\(\int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [160]

Optimal result

Integrand size = 24, antiderivative size = 197 \[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d} \] Output:

1/2*b^2*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(11/4)/(a^(1/ 
2)-b^(1/2))^(1/2)/d+1/2*b^2*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1 
/4))/a^(11/4)/(a^(1/2)+b^(1/2))^(1/2)/d-(a+b)*cot(d*x+c)/a^2/d-1/3*(3*a+b) 
*cot(d*x+c)^3/a^2/d-3/5*cot(d*x+c)^5/a/d-1/7*cot(d*x+c)^7/a/d
 

Mathematica [A] (verified)

Time = 8.88 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.41 \[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\left (\sqrt {a} \sqrt {b}+b\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}{2 a^{5/2} \sqrt {a+\sqrt {a} \sqrt {b}} d}-\frac {b^2 \text {arctanh}\left (\frac {\left (\sqrt {a} \sqrt {b}-b\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}{2 a^{5/2} \sqrt {-a+\sqrt {a} \sqrt {b}} d}-\frac {2 (24 a \cos (c+d x)+35 b \cos (c+d x)) \csc (c+d x)}{105 a^2 d}+\frac {(-24 a \cos (c+d x)-35 b \cos (c+d x)) \csc ^3(c+d x)}{105 a^2 d}-\frac {6 \cot (c+d x) \csc ^4(c+d x)}{35 a d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d} \] Input:

Integrate[Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]
 

Output:

(b^2*ArcTan[((Sqrt[a]*Sqrt[b] + b)*Tan[c + d*x])/(Sqrt[a + Sqrt[a]*Sqrt[b] 
]*Sqrt[b])])/(2*a^(5/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*d) - (b^2*ArcTanh[((Sqrt 
[a]*Sqrt[b] - b)*Tan[c + d*x])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(2*a 
^(5/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*d) - (2*(24*a*Cos[c + d*x] + 35*b*Cos[c 
+ d*x])*Csc[c + d*x])/(105*a^2*d) + ((-24*a*Cos[c + d*x] - 35*b*Cos[c + d* 
x])*Csc[c + d*x]^3)/(105*a^2*d) - (6*Cot[c + d*x]*Csc[c + d*x]^4)/(35*a*d) 
 - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3696, 1610, 2009}

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.

\(\displaystyle \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (c+d x)^8 \left (a-b \sin (c+d x)^4\right )}dx\)

\(\Big \downarrow \) 3696

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

\(\Big \downarrow \) 1610

\(\displaystyle \frac {\int \left (\frac {\cot ^8(c+d x)}{a}+\frac {3 \cot ^6(c+d x)}{a}+\frac {(3 a+b) \cot ^4(c+d x)}{a^2}+\frac {(a+b) \cot ^2(c+d x)}{a^2}+\frac {b^2 \left (\tan ^2(c+d x)+1\right )}{a^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}\right )d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^2 \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2}-\frac {(a+b) \cot (c+d x)}{a^2}-\frac {\cot ^7(c+d x)}{7 a}-\frac {3 \cot ^5(c+d x)}{5 a}}{d}\)

Input:

Int[Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]
 

Output:

((b^2*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(11/4)* 
Sqrt[Sqrt[a] - Sqrt[b]]) + (b^2*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d* 
x])/a^(1/4)])/(2*a^(11/4)*Sqrt[Sqrt[a] + Sqrt[b]]) - ((a + b)*Cot[c + d*x] 
)/a^2 - ((3*a + b)*Cot[c + d*x]^3)/(3*a^2) - (3*Cot[c + d*x]^5)/(5*a) - Co 
t[c + d*x]^7/(7*a))/d
 

Defintions of rubi rules used

rule 1610

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a 
+ b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 
*a*c, 0] && IntegerQ[q] && IntegerQ[m]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3696

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( 
p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 
)/f   Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) 
^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & 
& IntegerQ[m/2] && IntegerQ[p]
 

