Integrand size = 22, antiderivative size = 231 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=-\frac {\left (5 \sqrt {a}-4 \sqrt {b}\right ) \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^2 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\left (5 \sqrt {a}+4 \sqrt {b}\right ) \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^2 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \] Output:
-1/8*(5*a^(1/2)-4*b^(1/2))*b^(1/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1 /2))^(1/2))/a^2/(a^(1/2)-b^(1/2))^(3/2)/d-arctanh(cos(d*x+c))/a^2/d+1/8*(5 *a^(1/2)+4*b^(1/2))*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^( 1/2))/a^2/(a^(1/2)+b^(1/2))^(3/2)/d-1/4*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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 3.79 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.60 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {\frac {16 a b (-5 \cos (c+d x)+\cos (3 (c+d x)))}{(a-b) (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}-32 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {i b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-10 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+8 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+5 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-4 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+38 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-24 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-19 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+12 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-38 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+24 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+19 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-12 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+10 a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-8 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-5 i a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+4 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a-b}}{32 a^2 d} \] Input:
Integrate[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^2,x]
Output:
((16*a*b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*C os[2*(c + d*x)] - b*Cos[4*(c + d*x)])) - 32*Log[Cos[(c + d*x)/2]] + 32*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 & , (-10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 8*b* ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (5*I)*a*Log[1 - 2*Cos[c + d*x]* #1 + #1^2] - (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 38*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (19*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (12*I)*b* Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 38*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + ( 19*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (12*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^ 6 - 8*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (5*I)*a*Log[1 - 2* Cos[c + d*x]*#1 + #1^2]*#1^6 + (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*# 1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(32 *a^2*d)
Time = 0.54 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.35, 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.
| \(\displaystyle \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x) \left (a-b \sin (c+d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right ) \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle -\frac {\int \left (\frac {b-b \cos ^2(c+d x)}{a^2 \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}+\frac {b-b \cos ^2(c+d x)}{a \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )^2}-\frac {1}{a^2 \left (\cos ^2(c+d x)-1\right )}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cos (c+d x))}{a^2}+\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}}{d}\) |
Input:
Int[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^2,x]
Output:
-(((b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^( 3/2)*(Sqrt[a] - Sqrt[b])^(3/2)) + (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/S qrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[Cos[c + d*x]]/a^2 - (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b ]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)) - (b^(1/4)*ArcTanh[(b^(1/4)*Cos [c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + Sqrt[b]]) + (b* Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a*(a - b)*(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 = 2.76 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {-\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{2 a^{2}}+\frac {b \left (\frac {\frac {a \cos \left (d x +c \right )^{3}}{4 a -4 b}-\frac {a \cos \left (d x +c \right )}{2 \left (a -b \right )}}{a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b +a b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 a -4 b}\right )}{a^{2}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{2}}}{d}\) | \(243\) |
| default | \(\frac {-\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{2 a^{2}}+\frac {b \left (\frac {\frac {a \cos \left (d x +c \right )^{3}}{4 a -4 b}-\frac {a \cos \left (d x +c \right )}{2 \left (a -b \right )}}{a -b +2 b \cos \left (d x +c \right )^{2}-b \cos \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b +a b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (-5 a \sqrt {a b}+4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{4 a -4 b}\right )}{a^{2}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{2}}}{d}\) | \(243\) |
| risch | \(\frac {b \left ({\mathrm e}^{7 i \left (d x +c \right )}-5 \,{\mathrm e}^{5 i \left (d x +c \right )}-5 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 a \left (-a +b \right ) d \left ({\mathrm e}^{8 i \left (d x +c \right )} b -4 \,{\mathrm e}^{6 i \left (d x +c \right )} b -16 \,{\mathrm e}^{4 i \left (d x +c \right )} a +6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}+2 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1048576 a^{11} d^{4}-3145728 a^{10} b \,d^{4}+3145728 a^{9} b^{2} d^{4}-1048576 a^{8} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (-71680 a^{6} b \,d^{2}+96256 a^{5} b^{2} d^{2}-32768 a^{4} b^{3} d^{2}\right ) \textit {\_Z}^{2}-625 a^{2} b +800 b^{2} a -256 b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {327680 i a^{10} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {1179648 i a^{9} d^{3} b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {1572864 i a^{8} d^{3} b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {917504 i a^{7} d^{3} b^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {196608 i a^{6} b^{4} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {20800 i a^{5} d b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {39296 i a^{4} d \,b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {24640 i a^{3} d \,b^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {5120 i a^{2} b^{4} d}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\) | \(644\) |
Input:
int(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2/a^2*ln(cos(d*x+c)+1)+b/a^2*((1/4*a/(a-b)*cos(d*x+c)^3-1/2*a/(a-b )*cos(d*x+c))/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)+1/4/(a-b)*b*(-1/2*(-5* a*(a*b)^(1/2)+4*(a*b)^(1/2)*b+a*b)/(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*(-5*a*(a*b)^(1/2)+4*( a*b)^(1/2)*b-a*b)/(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^2*ln(cos(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 2711 vs. \(2 (182) = 364\).
