Integrand size = 24, antiderivative size = 149 \[ \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d} \]
-cot(d*x+c)/a/d-1/3*cot(d*x+c)^3/a/d+1/2*b*arctan((a^(1/2)-b^(1/2))^(1/2)* tan(d*x+c)/a^(1/4))/a^(7/4)/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*b*arctan((a^(1/2 )+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(7/4)/d/(a^(1/2)+b^(1/2))^(1/2)
Time = 1.64 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11 \[ \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {3 b \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {3 b \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}-4 \sqrt {a} \cot (c+d x)-2 \sqrt {a} \cot (c+d x) \csc ^2(c+d x)}{6 a^{3/2} d} \]
((3*b*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]] )/Sqrt[a + Sqrt[a]*Sqrt[b]] - (3*b*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d* x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] - 4*Sqrt[a]*Co t[c + d*x] - 2*Sqrt[a]*Cot[c + d*x]*Csc[c + d*x]^2)/(6*a^(3/2)*d)
Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^4 \left (a-b \sin (c+d x)^4\right )}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\cot ^4(c+d x) \left (\tan ^2(c+d x)+1\right )^3}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1610 |
| \(\displaystyle \frac {\int \left (\frac {\cot ^4(c+d x)}{a}+\frac {\cot ^2(c+d x)}{a}+\frac {b \left (\tan ^2(c+d x)+1\right )}{a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cot ^3(c+d x)}{3 a}-\frac {\cot (c+d x)}{a}}{d}\) |
((b*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqr t[Sqrt[a] - Sqrt[b]]) + (b*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a ^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] + Sqrt[b]]) - Cot[c + d*x]/a - Cot[c + d* x]^3/(3*a))/d
3.3.9.3.1 Defintions of rubi rules used
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]
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]
Time = 1.44 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a}-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {1}{a \tan \left (d x +c \right )}}{d}\) | \(177\) |
| default | \(\frac {\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a}-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {1}{a \tan \left (d x +c \right )}}{d}\) | \(177\) |
| risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16777216 a^{8} d^{4}-16777216 a^{7} b \,d^{4}\right ) \textit {\_Z}^{4}+8192 a^{4} b^{2} d^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {524288 i d^{3} a^{7}}{b^{4}}-\frac {524288 i a^{6} d^{3}}{b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {8192 d^{2} a^{5}}{b^{3}}+\frac {8192 d^{2} a^{4}}{b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {128 i d \,a^{3}}{b^{2}}+\frac {128 i a^{2} d}{b}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\) | \(182\) |
1/d*(1/a*b*(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)))-1/3/a/tan(d*x+c)^3-1/a/tan(d* x+c))
Leaf count of result is larger than twice the leaf count of optimal. 1365 vs. \(2 (111) = 222\).
Time = 0.41 (sec) , antiderivative size = 1365, normalized size of antiderivative = 9.16 \[ \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
-1/24*(3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^ 9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(1/4*b^4*cos(d *x + c)^2 - 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + 1/2*(a^2*b^3* d*cos(d*x + c)*sin(d*x + c) - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b ^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) - 3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(1/4*b^4*cos(d* x + c)^2 - 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - 1/2*(a^2*b^3*d *cos(d*x + c)*sin(d*x + c) - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^ 5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) - 3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(-1/4*b^4*cos(d*x + c)^2 + 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + 1/2*(a^2*b^3*d* cos(d*x + c)*sin(d*x + c) + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a ^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b...
