\(\int \frac {\cot ^5(e+f x)}{(a+b \sec ^2(e+f x))^{3/2}} \, dx\) [420]

Optimal result

Integrand size = 25, antiderivative size = 213 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}-\frac {\left (8 a^2+28 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a+b}}\right )}{8 (a+b)^{7/2} f}+\frac {b \left (4 a^2+11 a b-8 b^2\right )}{8 a (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(4 a+9 b) \cot ^2(e+f x)}{8 (a+b)^2 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\cot ^4(e+f x)}{4 (a+b) f \sqrt {a+b \sec ^2(e+f x)}} \] Output:

arctanh((a+b*sec(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/f-1/8*(8*a^2+28*a*b+35*b 
^2)*arctanh((a+b*sec(f*x+e)^2)^(1/2)/(a+b)^(1/2))/(a+b)^(7/2)/f+1/8*b*(4*a 
^2+11*a*b-8*b^2)/a/(a+b)^3/f/(a+b*sec(f*x+e)^2)^(1/2)+1/8*(4*a+9*b)*cot(f* 
x+e)^2/(a+b)^2/f/(a+b*sec(f*x+e)^2)^(1/2)-1/4*cot(f*x+e)^4/(a+b)/f/(a+b*se 
c(f*x+e)^2)^(1/2)
 

Mathematica [F]

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

Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]
 

Output:

Integrate[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x]
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.16, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4627, 25, 354, 114, 27, 168, 27, 169, 27, 174, 73, 221}

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.

\(\displaystyle \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4627

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\frac {\frac {4 a+9 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}-\frac {\int -\frac {\cos (e+f x) \left (8 (a+b)^2+3 b (4 a+9 b) \sec ^2(e+f x)\right )}{2 \left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec ^2(e+f x)}{a+b}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\int \frac {\cos (e+f x) \left (8 (a+b)^2+3 b (4 a+9 b) \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec ^2(e+f x)}{2 (a+b)}+\frac {4 a+9 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{2 f}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {\frac {\frac {\frac {2 \int \frac {\cos (e+f x) \left (8 (a+b)^3+b \left (4 a^2+11 b a-8 b^2\right ) \sec ^2(e+f x)\right )}{2 \left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (4 a^2+11 a b-8 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}+\frac {4 a+9 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\cos (e+f x) \left (8 (a+b)^3+b \left (4 a^2+11 b a-8 b^2\right ) \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (4 a^2+11 a b-8 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}+\frac {4 a+9 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{2 f}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {\frac {\frac {\frac {a \left (8 a^2+28 a b+35 b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)+8 (a+b)^3 \int \frac {\cos (e+f x)}{\sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (4 a^2+11 a b-8 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}+\frac {4 a+9 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{2 f}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {2 a \left (8 a^2+28 a b+35 b^2\right ) \int \frac {1}{\frac {a+b}{b}-\frac {\sec ^4(e+f x)}{b}}d\sqrt {b \sec ^2(e+f x)+a}}{b}+\frac {16 (a+b)^3 \int \frac {1}{\frac {\sec ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \sec ^2(e+f x)+a}}{b}}{a (a+b)}-\frac {2 b \left (4 a^2+11 a b-8 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}+\frac {4 a+9 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{2 f}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\frac {\frac {\frac {2 a \left (8 a^2+28 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}-\frac {16 (a+b)^3 \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a+b)}-\frac {2 b \left (4 a^2+11 a b-8 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{2 (a+b)}+\frac {4 a+9 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \sqrt {a+b \sec ^2(e+f x)}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \sqrt {a+b \sec ^2(e+f x)}}}{2 f}\)

Input:

Int[Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2),x]
 

Output:

-1/2*(1/(2*(a + b)*(1 - Sec[e + f*x]^2)^2*Sqrt[a + b*Sec[e + f*x]^2]) + (( 
4*a + 9*b)/((a + b)*(1 - Sec[e + f*x]^2)*Sqrt[a + b*Sec[e + f*x]^2]) + ((( 
-16*(a + b)^3*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (2*a* 
(8*a^2 + 28*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]]) 
/Sqrt[a + b])/(a*(a + b)) - (2*b*(4*a^2 + 11*a*b - 8*b^2))/(a*(a + b)*Sqrt 
[a + b*Sec[e + f*x]^2]))/(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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

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])
 

rule 168

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] && ILtQ[m, -1]
 

rule 169

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]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 3042

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

rule 4627

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( 
f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Si 
mp[1/f   Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x), 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] || IGtQ[p, 0] || Integers 
Q[2*n, p])
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4961\) vs. \(2(187)=374\).

Time = 1.21 (sec) , antiderivative size = 4962, normalized size of antiderivative = 23.30

method	result	size
default	\(\text {Expression too large to display}\)	\(4962\)

