\(\int \frac {1}{(a+b \cot ^2(c+d x))^{7/2}} \, dx\) [37]

Optimal result

Integrand size = 16, antiderivative size = 190 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^{7/2} d}+\frac {b \cot (c+d x)}{5 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{5/2}}+\frac {(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a+b \cot ^2(c+d x)}} \] Output:

-arctan((a-b)^(1/2)*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2))/(a-b)^(7/2)/d+1/5 
*b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)^(5/2)+1/15*(9*a-4*b)*b*cot(d*x+ 
c)/a^2/(a-b)^2/d/(a+b*cot(d*x+c)^2)^(3/2)+1/15*b*(33*a^2-26*a*b+8*b^2)*cot 
(d*x+c)/a^3/(a-b)^3/d/(a+b*cot(d*x+c)^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.51 (sec) , antiderivative size = 2553, normalized size of antiderivative = 13.44 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*Cot[c + d*x]^2)^(-7/2),x]
 

Output:

-1/4725*(Cot[c + d*x]*(-33075*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]] + ( 
99225*(a - b)*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2)/a - 
 (99225*(a - b)^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4) 
/a^2 + (33075*(a - b)^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d 
*x]^6)/a^3 - (66150*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x 
]^2)/a + (198450*(a - b)*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c 
+ d*x]^2*Cot[c + d*x]^2)/a^2 + (66150*(a - b)^3*b*ArcSin[Sqrt[((a - b)*Cos 
[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^2)/a^4 - (52920*b^2*ArcSin[Sq 
rt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^4)/a^2 + (158760*(a - b)*b^2* 
ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^4)/a^ 
3 - (158760*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + 
 d*x]^4*Cot[c + d*x]^4)/a^4 + (52920*(a - b)^3*b^2*ArcSin[Sqrt[((a - b)*Co 
s[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^4)/a^5 - (15120*b^3*ArcSin[S 
qrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^6)/a^3 + (45360*(a - b)*b^3* 
ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^6)/a^ 
4 - (45360*(a - b)^2*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + 
d*x]^4*Cot[c + d*x]^6)/a^5 + (15120*(a - b)^3*b^3*ArcSin[Sqrt[((a - b)*Cos 
[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^6)/a^6 - 77175*(((a - b)*Cos[ 
c + d*x]^2)/a)^(3/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + 507 
15*(((a - b)*Cos[c + d*x]^2)/a)^(5/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c...
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4144, 316, 402, 402, 27, 291, 216}

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.

\(\displaystyle \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+b \tan \left (c+d x+\frac {\pi }{2}\right )^2\right )^{7/2}}dx\)

\(\Big \downarrow \) 4144

\(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )^{7/2}}d\cot (c+d x)}{d}\)

\(\Big \downarrow \) 316

\(\displaystyle -\frac {\frac {\int \frac {-4 b \cot ^2(c+d x)+5 a-4 b}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )^{5/2}}d\cot (c+d x)}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle -\frac {\frac {\frac {\int \frac {15 a^2-18 b a+8 b^2-2 (9 a-4 b) b \cot ^2(c+d x)}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )^{3/2}}d\cot (c+d x)}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {15 a^3}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)}{a (a-b)}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\frac {15 a^2 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)}{a-b}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\)

\(\Big \downarrow \) 291

\(\displaystyle -\frac {\frac {\frac {\frac {15 a^2 \int \frac {1}{1-\frac {(b-a) \cot ^2(c+d x)}{b \cot ^2(c+d x)+a}}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a}}}{a-b}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {\frac {\frac {15 a^2 \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^{3/2}}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\)

Input:

Int[(a + b*Cot[c + d*x]^2)^(-7/2),x]
 

Output:

-((-1/5*(b*Cot[c + d*x])/(a*(a - b)*(a + b*Cot[c + d*x]^2)^(5/2)) + (-1/3* 
((9*a - 4*b)*b*Cot[c + d*x])/(a*(a - b)*(a + b*Cot[c + d*x]^2)^(3/2)) + (( 
15*a^2*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(a - 
 b)^(3/2) - (b*(33*a^2 - 26*a*b + 8*b^2)*Cot[c + d*x])/(a*(a - b)*Sqrt[a + 
 b*Cot[c + d*x]^2]))/(3*a*(a - b)))/(5*a*(a - b)))/d)
 

