Integrand size = 22, antiderivative size = 617 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac {b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )} \]
-arctanh(cos(d*x+c))/a^3/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)^2-1/4*b*cos(d*x+c)*(2-cos(d*x+c)^2)/a^2 /(a-b)/d/(a-b+2*b*cos(d*x+c)^2-b*cos(d*x+c)^4)-1/32*b*cos(d*x+c)*(11*a+b-( 5*a+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)-1 /64*b^(1/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*(5*a^(1/2)- 2*b^(1/2))/a^(5/2)/d/(a^(1/2)-b^(1/2))^(5/2)-1/8*b^(1/4)*arctan(b^(1/4)*co s(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(5/2)/d/(a^(1/2)-b^(1/2))^(3/2)+1/8*b^ (1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(5/2)/d/(a^(1/ 2)+b^(1/2))^(3/2)+1/64*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2) )^(1/2))*(5*a^(1/2)+2*b^(1/2))/a^(5/2)/d/(a^(1/2)+b^(1/2))^(5/2)-1/2*b^(1/ 4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^3/d/(a^(1/2)-b^(1/ 2))^(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/ a^3/d/(a^(1/2)+b^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 10.95 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.49 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\frac {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)))}+\frac {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 \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+256 \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 {-90 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+142 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-64 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+45 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-71 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+32 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+398 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-506 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+192 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-199 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+253 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-96 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-398 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+506 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-192 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+199 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-253 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+96 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+90 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-142 a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6+64 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-45 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6+71 i a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6-32 i b^2 \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)^2}}{256 a^3 d} \]
((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 = 0.94 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.96, 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 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \left (5 \sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\sqrt [4]{b} \left (5 \sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cos (c+d x))}{a^3}+\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}+\frac {b \cos (c+d x) \left (-\left ((5 a+b) \cos ^2(c+d x)\right )+11 a+b\right )}{32 a^2 (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}+\frac {b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2}}{d}\) |
-((((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)
3.3.29.3.1 Defintions of rubi rules used
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 = 7.50 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {-\frac {a b \left (13 a -7 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (53 a -29 b \right ) a b \left (\cos ^{5}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}+\frac {a \left (17 a^{2}-78 a b +37 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}-\frac {5 \left (7 a -3 b \right ) a \cos \left (d x +c \right )}{32 \left (a -b \right )}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}+\frac {b \left (\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}+16 a^{2} b -10 a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{32 a^{2}-64 a b +32 b^{2}}\right )}{a^{3}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{3}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\) | \(384\) |
| default | \(\frac {\frac {b \left (\frac {-\frac {a b \left (13 a -7 b \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (53 a -29 b \right ) a b \left (\cos ^{5}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}+\frac {a \left (17 a^{2}-78 a b +37 b^{2}\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{32 a^{2}-64 a b +32 b^{2}}-\frac {5 \left (7 a -3 b \right ) a \cos \left (d x +c \right )}{32 \left (a -b \right )}}{{\left (a -b +2 b \left (\cos ^{2}\left (d x +c \right )\right )-b \left (\cos ^{4}\left (d x +c \right )\right )\right )}^{2}}+\frac {b \left (\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}-16 a^{2} b +10 a \,b^{2}\right ) \arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\left (-45 a^{2} \sqrt {a b}+71 a b \sqrt {a b}-32 b^{2} \sqrt {a b}+16 a^{2} b -10 a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{32 a^{2}-64 a b +32 b^{2}}\right )}{a^{3}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{3}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\) | \(384\) |
| risch | \(\text {Expression too large to display}\) | \(1530\) |
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)*a*b/(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*a*b^2)/(a*b)^(1/2)/b/((( a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((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*a*b ^2)/(a*b)^(1/2)/b/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^( 1/2)+b)*b)^(1/2))))-1/2/a^3*ln(1+cos(d*x+c))+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. \(2 (481) = 962\).
Time = 2.20 (sec) , antiderivative size = 5020, normalized size of antiderivative = 8.14 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
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 -...
\[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\csc \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
Time = 20.97 (sec) , antiderivative size = 12247, normalized size of antiderivative = 19.85 \[ \int \frac {\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
- ((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*...