\(\int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx\) [387]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {(a+b)^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{9/2} d}+\frac {(a+b)^3 \cot (c+d x)}{a^4 d}-\frac {(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac {(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^7(c+d x)}{7 a d} \] Output:

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

Mathematica [A] (verified)

Time = 3.67 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.15 \[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {(a+b)^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{9/2} d}+\frac {\cot (c+d x) \left (176 a^3+406 a^2 b+350 a b^2+105 b^3-a \left (122 a^2+112 a b+35 b^2\right ) \csc ^2(c+d x)+3 a^2 (22 a+7 b) \csc ^4(c+d x)-15 a^3 \csc ^6(c+d x)\right )}{105 a^4 d} \] Input:

Integrate[Cot[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]
 

Output:

((a + b)^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(9/2)*d) + ( 
Cot[c + d*x]*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3 - a*(122*a^2 + 112 
*a*b + 35*b^2)*Csc[c + d*x]^2 + 3*a^2*(22*a + 7*b)*Csc[c + d*x]^4 - 15*a^3 
*Csc[c + d*x]^6))/(105*a^4*d)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3674, 264, 264, 264, 264, 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.

\(\displaystyle \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (c+d x)^8 \left (a+b \sin (c+d x)^2\right )}dx\)

\(\Big \downarrow \) 3674

\(\displaystyle \frac {\int \frac {\cot ^8(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {-\frac {(a+b) \int \frac {\cot ^6(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{a}-\frac {\cot ^7(c+d x)}{7 a}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {-\frac {(a+b) \left (-\frac {(a+b) \int \frac {\cot ^4(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{a}-\frac {\cot ^5(c+d x)}{5 a}\right )}{a}-\frac {\cot ^7(c+d x)}{7 a}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {-\frac {(a+b) \left (-\frac {(a+b) \left (-\frac {(a+b) \int \frac {\cot ^2(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{a}-\frac {\cot ^3(c+d x)}{3 a}\right )}{a}-\frac {\cot ^5(c+d x)}{5 a}\right )}{a}-\frac {\cot ^7(c+d x)}{7 a}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {-\frac {(a+b) \left (-\frac {(a+b) \left (-\frac {(a+b) \left (-\frac {(a+b) \int \frac {1}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{a}-\frac {\cot (c+d x)}{a}\right )}{a}-\frac {\cot ^3(c+d x)}{3 a}\right )}{a}-\frac {\cot ^5(c+d x)}{5 a}\right )}{a}-\frac {\cot ^7(c+d x)}{7 a}}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {(a+b) \left (-\frac {(a+b) \left (-\frac {(a+b) \left (-\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\cot (c+d x)}{a}\right )}{a}-\frac {\cot ^3(c+d x)}{3 a}\right )}{a}-\frac {\cot ^5(c+d x)}{5 a}\right )}{a}-\frac {\cot ^7(c+d x)}{7 a}}{d}\)

Input:

Int[Cot[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]
 

Output:

(-1/7*Cot[c + d*x]^7/a - ((a + b)*(-1/5*Cot[c + d*x]^5/a - ((a + b)*(-1/3* 
Cot[c + d*x]^3/a - ((a + b)*(-((Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[c + d* 
x])/Sqrt[a]])/a^(3/2)) - Cot[c + d*x]/a))/a))/a))/a)/d
 

Defintions of rubi rules used

rule 218

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]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 3042

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

rule 3674

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*((d_.)*tan[(e_.) + (f_.) 
*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[f 
f/f   Subst[Int[(d*ff*x)^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(p + 1) 
), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ 
[p]
 

Maple [A] (verified)