Defintions of rubi rules used

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[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 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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 4144

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> 
With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f)   Subst[Int[(a + b* 
(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, 
 b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || 
EqQ[n^2, 16])
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\frac {\frac {b \left (\frac {\cot \left (d x +c \right )}{3 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \cot \left (d x +c \right )}{3 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{2}}+\frac {b \left (\frac {\cot \left (d x +c \right )}{5 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 \cot \left (d x +c \right )}{15 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {8 \cot \left (d x +c \right )}{15 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{a}\right )}{a -b}+\frac {b \cot \left (d x +c \right )}{\left (a -b \right )^{3} a \sqrt {a +b \cot \left (d x +c \right )^{2}}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{4} b^{2}}}{d}\)	\(253\)
default	\(\frac {\frac {b \left (\frac {\cot \left (d x +c \right )}{3 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \cot \left (d x +c \right )}{3 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{2}}+\frac {b \left (\frac {\cot \left (d x +c \right )}{5 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 \cot \left (d x +c \right )}{15 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {8 \cot \left (d x +c \right )}{15 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{a}\right )}{a -b}+\frac {b \cot \left (d x +c \right )}{\left (a -b \right )^{3} a \sqrt {a +b \cot \left (d x +c \right )^{2}}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{4} b^{2}}}{d}\)	\(253\)

Input:

int(1/(a+b*cot(d*x+c)^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/(a-b)^2*b*(1/3*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(3/2)+2/3/a^2*cot(d* 
x+c)/(a+b*cot(d*x+c)^2)^(1/2))+1/(a-b)*b*(1/5*cot(d*x+c)/a/(a+b*cot(d*x+c) 
^2)^(5/2)+4/5/a*(1/3*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(3/2)+2/3/a^2*cot(d*x 
+c)/(a+b*cot(d*x+c)^2)^(1/2)))+b/(a-b)^3*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^( 
1/2)-1/(a-b)^4*(b^4*(a-b))^(1/2)/b^2*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a 
+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 752 vs. \(2 (172) = 344\).

Time = 0.21 (sec) , antiderivative size = 1504, normalized size of antiderivative = 7.92 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*cot(d*x+c)^2)^(7/2),x, algorithm="fricas")
 

Output:

[-1/60*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 - (a^6 - 3*a^5*b + 3*a^4*b 
^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 + 3*(a^6 - a^5*b - a^4*b^2 + a^3*b^3)*cos 
(2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c))*sq 
rt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + 2*((a - b)*cos( 
2*d*x + 2*c) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(co 
s(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2* 
d*x + 2*c)) + 4*(45*a^5*b - 15*a^4*b^2 - 47*a^3*b^3 + 11*a^2*b^4 + 14*a*b^ 
5 - 8*b^6 + (45*a^5*b - 165*a^4*b^2 + 233*a^3*b^3 - 159*a^2*b^4 + 54*a*b^5 
 - 8*b^6)*cos(2*d*x + 2*c)^2 - 2*(45*a^5*b - 90*a^4*b^2 + 27*a^3*b^3 + 44* 
a^2*b^4 - 34*a*b^5 + 8*b^6)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2* 
c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^10 - 7*a^9*b + 2 
1*a^8*b^2 - 35*a^7*b^3 + 35*a^6*b^4 - 21*a^5*b^5 + 7*a^4*b^6 - a^3*b^7)*d* 
cos(2*d*x + 2*c)^3 - 3*(a^10 - 5*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 - 5*a^6*b^4 
 + 9*a^5*b^5 - 5*a^4*b^6 + a^3*b^7)*d*cos(2*d*x + 2*c)^2 + 3*(a^10 - 3*a^9 
*b + a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - a^5*b^5 + 3*a^4*b^6 - a^3*b^7)*d*co 
s(2*d*x + 2*c) - (a^10 - a^9*b - 3*a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 - 3*a^5 
*b^5 - a^4*b^6 + a^3*b^7)*d), 1/30*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^ 
3 - (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 + 3*(a^6 - a^ 
5*b - a^4*b^2 + a^3*b^3)*cos(2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a 
^3*b^3)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-((a - b)*cos(2*d*x + 2*c)...
 

