Integrand size = 24, antiderivative size = 127 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {x}{b}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b d}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b d} \]
-x/b+1/2*a^(1/4)*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b/d/(a ^(1/2)-b^(1/2))^(1/2)+1/2*a^(1/4)*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c )/a^(1/4))/b/d/(a^(1/2)+b^(1/2))^(1/2)
Time = 2.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-2 (c+d x)+\frac {\sqrt {a} \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}}{2 b d} \]
(-2*(c + d*x) + (Sqrt[a]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (Sqrt[a]*ArcTanh[((Sqrt[a ] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]* Sqrt[b]])/(2*b*d)
Time = 0.36 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4}{a-b \sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1610 |
| \(\displaystyle \frac {\int \left (\frac {a \left (\tan ^2(c+d x)+1\right )}{b \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {1}{b \left (\tan ^2(c+d x)+1\right )}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\arctan (\tan (c+d x))}{b}}{d}\) |
(-(ArcTan[Tan[c + d*x]]/b) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[ c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b) + (a^(1/4)*ArcTan[(Sqrt[ Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b))/ d
3.3.5.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.72 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01
| method | result | size |
| risch | \(-\frac {x}{b}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,b^{4} d^{4}-b^{5} d^{4}\right ) \textit {\_Z}^{4}+32 a \,b^{2} d^{2} \textit {\_Z}^{2}+256 a \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {1}{32} i a \,b^{2} d^{3}-\frac {1}{32} i b^{3} d^{3}\right ) \textit {\_R}^{3}+\left (-\frac {1}{8} b \,d^{2} a +\frac {1}{8} b^{2} d^{2}\right ) \textit {\_R}^{2}+\left (\frac {1}{2} i a d +\frac {1}{2} i d b \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )}{16}\) | \(128\) |
| derivativedivides | \(\frac {-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\) | \(163\) |
| default | \(\frac {-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\) | \(163\) |
-x/b+1/16*sum(_R*ln(exp(2*I*(d*x+c))+(1/32*I*a*b^2*d^3-1/32*I*b^3*d^3)*_R^ 3+(-1/8*b*d^2*a+1/8*b^2*d^2)*_R^2+(1/2*I*a*d+1/2*I*d*b)*_R-2/b*a-1),_R=Roo tOf((a*b^4*d^4-b^5*d^4)*_Z^4+32*a*b^2*d^2*_Z^2+256*a))
Leaf count of result is larger than twice the leaf count of optimal. 1125 vs. \(2 (91) = 182\).
Time = 0.40 (sec) , antiderivative size = 1125, normalized size of antiderivative = 8.86 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
1/8*(b*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(1/4*cos(d*x + c)^2 + 1/2*((a*b^2 - b^3)*d^3*sq rt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) - b*d*cos( d*x + c)*sin(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1/4) - b*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/(( a*b^2 - b^3)*d^2))*log(1/4*cos(d*x + c)^2 - 1/2*((a*b^2 - b^3)*d^3*sqrt(a/ ((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) - b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^ 5)*d^4)) + a)/((a*b^2 - b^3)*d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1/4) + b*sqr t(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-1/4*cos(d*x + c)^2 + 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2 *b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) + b*d*cos(d*x + c)*s in(d*x + c))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4 )) - a)/((a*b^2 - b^3)*d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a* b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1/4) - b*sqrt(((a* b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3) *d^2))*log(-1/4*cos(d*x + c)^2 - 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^...
