\(\int \frac {\csc ^5(e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [124]

Optimal result

Integrand size = 25, antiderivative size = 234 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\left (3 a^2-24 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{8 (a+b)^{9/2} f}-\frac {(5 a-2 b) \cot (e+f x) \csc (e+f x)}{8 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-12 b) b \sec (e+f x)}{24 (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-10 b) b \sec (e+f x)}{24 (a+b)^4 f \sqrt {a+b \sec ^2(e+f x)}} \] Output:

-1/8*(3*a^2-24*a*b+8*b^2)*arctanh((a+b)^(1/2)*sec(f*x+e)/(a+b*sec(f*x+e)^2 
)^(1/2))/(a+b)^(9/2)/f-1/8*(5*a-2*b)*cot(f*x+e)*csc(f*x+e)/(a+b)^2/f/(a+b* 
sec(f*x+e)^2)^(3/2)-1/4*cot(f*x+e)^3*csc(f*x+e)/(a+b)/f/(a+b*sec(f*x+e)^2) 
^(3/2)-1/24*(23*a-12*b)*b*sec(f*x+e)/(a+b)^3/f/(a+b*sec(f*x+e)^2)^(3/2)-5/ 
24*(11*a-10*b)*b*sec(f*x+e)/(a+b)^4/f/(a+b*sec(f*x+e)^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.55 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a+2 b+a \cos (2 (e+f x))) \left (3 (a+b) (3 a-4 b+(a+8 b) \cos (2 (e+f x))) \csc ^4(e+f x)-2 \left (3 a^2-24 a b+8 b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1-\frac {a \sin ^2(e+f x)}{a+b}\right )\right ) \sec ^5(e+f x)}{96 (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \] Input:

Integrate[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]
 

Output:

-1/96*((a + 2*b + a*Cos[2*(e + f*x)])*(3*(a + b)*(3*a - 4*b + (a + 8*b)*Co 
s[2*(e + f*x)])*Csc[e + f*x]^4 - 2*(3*a^2 - 24*a*b + 8*b^2)*Hypergeometric 
2F1[-3/2, 1, -1/2, 1 - (a*Sin[e + f*x]^2)/(a + b)])*Sec[e + f*x]^5)/((a + 
b)^3*f*(a + b*Sec[e + f*x]^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4622, 25, 372, 402, 25, 402, 25, 27, 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.

\(\displaystyle \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (e+f x)^5 \left (a+b \sec (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 4622

\(\displaystyle \frac {\int -\frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {\sec ^4(e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{f}\)

\(\Big \downarrow \) 372

\(\displaystyle \frac {\frac {\int \frac {2 (2 a-b) \sec ^2(e+f x)+a}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {\int -\frac {a (3 a-4 b)-4 (5 a-2 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{2 (a+b)}+\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {a (3 a-4 b)-4 (5 a-2 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {b (23 a-12 b) \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {a \left (a (9 a-26 b)-2 (23 a-12 b) b \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{3 a (a+b)}}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int \frac {a \left (a (9 a-26 b)-2 (23 a-12 b) b \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{3 a (a+b)}+\frac {b (23 a-12 b) \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int \frac {a (9 a-26 b)-2 (23 a-12 b) b \sec ^2(e+f x)}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec (e+f x)}{3 (a+b)}+\frac {b (23 a-12 b) \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {5 b (11 a-10 b) \sec (e+f x)}{(a+b) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\int -\frac {3 a \left (3 a^2-24 b a+8 b^2\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec (e+f x)}{a (a+b)}}{3 (a+b)}+\frac {b (23 a-12 b) \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {3 \left (3 a^2-24 a b+8 b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec (e+f x)}{a+b}+\frac {5 b (11 a-10 b) \sec (e+f x)}{(a+b) \sqrt {a+b \sec ^2(e+f x)}}}{3 (a+b)}+\frac {b (23 a-12 b) \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {3 \left (3 a^2-24 a b+8 b^2\right ) \int \frac {1}{1-\frac {(a+b) \sec ^2(e+f x)}{b \sec ^2(e+f x)+a}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a}}}{a+b}+\frac {5 b (11 a-10 b) \sec (e+f x)}{(a+b) \sqrt {a+b \sec ^2(e+f x)}}}{3 (a+b)}+\frac {b (23 a-12 b) \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {(5 a-2 b) \sec (e+f x)}{2 (a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {3 \left (3 a^2-24 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{(a+b)^{3/2}}+\frac {5 b (11 a-10 b) \sec (e+f x)}{(a+b) \sqrt {a+b \sec ^2(e+f x)}}}{3 (a+b)}+\frac {b (23 a-12 b) \sec (e+f x)}{3 (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}}{4 (a+b)}-\frac {\sec (e+f x)}{4 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{f}\)

