Integrand size = 22, antiderivative size = 319 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\sqrt [4]{b} \left (45 a-74 \sqrt {a} \sqrt {b}+32 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^3 \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \left (45 a+74 \sqrt {a} \sqrt {b}+32 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^3 \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (3 (9 a-5 b)-(13 a-7 b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \] Output:
-1/64*b^(1/4)*(45*a-74*a^(1/2)*b^(1/2)+32*b)*arctan(b^(1/4)*cos(d*x+c)/(a^ (1/2)-b^(1/2))^(1/2))/a^3/(a^(1/2)-b^(1/2))^(5/2)/d-arctanh(cos(d*x+c))/a^ 3/d+1/64*b^(1/4)*(45*a+74*a^(1/2)*b^(1/2)+32*b)*arctanh(b^(1/4)*cos(d*x+c) /(a^(1/2)+b^(1/2))^(1/2))/a^3/(a^(1/2)+b^(1/2))^(5/2)/d-1/8*b*cos(d*x+c)*( 2-cos(d*x+c)^2)/a/(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)^2-1/32*b*c os(d*x+c)*(27*a-15*b-(13*a-7*b)*cos(d*x+c)^2)/a^2/(a-b)^2/d/(a-b+2*b*cos(d *x+c)^2-b*cos(d*x+c)^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 11.40 (sec) , antiderivative size = 920, normalized size of antiderivative = 2.88 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]
Output:
((32*a*b*Cos[c + d*x]*(-41*a + 23*b + (13*a - 7*b)*Cos[2*(c + d*x)]))/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (512*a^2 *b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(-8*a + 3*b - 4*b*Cos[2* (c + d*x)] + b*Cos[4*(c + d*x)])^2) - 256*Log[Cos[(c + d*x)/2]] + 256*Log[ Sin[(c + d*x)/2]] - (I*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b *#1^6 + b*#1^8 & , (-90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 142 *a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 64*b^2*ArcTan[Sin[c + d*x] /(Cos[c + d*x] - #1)] + (45*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (71 *I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (32*I)*b^2*Log[1 - 2*Cos[c + d *x]*#1 + #1^2] + 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 5 06*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 192*b^2*ArcTan[Sin[ c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (199*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (96*I )*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 398*a^2*ArcTan[Sin[c + d*x] /(Cos[c + d*x] - #1)]*#1^4 + 506*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - # 1)]*#1^4 - 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (199*I) *a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (96*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 142*a*b*ArcTan[S in[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 64*b^2*ArcTan[Sin[c + d*x]/(Cos...
Time = 0.98 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 3694, 1567, 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.
| \(\displaystyle \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x) \left (a-b \sin (c+d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right ) \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^3}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle -\frac {\int \left (\frac {b-b \cos ^2(c+d x)}{a^3 \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}+\frac {b-b \cos ^2(c+d x)}{a^2 \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}+\frac {b-b \cos ^2(c+d x)}{a \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^3}-\frac {1}{a^3 \left (\cos ^2(c+d x)-1\right )}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \left (5 \sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\sqrt [4]{b} \left (5 \sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cos (c+d x))}{a^3}+\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}+\frac {b \cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{32 a^2 (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}+\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
Input:
Int[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]
Output:
-((((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqr t[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)) + (b^(1/4)*ArcTan [(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] - Sq rt[b])^(3/2)) + (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt [b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[Cos[c + d*x]]/a^3 - (b^(1 /4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(5/2)*(S qrt[a] + Sqrt[b])^(3/2)) - (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sq rt[a] + Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]) - ((5*Sqrt[a] + 2*Sqrt[ b])*b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a ^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)) + (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/ (8*a*(a - b)*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a^2*(a - b)*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)) + (b*Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2 ))/(32*a^2*(a - b)^2*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)))/d)