\[ \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc ^{4}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{4}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
-4/3*(12*(a*b*d*cos(6*d*x + 6*c)^2 + 9*a*b*d*cos(4*d*x + 4*c)^2 + 9*a*b*d* cos(2*d*x + 2*c)^2 + a*b*d*sin(6*d*x + 6*c)^2 + 9*a*b*d*sin(4*d*x + 4*c)^2 - 18*a*b*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*b*d*sin(2*d*x + 2*c)^2 - 6*a*b*d*cos(2*d*x + 2*c) + a*b*d - 2*(3*a*b*d*cos(4*d*x + 4*c) - 3*a*b* d*cos(2*d*x + 2*c) + a*b*d)*cos(6*d*x + 6*c) - 6*(3*a*b*d*cos(2*d*x + 2*c) - a*b*d)*cos(4*d*x + 4*c) - 6*(a*b*d*sin(4*d*x + 4*c) - a*b*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate((b*cos(8*d*x + 8*c)*cos(4*d*x + 4*c) - 4*b*cos(6*d*x + 6*c)*cos(4*d*x + 4*c) - 2*(8*a - 3*b)*cos(4*d*x + 4*c)^2 + b*sin(8*d*x + 8*c)*sin(4*d*x + 4*c) - 4*b*sin(6*d*x + 6*c)*sin(4*d*x + 4* c) - 2*(8*a - 3*b)*sin(4*d*x + 4*c)^2 - 4*b*sin(4*d*x + 4*c)*sin(2*d*x + 2 *c) - (4*b*cos(2*d*x + 2*c) - b)*cos(4*d*x + 4*c))/(a*b^2*cos(8*d*x + 8*c) ^2 + 16*a*b^2*cos(6*d*x + 6*c)^2 + 16*a*b^2*cos(2*d*x + 2*c)^2 + a*b^2*sin (8*d*x + 8*c)^2 + 16*a*b^2*sin(6*d*x + 6*c)^2 + 16*a*b^2*sin(2*d*x + 2*c)^ 2 - 8*a*b^2*cos(2*d*x + 2*c) + a*b^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*cos (4*d*x + 4*c)^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*sin(4*d*x + 4*c)^2 + 16* (8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(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(8*d*x + 8*c) + 8*(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(6*d*x + 6*c) - 4*(8*a^2*b - 3*a*b^ 2 - 4*(8*a^2*b - 3*a*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*a*b...
Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (111) = 222\).
Time = 0.70 (sec) , antiderivative size = 937, normalized size of antiderivative = 6.29 \[ \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {3 \, {\left ({\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} a^{2} {\left | a - b \right |} + {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{4}\right )} {\left | a - b \right |} {\left | a \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3}\right )} {\left | a - b \right |}\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^{2} + \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} + \frac {3 \, {\left ({\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} a^{2} {\left | a - b \right |} + {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{4}\right )} {\left | a - b \right |} {\left | a \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3}\right )} {\left | a - b \right |}\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^{2} - \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} - \frac {2 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )}}{a \tan \left (d x + c\right )^{3}}}{6 \, d} \]
1/6*(3*((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^ 2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)* (a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a* b^4)*abs(a - b)*abs(a) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)* a^4*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 + sqrt(a^4 - (a^2 - a*b)*a^ 2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^ 4 + a^3*b^5)*abs(a)) + 3*((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b) *a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) + (3*sqrt(a^2 - a *b - sqrt(a*b)*(a - b))*a^4*b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3* b^2 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 + sqrt(a^2 - a*b - sqr t(a*b)*(a - b))*a*b^4)*abs(a - b)*abs(a) - (3*sqrt(a^2 - a*b - sqrt(a*b)*( a - b))*sqrt(a*b)*a^4*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)* a^3*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3)*abs(a - b ))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - sqrt(a^ 4 - (a^2 - a*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - ...
Time = 15.91 (sec) , antiderivative size = 1670, normalized size of antiderivative = 11.21 \[ \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
(atan((((((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 + tan(c + d*x)*(((a^7*b^5)^(1/2) + a^4*b^2)/(16 *(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2)) - tan(c + d* x)*(4*a^3*b^5 - 4*a^5*b^3))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8) ))^(1/2)*1i - ((((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(16* a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 - tan(c + d*x)*(((a^7*b^5)^(1/2) + a^4*b ^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2)) + tan (c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*1i)/(((((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/ 2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 + tan(c + d*x)*(((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2) ) - tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(((a^7*b^5)^(1/2) + a^4*b^2)/(16 *(a^7*b - a^8)))^(1/2) + ((((a^7*b^5)^(1/2) + a^4*b^2)/(16*(a^7*b - a^8))) ^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 - tan(c + d*x)*(((a^7*b^5)^(1 /2) + a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*a^9*b + 64*a^7*b^3 - 128*a^8* b^2)) + tan(c + d*x)*(4*a^3*b^5 - 4*a^5*b^3))*(((a^7*b^5)^(1/2) + a^4*b^2) /(16*(a^7*b - a^8)))^(1/2) - 2*a^2*b^5 + 2*a^3*b^4))*(((a^7*b^5)^(1/2) + a ^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*2i)/d + (atan((((-((a^7*b^5)^(1/2) - a^4 *b^2)/(16*(a^7*b - a^8)))^(1/2)*(16*a^5*b^4 - 32*a^6*b^3 + 16*a^7*b^2 + ta n(c + d*x)*(-((a^7*b^5)^(1/2) - a^4*b^2)/(16*(a^7*b - a^8)))^(1/2)*(64*...