Input:

Int[Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2),x]
 

Output:

(-1/4*Sec[e + f*x]/((a + b)*(1 - Sec[e + f*x]^2)^2*(a + b*Sec[e + f*x]^2)^ 
(3/2)) + (((5*a - 2*b)*Sec[e + f*x])/(2*(a + b)*(1 - Sec[e + f*x]^2)*(a + 
b*Sec[e + f*x]^2)^(3/2)) - (((23*a - 12*b)*b*Sec[e + f*x])/(3*(a + b)*(a + 
 b*Sec[e + f*x]^2)^(3/2)) + ((3*(3*a^2 - 24*a*b + 8*b^2)*ArcTanh[(Sqrt[a + 
 b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(a + b)^(3/2) + (5*(11*a - 
10*b)*b*Sec[e + f*x])/((a + b)*Sqrt[a + b*Sec[e + f*x]^2]))/(3*(a + b)))/( 
2*(a + b)))/(4*(a + b)))/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 291

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]
 

rule 372

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]
 

rule 402

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4622

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])
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3312\) vs. \(2(210)=420\).

Time = 193.19 (sec) , antiderivative size = 3313, normalized size of antiderivative = 14.16

method	result	size
default	\(\text {Expression too large to display}\)	\(3313\)

Input:

int(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/48/f/(2*(-a*b)^(1/2)+a-b)^4/(2*(-a*b)^(1/2)-a+b)^4/(a+b)^(17/2)*(a^8+8*a 
^7*b+28*a^6*b^2+56*a^5*b^3+70*a^4*b^4+56*a^3*b^5+28*a^2*b^6+8*a*b^7+b^8)/( 
a+b*sec(f*x+e)^2)^(5/2)*((18*cos(f*x+e)^2-30)*(a+b)^(9/2)*a^4*cot(f*x+e)*c 
sc(f*x+e)^3+(-144*cos(f*x+e)^4+282*cos(f*x+e)^2-186)*(a+b)^(9/2)*a^3*b*sec 
(f*x+e)*csc(f*x+e)^4+(48*cos(f*x+e)^6-416*cos(f*x+e)^4+562*cos(f*x+e)^2-26 
6)*(a+b)^(9/2)*a^2*b^2*sec(f*x+e)^3*csc(f*x+e)^4+(112*cos(f*x+e)^6-448*cos 
(f*x+e)^4+398*cos(f*x+e)^2-110)*(a+b)^(9/2)*a*b^3*sec(f*x+e)^5*csc(f*x+e)^ 
4+(64*cos(f*x+e)^4-176*cos(f*x+e)^2+100)*(a+b)^(9/2)*b^4*sec(f*x+e)^5*csc( 
f*x+e)^4+ln(-4*((a+b)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*co 
s(f*x+e)+(a+b)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+cos(f*x+e 
)*a+b)/(-1+cos(f*x+e)))*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^8*(- 
9-9*sec(f*x+e))+ln(-4*((a+b)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^( 
1/2)*cos(f*x+e)+(a+b)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+co 
s(f*x+e)*a+b)/(-1+cos(f*x+e)))*a^7*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^( 
1/2)*b*(36+36*sec(f*x+e)-18*sec(f*x+e)^2-18*sec(f*x+e)^3)+ln(-4*((a+b)^(1/ 
2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+(a+b)^(1/2)*((b+ 
a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+cos(f*x+e)*a+b)/(-1+cos(f*x+e)))*a 
^6*b^2*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(210+210*sec(f*x+e)+72* 
sec(f*x+e)^2+72*sec(f*x+e)^3-9*sec(f*x+e)^4-9*sec(f*x+e)^5)+ln(-4*((a+b)^( 
1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+(a+b)^(1/2)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (210) = 420\).

Time = 0.39 (sec) , antiderivative size = 1316, normalized size of antiderivative = 5.62 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/48*(3*((3*a^4 - 24*a^3*b + 8*a^2*b^2)*cos(f*x + e)^8 - 2*(3*a^4 - 27*a^ 
3*b + 32*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^6 + (3*a^4 - 36*a^3*b + 107*a^2*b 
^2 - 56*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 3*a^2*b^2 - 24*a*b^3 + 8*b^4 + 2*( 
3*a^3*b - 27*a^2*b^2 + 32*a*b^3 - 8*b^4)*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*(3*a^4 - 21*a^3* 
b - 16*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^7 - (15*a^4 - 117*a^3*b + 4*a^2*b^2 
 + 104*a*b^3 - 32*b^4)*cos(f*x + e)^5 - (78*a^3*b - 71*a^2*b^2 - 61*a*b^3 
+ 88*b^4)*cos(f*x + e)^3 - 5*(11*a^2*b^2 + a*b^3 - 10*b^4)*cos(f*x + e))*s 
qrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 
 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5 
*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b 
- 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x 
 + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f* 
cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 
+ b^7)*f), 1/24*(3*((3*a^4 - 24*a^3*b + 8*a^2*b^2)*cos(f*x + e)^8 - 2*(3*a 
^4 - 27*a^3*b + 32*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^6 + (3*a^4 - 36*a^3*b + 
 107*a^2*b^2 - 56*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 3*a^2*b^2 - 24*a*b^3 + 8 
*b^4 + 2*(3*a^3*b - 27*a^2*b^2 + 32*a*b^3 - 8*b^4)*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)*c...
 