Time = 0.62 (sec) , antiderivative size = 2711, normalized size of antiderivative = 11.74 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")
Output:
-1/16*(4*a*b*cos(d*x + c)^3 - 8*a*b*cos(d*x + c) + ((a^3*b - a^2*b^2)*d*co s(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2 *b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15 *a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 35*a^2 *b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(( 625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450 *a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11* b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5* b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b ^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*( a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^ 7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 12 41*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10 *b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 35*a^2*b + 47*a*b^2 - 16* b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log((625*a^3*b - 1125...
Timed out. \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)**4)**2,x)
Output:
Timed out
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")
Output:
1/2*(4*a*b^2*cos(2*d*x + 2*c)*cos(d*x + c) - 20*a*b^2*sin(3*d*x + 3*c)*sin (2*d*x + 2*c) + 4*a*b^2*sin(2*d*x + 2*c)*sin(d*x + c) - a*b^2*cos(d*x + c) - (a*b^2*cos(7*d*x + 7*c) - 5*a*b^2*cos(5*d*x + 5*c) - 5*a*b^2*cos(3*d*x + 3*c) + a*b^2*cos(d*x + c))*cos(8*d*x + 8*c) + (4*a*b^2*cos(6*d*x + 6*c) + 4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*cos(4*d*x + 4*c ))*cos(7*d*x + 7*c) - 4*(5*a*b^2*cos(5*d*x + 5*c) + 5*a*b^2*cos(3*d*x + 3* c) - a*b^2*cos(d*x + c))*cos(6*d*x + 6*c) - 5*(4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*cos(4*d*x + 4*c))*cos(5*d*x + 5*c) - 2*(5*(8 *a^2*b - 3*a*b^2)*cos(3*d*x + 3*c) - (8*a^2*b - 3*a*b^2)*cos(d*x + c))*cos (4*d*x + 4*c) - 5*(4*a*b^2*cos(2*d*x + 2*c) - a*b^2)*cos(3*d*x + 3*c) - 2* ((a^3*b^2 - a^2*b^3)*d*cos(8*d*x + 8*c)^2 + 16*(a^3*b^2 - a^2*b^3)*d*cos(6 *d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b + 57*a^3*b^2 - 9*a^2*b^3)*d*cos(4*d* x + 4*c)^2 + 16*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c)^2 + (a^3*b^2 - a^2* b^3)*d*sin(8*d*x + 8*c)^2 + 16*(a^3*b^2 - a^2*b^3)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b + 57*a^3*b^2 - 9*a^2*b^3)*d*sin(4*d*x + 4*c)^2 + 16* (8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 1 6*(a^3*b^2 - a^2*b^3)*d*sin(2*d*x + 2*c)^2 - 8*(a^3*b^2 - a^2*b^3)*d*cos(2 *d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d - 2*(4*(a^3*b^2 - a^2*b^3)*d*cos(6*d*x + 6*c) + 2*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*d*cos(4*d*x + 4*c) + 4*(a^3 *b^2 - a^2*b^3)*d*cos(2*d*x + 2*c) - (a^3*b^2 - a^2*b^3)*d)*cos(8*d*x +...