Time = 10.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.32

method	result	size
derivativedivides	\(\frac {\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{4} \sqrt {a \left (a +b \right )}}-\frac {-a^{3}-3 a^{2} b -3 b^{2} a -b^{3}}{a^{4} \tan \left (d x +c \right )}-\frac {-a -b}{5 a^{2} \tan \left (d x +c \right )^{5}}-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a^{2}+2 a b +b^{2}}{3 a^{3} \tan \left (d x +c \right )^{3}}}{d}\)	\(155\)
default	\(\frac {\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{4} \sqrt {a \left (a +b \right )}}-\frac {-a^{3}-3 a^{2} b -3 b^{2} a -b^{3}}{a^{4} \tan \left (d x +c \right )}-\frac {-a -b}{5 a^{2} \tan \left (d x +c \right )^{5}}-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a^{2}+2 a b +b^{2}}{3 a^{3} \tan \left (d x +c \right )^{3}}}{d}\)	\(155\)
risch	\(\frac {2 i \left (105 b^{3}+176 a^{3}+406 a^{2} b +350 b^{2} a +6370 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-2940 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+1575 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-2100 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+2436 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+1575 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-812 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-630 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+5390 a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-7840 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-6860 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+5586 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+5040 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2212 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2030 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3080 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-1260 a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+420 a^{3} {\mathrm e}^{12 i \left (d x +c \right )}+630 a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}-2310 a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-630 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+3080 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+420 a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{105 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{2 a^{2} d}+\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) b}{2 a^{3} d}+\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) b^{2}}{2 a^{4} d}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) b^{3}}{2 a^{5} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{2 a^{2} d}-\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) b}{2 a^{3} d}-\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) b^{2}}{2 a^{4} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) b^{3}}{2 a^{5} d}\)	\(797\)

Input:

int(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

1/d*((a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)/a^4/(a*(a+b))^(1/2)*arctan((a+b)* 
tan(d*x+c)/(a*(a+b))^(1/2))-1/a^4*(-a^3-3*a^2*b-3*a*b^2-b^3)/tan(d*x+c)-1/ 
5*(-a-b)/a^2/tan(d*x+c)^5-1/7/a/tan(d*x+c)^7-1/3/a^3*(a^2+2*a*b+b^2)/tan(d 
*x+c)^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (103) = 206\).

Time = 0.12 (sec) , antiderivative size = 834, normalized size of antiderivative = 7.13 \[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="fricas")
 

Output:

[1/420*(4*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3)*cos(d*x + c)^7 - 28* 
(58*a^3 + 158*a^2*b + 145*a*b^2 + 45*b^3)*cos(d*x + c)^5 + 140*(10*a^3 + 2 
9*a^2*b + 28*a*b^2 + 9*b^3)*cos(d*x + c)^3 + 105*((a^3 + 3*a^2*b + 3*a*b^2 
 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^4 
- a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d* 
x + c)^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*( 
4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a 
^2 + a*b)*cos(d*x + c))*sqrt(-(a + b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2) 
/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))* 
sin(d*x + c) - 420*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))/((a^4*d*c 
os(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*s 
in(d*x + c)), 1/210*(2*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3)*cos(d*x 
 + c)^7 - 14*(58*a^3 + 158*a^2*b + 145*a*b^2 + 45*b^3)*cos(d*x + c)^5 + 70 
*(10*a^3 + 29*a^2*b + 28*a*b^2 + 9*b^3)*cos(d*x + c)^3 - 105*((a^3 + 3*a^2 
*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos 
(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + 
 b^3)*cos(d*x + c)^2)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 
 - a - b)*sqrt((a + b)/a)/((a + b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c 
) - 210*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))/((a^4*d*cos(d*x + c) 
^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x +...
 

Sympy [F]

\[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\int \frac {\cot ^{8}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \] Input:

integrate(cot(d*x+c)**8/(a+b*sin(d*x+c)**2),x)
 

Output:

Integral(cot(c + d*x)**8/(a + b*sin(c + d*x)**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{4}} + \frac {105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )^{6} - 35 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a^{3} + 21 \, {\left (a^{3} + a^{2} b\right )} \tan \left (d x + c\right )^{2}}{a^{4} \tan \left (d x + c\right )^{7}}}{105 \, d} \] Input:

integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="maxima")
 

Output:

1/105*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*arctan((a + b)*tan( 
d*x + c)/sqrt((a + b)*a))/(sqrt((a + b)*a)*a^4) + (105*(a^3 + 3*a^2*b + 3* 
a*b^2 + b^3)*tan(d*x + c)^6 - 35*(a^3 + 2*a^2*b + a*b^2)*tan(d*x + c)^4 - 
15*a^3 + 21*(a^3 + a^2*b)*tan(d*x + c)^2)/(a^4*tan(d*x + c)^7))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (103) = 206\).