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 7.80 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {-\frac {a b \left (13 a -7 b \right ) \cos \left (d x +c \right )^{7}}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (53 a -29 b \right ) b a \cos \left (d x +c \right )^{5}}{32 a^{2}-64 a b +32 b^{2}}+\frac {a \left (17 a^{2}-78 a b +37 b^{2}\right ) \cos \left (d x +c \right )^{3}}{32 a^{2}-64 a b +32 b^{2}}-\frac {5 \left (7 a -3 b \right ) a \cos \left (d x +c \right )}{32 \left (a -b \right )}}{\left (a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}\right )^{2}}+\frac {b \left (-\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}+16 a^{2} b -10 b^{2} a \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}-16 a^{2} b +10 b^{2} a \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{32 a^{2}-64 a b +32 b^{2}}\right )}{a^{3}}-\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{2 a^{3}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\) | \(384\) |
| default | \(\frac {\frac {b \left (\frac {-\frac {a b \left (13 a -7 b \right ) \cos \left (d x +c \right )^{7}}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (53 a -29 b \right ) b a \cos \left (d x +c \right )^{5}}{32 a^{2}-64 a b +32 b^{2}}+\frac {a \left (17 a^{2}-78 a b +37 b^{2}\right ) \cos \left (d x +c \right )^{3}}{32 a^{2}-64 a b +32 b^{2}}-\frac {5 \left (7 a -3 b \right ) a \cos \left (d x +c \right )}{32 \left (a -b \right )}}{\left (a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}\right )^{2}}+\frac {b \left (-\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}+16 a^{2} b -10 b^{2} a \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}-16 a^{2} b +10 b^{2} a \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{32 a^{2}-64 a b +32 b^{2}}\right )}{a^{3}}-\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{2 a^{3}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\) | \(384\) |
| risch | \(\text {Expression too large to display}\) | \(1530\) |
Input:
int(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(b/a^3*((-1/32*a*b*(13*a-7*b)/(a^2-2*a*b+b^2)*cos(d*x+c)^7+1/32*(53*a- 29*b)*b*a/(a^2-2*a*b+b^2)*cos(d*x+c)^5+1/32*a*(17*a^2-78*a*b+37*b^2)/(a^2- 2*a*b+b^2)*cos(d*x+c)^3-5/32*(7*a-3*b)*a/(a-b)*cos(d*x+c))/(a-b+2*b*cos(d* x+c)^2-b*cos(d*x+c)^4)^2+1/32/(a^2-2*a*b+b^2)*b*(-1/2*(-45*a^2*(a*b)^(1/2) +71*a*b*(a*b)^(1/2)-32*b^2*(a*b)^(1/2)+16*a^2*b-10*b^2*a)/(a*b)^(1/2)/b/(( (a*b)^(1/2)+b)*b)^(1/2)*arctanh(b*cos(d*x+c)/(((a*b)^(1/2)+b)*b)^(1/2))+1/ 2*(-45*a^2*(a*b)^(1/2)+71*a*b*(a*b)^(1/2)-32*b^2*(a*b)^(1/2)-16*a^2*b+10*b ^2*a)/(a*b)^(1/2)/b/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(b*cos(d*x+c)/(((a*b)^ (1/2)-b)*b)^(1/2))))-1/2/a^3*ln(cos(d*x+c)+1)+1/2/a^3*ln(cos(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 5020 vs. \(2 (263) = 526\).
Time = 1.84 (sec) , antiderivative size = 5020, normalized size of antiderivative = 15.74 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")
Output:
1/16*(8*(13*a^2*b^4 - 7*a*b^5)*cos(2*d*x + 2*c)*cos(d*x + c) - 8*(121*a^2* b^4 - 67*a*b^5)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 8*(13*a^2*b^4 - 7*a*b^ 5)*sin(2*d*x + 2*c)*sin(d*x + c) - ((13*a^2*b^4 - 7*a*b^5)*cos(15*d*x + 15 *c) - (121*a^2*b^4 - 67*a*b^5)*cos(13*d*x + 13*c) - (272*a^3*b^3 - 461*a^2 *b^4 + 159*a*b^5)*cos(11*d*x + 11*c) + (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a *b^5)*cos(9*d*x + 9*c) + (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(7*d* x + 7*c) - (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*cos(5*d*x + 5*c) - (121 *a^2*b^4 - 67*a*b^5)*cos(3*d*x + 3*c) + (13*a^2*b^4 - 7*a*b^5)*cos(d*x + c ))*cos(16*d*x + 16*c) - (13*a^2*b^4 - 7*a*b^5 - 8*(13*a^2*b^4 - 7*a*b^5)*c os(14*d*x + 14*c) - 4*(104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*cos(12*d*x + 12*c) + 8*(208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)*cos(10*d*x + 10*c) + 2*(1 664*a^4*b^2 - 2144*a^3*b^3 + 1127*a^2*b^4 - 245*a*b^5)*cos(8*d*x + 8*c) + 8*(208*a^3*b^3 - 203*a^2*b^4 + 49*a*b^5)*cos(6*d*x + 6*c) - 4*(104*a^3*b^3 - 147*a^2*b^4 + 49*a*b^5)*cos(4*d*x + 4*c) - 8*(13*a^2*b^4 - 7*a*b^5)*cos (2*d*x + 2*c))*cos(15*d*x + 15*c) - 8*((121*a^2*b^4 - 67*a*b^5)*cos(13*d*x + 13*c) + (272*a^3*b^3 - 461*a^2*b^4 + 159*a*b^5)*cos(11*d*x + 11*c) - (1 424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(9*d*x + 9*c) - (1424*a^3*b^3 - 1121*a^2*b^4 + 99*a*b^5)*cos(7*d*x + 7*c) + (272*a^3*b^3 - 461*a^2*b^4 + 1 59*a*b^5)*cos(5*d*x + 5*c) + (121*a^2*b^4 - 67*a*b^5)*cos(3*d*x + 3*c) - ( 13*a^2*b^4 - 7*a*b^5)*cos(d*x + c))*cos(14*d*x + 14*c) + (121*a^2*b^4 -...