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm="giac")
Output:
sage0*x
Time = 41.09 (sec) , antiderivative size = 7491, normalized size of antiderivative = 32.43 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(sin(c + d*x)*(a - b*sin(c + d*x)^4)^2),x)
Output:
((b*cos(c + d*x)^3)/(4*a*(a - b)) - (b*cos(c + d*x))/(2*a*(a - b)))/(d*(a - b + 2*b*cos(c + d*x)^2 - b*cos(c + d*x)^4)) - (atan(((((3072*a^3*b^7 - 1 0944*a^4*b^6 + 9776*a^5*b^5)/(256*(a^7 - 2*a^6*b + a^5*b^2)) - (((49152*a^ 7*b^7 - 155648*a^8*b^6 + 172032*a^9*b^5 - 65536*a^10*b^4)/(256*(a^7 - 2*a^ 6*b + a^5*b^2)) - (cos(c + d*x)*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/ 2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^ 10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2)*(98304*a^8*b^7 - 262144*a^9*b^6 + 229376*a^10*b^5 - 65536*a^11*b^4))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((2 5*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5 *b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)) )^(1/2) + (cos(c + d*x)*(18432*a^4*b^7 - 45440*a^5*b^6 + 29312*a^6*b^5))/( 128*(a^6 - 2*a^5*b + a^4*b^2)))*((25*a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/ 2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^ 10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2))*((25*a^2*(a^9*b)^(1/2) + 8*b^2 *(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b^2 - 29*a*b*(a^9*b)^(1/2) )/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^(1/2) + (cos(c + d*x)*(76 8*b^7 - 2048*a*b^6 + 1425*a^2*b^5))/(128*(a^6 - 2*a^5*b + a^4*b^2)))*((25* a^2*(a^9*b)^(1/2) + 8*b^2*(a^9*b)^(1/2) + 35*a^6*b + 16*a^4*b^3 - 47*a^5*b ^2 - 29*a*b*(a^9*b)^(1/2))/(256*(3*a^10*b - a^11 + a^8*b^3 - 3*a^9*b^2)))^ (1/2)*1i - (((3072*a^3*b^7 - 10944*a^4*b^6 + 9776*a^5*b^5)/(256*(a^7 - ...
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x)
Output:
(28*cos(c + d*x)*sin(c + d*x)**2*a**4*b + 128*cos(c + d*x)*sin(c + d*x)**2 *a**3*b**2 + 18176*cos(c + d*x)*sin(c + d*x)**2*a**2*b**3 - 8192*cos(c + d *x)*sin(c + d*x)**2*a*b**4 + 56*cos(c + d*x)*a**4*b - 6912*cos(c + d*x)*a* *3*b**2 - 14848*cos(c + d*x)*a**2*b**3 + 8192*cos(c + d*x)*a*b**4 - 451584 *int(tan((c + d*x)/2)**3/(21*tan((c + d*x)/2)**16*a**4 - 408*tan((c + d*x) /2)**16*a**3*b - 320*tan((c + d*x)/2)**16*a**2*b**2 + 168*tan((c + d*x)/2) **14*a**4 - 3264*tan((c + d*x)/2)**14*a**3*b - 2560*tan((c + d*x)/2)**14*a **2*b**2 + 588*tan((c + d*x)/2)**12*a**4 - 12096*tan((c + d*x)/2)**12*a**3 *b + 4096*tan((c + d*x)/2)**12*a**2*b**2 + 10240*tan((c + d*x)/2)**12*a*b* *3 + 1176*tan((c + d*x)/2)**10*a**4 - 25536*tan((c + d*x)/2)**10*a**3*b + 34304*tan((c + d*x)/2)**10*a**2*b**2 + 40960*tan((c + d*x)/2)**10*a*b**3 + 1470*tan((c + d*x)/2)**8*a**4 - 32592*tan((c + d*x)/2)**8*a**3*b + 61312* tan((c + d*x)/2)**8*a**2*b**2 - 43008*tan((c + d*x)/2)**8*a*b**3 - 81920*t an((c + d*x)/2)**8*b**4 + 1176*tan((c + d*x)/2)**6*a**4 - 25536*tan((c + d *x)/2)**6*a**3*b + 34304*tan((c + d*x)/2)**6*a**2*b**2 + 40960*tan((c + d* x)/2)**6*a*b**3 + 588*tan((c + d*x)/2)**4*a**4 - 12096*tan((c + d*x)/2)**4 *a**3*b + 4096*tan((c + d*x)/2)**4*a**2*b**2 + 10240*tan((c + d*x)/2)**4*a *b**3 + 168*tan((c + d*x)/2)**2*a**4 - 3264*tan((c + d*x)/2)**2*a**3*b - 2 560*tan((c + d*x)/2)**2*a**2*b**2 + 21*a**4 - 408*a**3*b - 320*a**2*b**2), x)*sin(c + d*x)**4*a**6*b**3*d + 15998976*int(tan((c + d*x)/2)**3/(21*t...