Timed out. \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{4}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
-(16*a*b*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))/(b^3*cos(8*d*x + 8*c)^2 + 16*b^3*cos(6* d*x + 6*c)^2 + 16*b^3*cos(2*d*x + 2*c)^2 + b^3*sin(8*d*x + 8*c)^2 + 16*b^3 *sin(6*d*x + 6*c)^2 + 16*b^3*sin(2*d*x + 2*c)^2 - 8*b^3*cos(2*d*x + 2*c) + b^3 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*cos(4*d*x + 4*c)^2 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*sin(4*d*x + 4*c)^2 + 16*(8*a*b^2 - 3*b^3)*sin(4*d*x + 4* c)*sin(2*d*x + 2*c) - 2*(4*b^3*cos(6*d*x + 6*c) + 4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(4*b^3*c os(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a*b^2 - 3*b^3 - 4*(8*a*b^2 - 3*b^3)*cos(2*d*x + 2*c))*cos(4*d* x + 4*c) - 4*(2*b^3*sin(6*d*x + 6*c) + 2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + x)/b
Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (91) = 182\).
Time = 0.72 (sec) , antiderivative size = 912, normalized size of antiderivative = 7.18 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {2 \, {\left (d x + c\right )}}{b} + \frac {{\left ({\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} b^{2} {\left | -a + b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} + 5 \, \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 | -a + b \right |} {\left | b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\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 b + \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} {\left | b \right |}} + \frac {{\left ({\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} b^{2} {\left | -a + b \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} + 5 \, \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 | -a + b \right |} {\left | b \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\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 b - \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} {\left | b \right |}}}{2 \, d} \]
-1/2*(2*(d*x + c)/b + ((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^ 2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b + s qrt(a*b)*(a - b))*a^3*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 + 5*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)*abs(-a + b)*abs(b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*s qrt(a*b)*a^2*b^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor ((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b + sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a^3*b^4 - 18*a^ 2*b^5 + 3*a*b^6 + b^7)*abs(b)) + ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*s qrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt( a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^ 2 - a*b - sqrt(a*b)*(a - b))*a^3*b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b)) *a^2*b^2 + 5*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)*abs(-a + b)*abs(b) - (3*sqrt(a^2 - a*b - sqrt(a*b) *(a - b))*sqrt(a*b)*a^2*b^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a *b)*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b) )*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b - sqrt(a^2 *b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a...
Time = 16.32 (sec) , antiderivative size = 2991, normalized size of antiderivative = 23.55 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
- atan((18*a^5*tan(c + d*x))/(18*a^5 - 50*a^4*b + 32*a^3*b^2) - (50*a^4*ta n(c + d*x))/(32*a^3*b - 50*a^4 + (18*a^5)/b) + (32*a^3*b*tan(c + d*x))/(32 *a^3*b - 50*a^4 + (18*a^5)/b))/(b*d) - (atan((((-(a*b^2 - (a*b^5)^(1/2))/( 16*(a*b^4 - b^5)))^(1/2)*(((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^( 1/2)*(320*a^3*b^5 - 64*a^2*b^6 - 448*a^4*b^4 + 192*a^5*b^3 + tan(c + d*x)* (-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b^7 - 768*a^3 *b^6 - 768*a^4*b^5 + 768*a^5*b^4)) + tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b ^4 + 80*a^4*b^3 + 144*a^5*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5 )))^(1/2) + 12*a^5*b - 16*a^2*b^4 + 28*a^3*b^3 - 24*a^4*b^2) + tan(c + d*x )*(18*a^4*b + 6*a^5 - 4*a^2*b^3 - 20*a^3*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/( 16*(a*b^4 - b^5)))^(1/2)*1i + ((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5) ))^(1/2)*(((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(64*a^2*b^6 - 320*a^3*b^5 + 448*a^4*b^4 - 192*a^5*b^3 + tan(c + d*x)*(-(a*b^2 - (a*b^ 5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b^7 - 768*a^3*b^6 - 768*a^4*b ^5 + 768*a^5*b^4)) + tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 80*a^4*b^3 + 144*a^5*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) - 12*a ^5*b + 16*a^2*b^4 - 28*a^3*b^3 + 24*a^4*b^2) + tan(c + d*x)*(18*a^4*b + 6* a^5 - 4*a^2*b^3 - 20*a^3*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5) ))^(1/2)*1i)/(6*a^3*b - 6*a^4 + ((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^ 5)))^(1/2)*(((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(320*a...