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")
Output:
sage0*x
Time = 42.53 (sec) , antiderivative size = 12247, normalized size of antiderivative = 38.39 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/(sin(c + d*x)*(a - b*sin(c + d*x)^4)^3),x)
Output:
- ((5*cos(c + d*x)*(7*a*b - 3*b^2))/(32*a^2*(a - b)) - (cos(c + d*x)^3*(17 *a^2*b - 78*a*b^2 + 37*b^3))/(32*a^2*(a - b)^2) - (cos(c + d*x)^5*(53*a*b^ 2 - 29*b^3))/(32*a^2*(a - b)^2) + (b*cos(c + d*x)^7*(13*a*b - 7*b^2))/(32* a^2*(a - b)^2))/(d*(a^2 - 2*a*b + b^2 + cos(c + d*x)^2*(4*a*b - 4*b^2) - c os(c + d*x)^4*(2*a*b - 6*b^2) - 4*b^2*cos(c + d*x)^6 + b^2*cos(c + d*x)^8) ) - (atan((((((((192*a^11*b^9 - 990*a^12*b^8 + 2050*a^13*b^7 - 2154*a^14*b ^6 + 1158*a^15*b^5 - 256*a^16*b^4)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11 *b^3 + 6*a^12*b^2)) - (cos(c + d*x)*(402653184*a^12*b^9 - 1879048192*a^13* b^8 + 3489660928*a^14*b^7 - 3221225472*a^15*b^6 + 1476395008*a^16*b^5 - 26 8435456*a^17*b^4))/(2097152*a^3*(a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6 *a^10*b^2)))/(2*a^3) + (cos(c + d*x)*(75497472*a^6*b^9 - 337215488*a^7*b^8 + 592748544*a^8*b^7 - 489406464*a^9*b^6 + 163684352*a^10*b^5))/(1048576*( a^12 - 4*a^11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/(2*a^3) - (12*a^5*b^ 9 - (4311*a^6*b^8)/64 + (307961*a^7*b^7)/2048 - (1290253*a^8*b^6)/8192 + ( 546059*a^9*b^5)/8192)/(2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12 *b^2)))*1i)/(2*a^3) - (cos(c + d*x)*(3145728*b^9 - 14417920*a*b^8 + 264532 64*a^2*b^7 - 23076232*a^3*b^6 + 8247825*a^4*b^5)*1i)/(1048576*(a^12 - 4*a^ 11*b + a^8*b^4 - 4*a^9*b^3 + 6*a^10*b^2)))/a^3 - ((((((192*a^11*b^9 - 990* a^12*b^8 + 2050*a^13*b^7 - 2154*a^14*b^6 + 1158*a^15*b^5 - 256*a^16*b^4)/( 2*(a^14 - 4*a^13*b + a^10*b^4 - 4*a^11*b^3 + 6*a^12*b^2)) + (cos(c + d*...
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x)
Output:
(158400*cos(c + d*x)*sin(c + d*x)**6*a**8*b**2 + 10268544*cos(c + d*x)*sin (c + d*x)**6*a**7*b**3 - 10014720*cos(c + d*x)*sin(c + d*x)**6*a**6*b**4 - 4975329280*cos(c + d*x)*sin(c + d*x)**6*a**5*b**5 + 19249758208*cos(c + d *x)*sin(c + d*x)**6*a**4*b**6 - 15023996928*cos(c + d*x)*sin(c + d*x)**6*a **3*b**7 + 4831838208*cos(c + d*x)*sin(c + d*x)**6*a**2*b**8 + 411840*cos( c + d*x)*sin(c + d*x)**4*a**8*b**2 - 122098944*cos(c + d*x)*sin(c + d*x)** 4*a**7*b**3 + 1288740864*cos(c + d*x)*sin(c + d*x)**4*a**6*b**4 + 13942325 248*cos(c + d*x)*sin(c + d*x)**4*a**5*b**5 - 77298925568*cos(c + d*x)*sin( c + d*x)**4*a**4*b**6 + 63048777728*cos(c + d*x)*sin(c + d*x)**4*a**3*b**7 - 20937965568*cos(c + d*x)*sin(c + d*x)**4*a**2*b**8 - 138600*cos(c + d*x )*sin(c + d*x)**2*a**9*b - 464640*cos(c + d*x)*sin(c + d*x)**2*a**8*b**2 + 456692736*cos(c + d*x)*sin(c + d*x)**2*a**7*b**3 - 206749696*cos(c + d*x) *sin(c + d*x)**2*a**6*b**4 - 23488888832*cos(c + d*x)*sin(c + d*x)**2*a**5 *b**5 + 129671102464*cos(c + d*x)*sin(c + d*x)**2*a**4*b**6 - 106032005120 *cos(c + d*x)*sin(c + d*x)**2*a**3*b**7 + 35433480192*cos(c + d*x)*sin(c + d*x)**2*a**2*b**8 - 277200*cos(c + d*x)*a**9*b - 929280*cos(c + d*x)*a**8 *b**2 - 1337972736*cos(c + d*x)*a**7*b**3 + 2445574144*cos(c + d*x)*a**6*b **4 + 6826754048*cos(c + d*x)*a**5*b**5 - 66270003200*cos(c + d*x)*a**4*b* *6 + 56371445760*cos(c + d*x)*a**3*b**7 - 19327352832*cos(c + d*x)*a**2*b* *8 + 11633990860800*int(tan((c + d*x)/2)**3/(5775*tan((c + d*x)/2)**24*...