Sympy [F]

\[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{\left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(1/(a+b*cot(d*x+c)**2)**(7/2),x)
 

Output:

Integral((a + b*cot(c + d*x)**2)**(-7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(a+b*cot(d*x+c)^2)^(7/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more 
details)Is
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3249 vs. \(2 (172) = 344\).

Time = 1.98 (sec) , antiderivative size = 3249, normalized size of antiderivative = 17.10 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*cot(d*x+c)^2)^(7/2),x, algorithm="giac")
 

Output:

1/15*(30*arctan(-1/2*(sqrt(b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(b*tan(1/2*d*x 
+ 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c)^2 + b) 
+ sqrt(b))/sqrt(a - b))/((a^3*sgn(sin(d*x + c)) - 3*a^2*b*sgn(sin(d*x + c) 
) + 3*a*b^2*sgn(sin(d*x + c)) - b^3*sgn(sin(d*x + c)))*sqrt(a - b)) - (((( 
((33*a^20*b^3*sgn(sin(d*x + c)) - 620*a^19*b^4*sgn(sin(d*x + c)) + 5525*a^ 
18*b^5*sgn(sin(d*x + c)) - 31050*a^17*b^6*sgn(sin(d*x + c)) + 123420*a^16* 
b^7*sgn(sin(d*x + c)) - 368832*a^15*b^8*sgn(sin(d*x + c)) + 859860*a^14*b^ 
9*sgn(sin(d*x + c)) - 1601400*a^13*b^10*sgn(sin(d*x + c)) + 2419950*a^12*b 
^11*sgn(sin(d*x + c)) - 2996760*a^11*b^12*sgn(sin(d*x + c)) + 3058198*a^10 
*b^13*sgn(sin(d*x + c)) - 2576860*a^9*b^14*sgn(sin(d*x + c)) + 1790100*a^8 
*b^15*sgn(sin(d*x + c)) - 1020000*a^7*b^16*sgn(sin(d*x + c)) + 472260*a^6* 
b^17*sgn(sin(d*x + c)) - 175032*a^5*b^18*sgn(sin(d*x + c)) + 50745*a^4*b^1 
9*sgn(sin(d*x + c)) - 11100*a^3*b^20*sgn(sin(d*x + c)) + 1725*a^2*b^21*sgn 
(sin(d*x + c)) - 170*a*b^22*sgn(sin(d*x + c)) + 8*b^23*sgn(sin(d*x + c)))* 
tan(1/2*d*x + 1/2*c)^2/(a^24 - 21*a^23*b + 210*a^22*b^2 - 1330*a^21*b^3 + 
5985*a^20*b^4 - 20349*a^19*b^5 + 54264*a^18*b^6 - 116280*a^17*b^7 + 203490 
*a^16*b^8 - 293930*a^15*b^9 + 352716*a^14*b^10 - 352716*a^13*b^11 + 293930 
*a^12*b^12 - 203490*a^11*b^13 + 116280*a^10*b^14 - 54264*a^9*b^15 + 20349* 
a^8*b^16 - 5985*a^7*b^17 + 1330*a^6*b^18 - 210*a^5*b^19 + 21*a^4*b^20 - a^ 
3*b^21) + 5*(60*a^21*b^2*sgn(sin(d*x + c)) - 1165*a^20*b^3*sgn(sin(d*x ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{7/2}} \,d x \] Input:

int(1/(a + b*cot(c + d*x)^2)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int(1/(a + b*cot(c + d*x)^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {\sqrt {\cot \left (d x +c \right )^{2} b +a}}{\cot \left (d x +c \right )^{8} b^{4}+4 \cot \left (d x +c \right )^{6} a \,b^{3}+6 \cot \left (d x +c \right )^{4} a^{2} b^{2}+4 \cot \left (d x +c \right )^{2} a^{3} b +a^{4}}d x \] Input:

int(1/(a+b*cot(d*x+c)^2)^(7/2),x)
 

Output:

int(sqrt(cot(c + d*x)**2*b + a)/(cot(c + d*x)**8*b**4 + 4*cot(c + d*x)**6* 
a*b**3 + 6*cot(c + d*x)**4*a**2*b**2 + 4*cot(c + d*x)**2*a**3*b + a**4),x)