(f*x + e)^3 - (13*a^2*b + 11*a*b^2 - 2*b^3)*cos(f*x + e))*sqrt ((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a ^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^ 3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b ^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f), 1/8*(3*((a^3 - 4*a^2*b)*cos(f*x + e)^6 - (2*a^3 - 9*a^2 *b + 4*a*b^2)*cos(f*x + e)^4 + a^2*b - 4*a*b^2 + (a^3 - 6*a^2*b + 8*a*b^2) *cos(f*x + e)^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + (3*(a^3 - 3*a^2*b - 4*a*b^2)*c os(f*x + e)^5 - (5*a^3 - 16*a^2*b - 17*a*b^2 + 4*b^3)*cos(f*x + e)^3 - (13 *a^2*b + 11*a*b^2 - 2*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f *x + e)^2))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e )^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f *x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f)]
\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\csc ^{5}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^5\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]