Input:

int(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/16/f/(a+b)^(15/2)/a^(3/2)/((-a*b)^(1/2)+a)/((-a*b)^(1/2)-a)/(b+a*cos(f* 
x+e)^2)/(a+b*sec(f*x+e)^2)^(3/2)*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1 
/2)*(a+b)^(7/2)*ln(4*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)*c 
os(f*x+e)+4*a^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+4*cos(f*x+ 
e)*a)*a^6*b^2*(160*cos(f*x+e)^2+160*cos(f*x+e)+160+160*sec(f*x+e)+16*sec(f 
*x+e)^2+16*sec(f*x+e)^3)+((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(a+b) 
^(7/2)*ln(4*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)*cos(f*x+e) 
+4*a^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+4*cos(f*x+e)*a)*a^5 
*b^3*(160*cos(f*x+e)^2+160*cos(f*x+e)+320+320*sec(f*x+e)+80*sec(f*x+e)^2+8 
0*sec(f*x+e)^3)+((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(a+b)^(7/2)*ln 
(4*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)*cos(f*x+e)+4*a^(1/2 
)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+4*cos(f*x+e)*a)*a^4*b^4*(80* 
cos(f*x+e)^2+80*cos(f*x+e)+320+320*sec(f*x+e)+160*sec(f*x+e)^2+160*sec(f*x 
+e)^3)+((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(a+b)^(7/2)*ln(4*((b+a* 
cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)*cos(f*x+e)+4*a^(1/2)*((b+a*c 
os(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+4*cos(f*x+e)*a)*a^3*b^5*(16*cos(f*x+e 
)^2+16*cos(f*x+e)+160+160*sec(f*x+e)+160*sec(f*x+e)^2+160*sec(f*x+e)^3)+(( 
b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(a+b)^(7/2)*ln(4*((b+a*cos(f*x+e 
)^2)/(1+cos(f*x+e))^2)^(1/2)*a^(1/2)*cos(f*x+e)+4*a^(1/2)*((b+a*cos(f*x+e) 
^2)/(1+cos(f*x+e))^2)^(1/2)+4*cos(f*x+e)*a)*a^2*b^6*(32+32*sec(f*x+e)+8...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (187) = 374\).

Time = 4.40 (sec) , antiderivative size = 3501, normalized size of antiderivative = 16.44 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(cot(f*x+e)**5/(a+b*sec(f*x+e)**2)**(3/2),x)
 

Output:

Integral(cot(e + f*x)**5/(a + b*sec(e + f*x)**2)**(3/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1771 vs. \(2 (187) = 374\).

Time = 2.18 (sec) , antiderivative size = 1771, normalized size of antiderivative = 8.31 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/64*(((((a^10*b + 7*a^9*b^2 + 21*a^8*b^3 + 35*a^7*b^4 + 35*a^6*b^5 + 21*a 
^5*b^6 + 7*a^4*b^7 + a^3*b^8)*tan(1/2*f*x + 1/2*e)^2/(a^11*b*sgn(cos(f*x + 
 e)) + 8*a^10*b^2*sgn(cos(f*x + e)) + 28*a^9*b^3*sgn(cos(f*x + e)) + 56*a^ 
8*b^4*sgn(cos(f*x + e)) + 70*a^7*b^5*sgn(cos(f*x + e)) + 56*a^6*b^6*sgn(co 
s(f*x + e)) + 28*a^5*b^7*sgn(cos(f*x + e)) + 8*a^4*b^8*sgn(cos(f*x + e)) + 
 a^3*b^9*sgn(cos(f*x + e))) - (13*a^10*b + 101*a^9*b^2 + 333*a^8*b^3 + 605 
*a^7*b^4 + 655*a^6*b^5 + 423*a^5*b^6 + 151*a^4*b^7 + 23*a^3*b^8)/(a^11*b*s 
gn(cos(f*x + e)) + 8*a^10*b^2*sgn(cos(f*x + e)) + 28*a^9*b^3*sgn(cos(f*x + 
 e)) + 56*a^8*b^4*sgn(cos(f*x + e)) + 70*a^7*b^5*sgn(cos(f*x + e)) + 56*a^ 
6*b^6*sgn(cos(f*x + e)) + 28*a^5*b^7*sgn(cos(f*x + e)) + 8*a^4*b^8*sgn(cos 
(f*x + e)) + a^3*b^9*sgn(cos(f*x + e))))*tan(1/2*f*x + 1/2*e)^2 + (23*a^10 
*b + 145*a^9*b^2 + 331*a^8*b^3 + 349*a^7*b^4 + 245*a^6*b^5 + 323*a^5*b^6 + 
 425*a^4*b^7 + 271*a^3*b^8 + 64*a^2*b^9)/(a^11*b*sgn(cos(f*x + e)) + 8*a^1 
0*b^2*sgn(cos(f*x + e)) + 28*a^9*b^3*sgn(cos(f*x + e)) + 56*a^8*b^4*sgn(co 
s(f*x + e)) + 70*a^7*b^5*sgn(cos(f*x + e)) + 56*a^6*b^6*sgn(cos(f*x + e)) 
+ 28*a^5*b^7*sgn(cos(f*x + e)) + 8*a^4*b^8*sgn(cos(f*x + e)) + a^3*b^9*sgn 
(cos(f*x + e))))*tan(1/2*f*x + 1/2*e)^2 - (11*a^10*b + 91*a^9*b^2 + 315*a^ 
8*b^3 + 659*a^7*b^4 + 985*a^6*b^5 + 1081*a^5*b^6 + 801*a^4*b^7 + 345*a^3*b 
^8 + 64*a^2*b^9)/(a^11*b*sgn(cos(f*x + e)) + 8*a^10*b^2*sgn(cos(f*x + e)) 
+ 28*a^9*b^3*sgn(cos(f*x + e)) + 56*a^8*b^4*sgn(cos(f*x + e)) + 70*a^7*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Hanged} \] Input:

int(cot(e + f*x)^5/(a + b/cos(e + f*x)^2)^(3/2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

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

int(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x)
 

Output:

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