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {3 a +b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {3}{5 a \tan \left (d x +c \right )^{5}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\)	\(213\)
default	\(\frac {-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {3 a +b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {3}{5 a \tan \left (d x +c \right )^{5}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\)	\(213\)
risch	\(\frac {4 i \left (105 \,{\mathrm e}^{10 i \left (d x +c \right )} b -455 \,{\mathrm e}^{8 i \left (d x +c \right )} b +840 \,{\mathrm e}^{6 i \left (d x +c \right )} a +770 \,{\mathrm e}^{6 i \left (d x +c \right )} b -504 \,{\mathrm e}^{4 i \left (d x +c \right )} a -630 b \,{\mathrm e}^{4 i \left (d x +c \right )}+168 a \,{\mathrm e}^{2 i \left (d x +c \right )}+245 b \,{\mathrm e}^{2 i \left (d x +c \right )}-24 a -35 b \right )}{105 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+256 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1099511627776 a^{12} d^{4}-1099511627776 a^{11} b \,d^{4}\right ) \textit {\_Z}^{4}+2097152 a^{6} b^{4} d^{2} \textit {\_Z}^{2}+b^{8}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {2147483648 i d^{3} a^{10}}{b^{7}}-\frac {2147483648 i d^{3} a^{9}}{b^{6}}\right ) \textit {\_R}^{3}+\left (-\frac {2097152 d^{2} a^{7}}{b^{5}}+\frac {2097152 d^{2} a^{6}}{b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {2048 i d \,a^{4}}{b^{3}}+\frac {2048 i d \,a^{3}}{b^{2}}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\)	\(272\)

Input:

int(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1585 vs. \(2 (155) = 310\).

Time = 0.24 (sec) , antiderivative size = 1585, normalized size of antiderivative = 8.05 \[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
 

Output:

-1/840*(16*(24*a + 35*b)*cos(d*x + c)^7 - 56*(24*a + 35*b)*cos(d*x + c)^5 
+ 560*(3*a + 4*b)*cos(d*x + c)^3 + 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos 
(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 + (a^6 - a^5*b)*s 
qrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log( 
1/4*b^7*cos(d*x + c)^2 - 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x 
+ c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d 
^4)) + 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) - (a^10 - a^9*b)*sqrt(b^9/ 
((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-( 
b^4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^ 
6 - a^5*b)*d^2)))*sin(d*x + c) - 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d 
*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 + (a^6 - a^5*b)*sqr 
t(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(1/ 
4*b^7*cos(d*x + c)^2 - 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + 
c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4 
)) - 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) - (a^10 - a^9*b)*sqrt(b^9/(( 
a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^ 
4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 
- a^5*b)*d^2)))*sin(d*x + c) - 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x 
 + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt( 
b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(-...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(csc(d*x+c)**8/(a-b*sin(d*x+c)**4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{8}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \] Input:

integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
 

Output:

4/105*(735*b*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) - 7*(15*b*sin(10*d*x + 10*c 
) - 65*b*sin(8*d*x + 8*c) + 10*(12*a + 11*b)*sin(6*d*x + 6*c) - 18*(4*a + 
5*b)*sin(4*d*x + 4*c) + (24*a + 35*b)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) 
 + 49*(15*b*sin(10*d*x + 10*c) - 65*b*sin(8*d*x + 8*c) + 10*(12*a + 11*b)* 
sin(6*d*x + 6*c) - 18*(4*a + 5*b)*sin(4*d*x + 4*c) + (24*a + 35*b)*sin(2*d 
*x + 2*c))*cos(12*d*x + 12*c) + 147*(40*b*sin(8*d*x + 8*c) - 5*(24*a + 17* 
b)*sin(6*d*x + 6*c) + 3*(24*a + 25*b)*sin(4*d*x + 4*c) - 6*(4*a + 5*b)*sin 
(2*d*x + 2*c))*cos(10*d*x + 10*c) + 245*(15*(8*a + 3*b)*sin(6*d*x + 6*c) - 
 3*(24*a + 17*b)*sin(4*d*x + 4*c) + 2*(12*a + 11*b)*sin(2*d*x + 2*c))*cos( 
8*d*x + 8*c) + 245*(24*b*sin(4*d*x + 4*c) - 13*b*sin(2*d*x + 2*c))*cos(6*d 
*x + 6*c) - 420*(a^2*b^2*d*cos(14*d*x + 14*c)^2 + 49*a^2*b^2*d*cos(12*d*x 
+ 12*c)^2 + 441*a^2*b^2*d*cos(10*d*x + 10*c)^2 + 1225*a^2*b^2*d*cos(8*d*x 
+ 8*c)^2 + 1225*a^2*b^2*d*cos(6*d*x + 6*c)^2 + 441*a^2*b^2*d*cos(4*d*x + 4 
*c)^2 + 49*a^2*b^2*d*cos(2*d*x + 2*c)^2 + a^2*b^2*d*sin(14*d*x + 14*c)^2 + 
 49*a^2*b^2*d*sin(12*d*x + 12*c)^2 + 441*a^2*b^2*d*sin(10*d*x + 10*c)^2 + 
1225*a^2*b^2*d*sin(8*d*x + 8*c)^2 + 1225*a^2*b^2*d*sin(6*d*x + 6*c)^2 + 44 
1*a^2*b^2*d*sin(4*d*x + 4*c)^2 - 294*a^2*b^2*d*sin(4*d*x + 4*c)*sin(2*d*x 
+ 2*c) + 49*a^2*b^2*d*sin(2*d*x + 2*c)^2 - 14*a^2*b^2*d*cos(2*d*x + 2*c) + 
 a^2*b^2*d - 2*(7*a^2*b^2*d*cos(12*d*x + 12*c) - 21*a^2*b^2*d*cos(10*d*x + 
 10*c) + 35*a^2*b^2*d*cos(8*d*x + 8*c) - 35*a^2*b^2*d*cos(6*d*x + 6*c) ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (155) = 310\).

Time = 0.66 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.37 \[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {105 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} + \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} + \frac {105 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} - \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} - \frac {2 \, {\left (105 \, a \tan \left (d x + c\right )^{6} + 105 \, b \tan \left (d x + c\right )^{6} + 105 \, a \tan \left (d x + c\right )^{4} + 35 \, b \tan \left (d x + c\right )^{4} + 63 \, a \tan \left (d x + c\right )^{2} + 15 \, a\right )}}{a^{2} \tan \left (d x + c\right )^{7}}}{210 \, d} \] Input:

integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="giac")
 

Output:

1/210*(105*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 6*sqrt(a^2 - a 
*b + sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*( 
pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 + sqrt(a^6 - 
(a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3*a^7 - 12*a^6*b + 14*a^5 
*b^2 - 4*a^4*b^3 - a^3*b^4) + 105*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a 
^2*b^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b - sq 
rt(a*b)*(a - b))*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/ 
sqrt((a^3 - sqrt(a^6 - (a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3* 
a^7 - 12*a^6*b + 14*a^5*b^2 - 4*a^4*b^3 - a^3*b^4) - 2*(105*a*tan(d*x + c) 
^6 + 105*b*tan(d*x + c)^6 + 105*a*tan(d*x + c)^4 + 35*b*tan(d*x + c)^4 + 6 
3*a*tan(d*x + c)^2 + 15*a)/(a^2*tan(d*x + c)^7))/d
 

Mupad [B] (verification not implemented)

Time = 39.01 (sec) , antiderivative size = 1704, normalized size of antiderivative = 8.65 \[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \] Input:

int(1/(sin(c + d*x)^8*(a - b*sin(c + d*x)^4)),x)
 

Output:

(atan((((((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9* 
b^5 - 32*a^10*b^4 + 16*a^11*b^3 + tan(c + d*x)*(((a^11*b^9)^(1/2) + a^6*b^ 
4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2)) - 
 tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*( 
a^11*b - a^12)))^(1/2)*1i - ((((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a 
^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 - tan(c + d*x)*(((a^1 
1*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b 
^3 - 128*a^13*b^2)) + tan(c + d*x)*(4*a^6*b^7 - 4*a^8*b^5))*(((a^11*b^9)^( 
1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*1i)/(((((a^11*b^9)^(1/2) + a^6 
*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 
+ tan(c + d*x)*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*( 
64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2)) - tan(c + d*x)*(4*a^6*b^7 - 4*a^8 
*b^5))*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2) + ((((a^1 
1*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10* 
b^4 + 16*a^11*b^3 - tan(c + d*x)*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b 
 - a^12)))^(1/2)*(64*a^14*b + 64*a^12*b^3 - 128*a^13*b^2)) + tan(c + d*x)* 
(4*a^6*b^7 - 4*a^8*b^5))*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11*b - a^12) 
))^(1/2) - 2*a^4*b^8 + 2*a^5*b^7))*(((a^11*b^9)^(1/2) + a^6*b^4)/(16*(a^11 
*b - a^12)))^(1/2)*2i)/d + (atan((((-((a^11*b^9)^(1/2) - a^6*b^4)/(16*(a^1 
1*b - a^12)))^(1/2)*(16*a^9*b^5 - 32*a^10*b^4 + 16*a^11*b^3 + tan(c + d...
 

Reduce [F]

\[ \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {too large to display} \] Input:

int(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x)
 

Output:

( - 48*cos(c + d*x)*sin(c + d*x)**6*tan((c + d*x)/2)**7*a**5 - 70*cos(c + 
d*x)*sin(c + d*x)**6*tan((c + d*x)/2)**7*a**4*b + 81408*cos(c + d*x)*sin(c 
 + d*x)**6*tan((c + d*x)/2)**7*a**2*b**3 + 90112*cos(c + d*x)*sin(c + d*x) 
**6*tan((c + d*x)/2)**7*a*b**4 - 24*cos(c + d*x)*sin(c + d*x)**4*tan((c + 
d*x)/2)**7*a**5 - 35*cos(c + d*x)*sin(c + d*x)**4*tan((c + d*x)/2)**7*a**4 
*b - 295296*cos(c + d*x)*sin(c + d*x)**4*tan((c + d*x)/2)**7*a**2*b**3 - 3 
85024*cos(c + d*x)*sin(c + d*x)**4*tan((c + d*x)/2)**7*a*b**4 - 18*cos(c + 
 d*x)*sin(c + d*x)**2*tan((c + d*x)/2)**7*a**5 + 534528*cos(c + d*x)*sin(c 
 + d*x)**2*tan((c + d*x)/2)**7*a**2*b**3 + 786432*cos(c + d*x)*sin(c + d*x 
)**2*tan((c + d*x)/2)**7*a*b**4 - 15*cos(c + d*x)*tan((c + d*x)/2)**7*a**5 
 - 307200*cos(c + d*x)*tan((c + d*x)/2)**7*a**2*b**3 - 491520*cos(c + d*x) 
*tan((c + d*x)/2)**7*a*b**4 + 33600*int(tan((c + d*x)/2)**6/(tan((c + d*x) 
/2)**8*a + 4*tan((c + d*x)/2)**6*a + 6*tan((c + d*x)/2)**4*a - 16*tan((c + 
 d*x)/2)**4*b + 4*tan((c + d*x)/2)**2*a + a),x)*sin(c + d*x)**7*tan((c + d 
*x)/2)**7*a**3*b**3*d + 215040*int(tan((c + d*x)/2)**6/(tan((c + d*x)/2)** 
8*a + 4*tan((c + d*x)/2)**6*a + 6*tan((c + d*x)/2)**4*a - 16*tan((c + d*x) 
/2)**4*b + 4*tan((c + d*x)/2)**2*a + a),x)*sin(c + d*x)**7*tan((c + d*x)/2 
)**7*a**2*b**4*d + 210000*int(tan((c + d*x)/2)**4/(tan((c + d*x)/2)**8*a + 
 4*tan((c + d*x)/2)**6*a + 6*tan((c + d*x)/2)**4*a - 16*tan((c + d*x)/2)** 
4*b + 4*tan((c + d*x)/2)**2*a + a),x)*sin(c + d*x)**7*tan((c + d*x)/2)*...