Time = 0.76 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.03 \[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{\sqrt {a^{2} + a b} a^{4}} + \frac {105 \, a^{3} \tan \left (d x + c\right )^{6} + 315 \, a^{2} b \tan \left (d x + c\right )^{6} + 315 \, a b^{2} \tan \left (d x + c\right )^{6} + 105 \, b^{3} \tan \left (d x + c\right )^{6} - 35 \, a^{3} \tan \left (d x + c\right )^{4} - 70 \, a^{2} b \tan \left (d x + c\right )^{4} - 35 \, a b^{2} \tan \left (d x + c\right )^{4} + 21 \, a^{3} \tan \left (d x + c\right )^{2} + 21 \, a^{2} b \tan \left (d x + c\right )^{2} - 15 \, a^{3}}{a^{4} \tan \left (d x + c\right )^{7}}}{105 \, d} \] Input:

integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="giac")
 

Output:

1/105*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(pi*floor((d*x + c) 
/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt( 
a^2 + a*b)))/(sqrt(a^2 + a*b)*a^4) + (105*a^3*tan(d*x + c)^6 + 315*a^2*b*t 
an(d*x + c)^6 + 315*a*b^2*tan(d*x + c)^6 + 105*b^3*tan(d*x + c)^6 - 35*a^3 
*tan(d*x + c)^4 - 70*a^2*b*tan(d*x + c)^4 - 35*a*b^2*tan(d*x + c)^4 + 21*a 
^3*tan(d*x + c)^2 + 21*a^2*b*tan(d*x + c)^2 - 15*a^3)/(a^4*tan(d*x + c)^7) 
)/d
 

Mupad [B] (verification not implemented)

Time = 37.90 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )\,{\left (a+b\right )}^{7/2}}{a^{9/2}\,d}-\frac {\frac {1}{7\,a}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a+b\right )}{5\,a^2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,{\left (a+b\right )}^2}{3\,a^3}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^6\,{\left (a+b\right )}^3}{a^4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^7} \] Input:

int(cot(c + d*x)^8/(a + b*sin(c + d*x)^2),x)
 

Output:

(atan((tan(c + d*x)*(a + b)^(1/2))/a^(1/2))*(a + b)^(7/2))/(a^(9/2)*d) - ( 
1/(7*a) - (tan(c + d*x)^2*(a + b))/(5*a^2) + (tan(c + d*x)^4*(a + b)^2)/(3 
*a^3) - (tan(c + d*x)^6*(a + b)^3)/a^4)/(d*tan(c + d*x)^7)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1969, normalized size of antiderivative = 16.83 \[ \int \frac {\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx =\text {Too large to display} \] Input:

int(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x)
 

Output:

( - 210*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)* 
atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b))) 
*sin(c + d*x)**7*a**3 - 630*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqr 
t(a + b) + a + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt 
(a + b) + a + 2*b)))*sin(c + d*x)**7*a**2*b - 630*sqrt(b)*sqrt(a)*sqrt(a + 
 b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt( 
a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(c + d*x)**7*a*b**2 - 210*sq 
rt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan( 
(c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(c + d 
*x)**7*b**3 + 210*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan( 
(c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(c + d 
*x)**7*a**4 + 840*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan( 
(c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(c + d 
*x)**7*a**3*b + 1260*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((t 
an((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(c 
+ d*x)**7*a**2*b**2 + 840*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*at 
an((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*s 
in(c + d*x)**7*a*b**3 + 210*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)* 
atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b))) 
*sin(c + d*x)**7*b**4 + 352*cos(c + d*x)*sin(c + d*x)**6*a**5 + 812*cos...