Sympy [F]

\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\csc ^{5}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(csc(f*x+e)**5/(a+b*sec(f*x+e)**2)**(5/2),x)
 

Output:

Integral(csc(e + f*x)**5/(a + b*sec(e + f*x)**2)**(5/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3221 vs. \(2 (210) = 420\).

Time = 2.07 (sec) , antiderivative size = 3221, normalized size of antiderivative = 13.76 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

-1/192*(((((3*((a^17*b^2 + 15*a^16*b^3 + 105*a^15*b^4 + 455*a^14*b^5 + 136 
5*a^13*b^6 + 3003*a^12*b^7 + 5005*a^11*b^8 + 6435*a^10*b^9 + 6435*a^9*b^10 
 + 5005*a^8*b^11 + 3003*a^7*b^12 + 1365*a^6*b^13 + 455*a^5*b^14 + 105*a^4* 
b^15 + 15*a^3*b^16 + a^2*b^17)*tan(1/2*f*x + 1/2*e)^2/(a^18*b^2*sgn(cos(f* 
x + e)) + 16*a^17*b^3*sgn(cos(f*x + e)) + 120*a^16*b^4*sgn(cos(f*x + e)) + 
 560*a^15*b^5*sgn(cos(f*x + e)) + 1820*a^14*b^6*sgn(cos(f*x + e)) + 4368*a 
^13*b^7*sgn(cos(f*x + e)) + 8008*a^12*b^8*sgn(cos(f*x + e)) + 11440*a^11*b 
^9*sgn(cos(f*x + e)) + 12870*a^10*b^10*sgn(cos(f*x + e)) + 11440*a^9*b^11* 
sgn(cos(f*x + e)) + 8008*a^8*b^12*sgn(cos(f*x + e)) + 4368*a^7*b^13*sgn(co 
s(f*x + e)) + 1820*a^6*b^14*sgn(cos(f*x + e)) + 560*a^5*b^15*sgn(cos(f*x + 
 e)) + 120*a^4*b^16*sgn(cos(f*x + e)) + 16*a^3*b^17*sgn(cos(f*x + e)) + a^ 
2*b^18*sgn(cos(f*x + e))) + (5*a^17*b^2 + 61*a^16*b^3 + 329*a^15*b^4 + 100 
1*a^14*b^5 + 1729*a^13*b^6 + 1001*a^12*b^7 - 3003*a^11*b^8 - 9867*a^10*b^9 
 - 15873*a^9*b^10 - 17017*a^8*b^11 - 13013*a^7*b^12 - 7189*a^6*b^13 - 2821 
*a^5*b^14 - 749*a^4*b^15 - 121*a^3*b^16 - 9*a^2*b^17)/(a^18*b^2*sgn(cos(f* 
x + e)) + 16*a^17*b^3*sgn(cos(f*x + e)) + 120*a^16*b^4*sgn(cos(f*x + e)) + 
 560*a^15*b^5*sgn(cos(f*x + e)) + 1820*a^14*b^6*sgn(cos(f*x + e)) + 4368*a 
^13*b^7*sgn(cos(f*x + e)) + 8008*a^12*b^8*sgn(cos(f*x + e)) + 11440*a^11*b 
^9*sgn(cos(f*x + e)) + 12870*a^10*b^10*sgn(cos(f*x + e)) + 11440*a^9*b^11* 
sgn(cos(f*x + e)) + 8008*a^8*b^12*sgn(cos(f*x + e)) + 4368*a^7*b^13*sgn...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Hanged} \] Input:

int(1/(sin(e + f*x)^5*(a + b/cos(e + f*x)^2)^(5/2)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\sec \left (f x +e \right )^{2} b +a}\, \csc \left (f x +e \right )^{5}}{\sec \left (f x +e \right )^{6} b^{3}+3 \sec \left (f x +e \right )^{4} a \,b^{2}+3 \sec \left (f x +e \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x)
 

Output:

int((sqrt(sec(e + f*x)**2*b + a)*csc(e + f*x)**5)/(sec(e + f*x)**6*b**3 + 
3*sec(e + f*x)**4*a*b**2 + 3*sec(e + f*x)**2*a**2*b + a**3),x)