Integrand size = 24, antiderivative size = 155 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {x}{2 b}+\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/2} d}-\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/2} d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b d} \]
-1/2*x/b+1/2*cos(d*x+c)*sin(d*x+c)/b/d+1/2*a^(3/4)*arctan((a^(1/2)-b^(1/2) )^(1/2)*tan(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)-b^(1/2))^(1/2)-1/2*a^(3/4)* arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)+b^(1 /2))^(1/2)
Time = 1.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-2 \sqrt {b} (c+d x)-\frac {2 a \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {2 a \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\sqrt {b} \sin (2 (c+d x))}{4 b^{3/2} d} \]
(-2*Sqrt[b]*(c + d*x) - (2*a*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqr t[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (2*a*ArcTanh[((Sqrt[a ] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]* Sqrt[b]] + Sqrt[b]*Sin[2*(c + d*x)])/(4*b^(3/2)*d)
Time = 0.37 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^6}{a-b \sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tan ^6(c+d x)}{\left (\tan ^2(c+d x)+1\right )^2 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1610 |
| \(\displaystyle \frac {\int \left (\frac {a \tan ^2(c+d x)}{b \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {1}{b \left (\tan ^2(c+d x)+1\right )}+\frac {1}{b \left (\tan ^2(c+d x)+1\right )^2}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^{3/2} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^{3/2} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\arctan (\tan (c+d x))}{2 b}+\frac {\tan (c+d x)}{2 b \left (\tan ^2(c+d x)+1\right )}}{d}\) |
(-1/2*ArcTan[Tan[c + d*x]]/b + (a^(3/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Ta n[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/2)) - (a^(3/4)*ArcTa n[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[ b]]*b^(3/2)) + Tan[c + d*x]/(2*b*(1 + Tan[c + d*x]^2)))/d
3.3.4.3.1 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 = 1.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(-\frac {x}{2 b}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b d}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,b^{6} d^{4}-b^{7} d^{4}\right ) \textit {\_Z}^{4}+512 a^{2} b^{3} d^{2} \textit {\_Z}^{2}+65536 a^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {i b^{4} d^{3}}{2048 a}-\frac {i b^{5} d^{3}}{2048 a^{2}}\right ) \textit {\_R}^{3}+\left (-\frac {b^{2} d^{2}}{128}+\frac {b^{3} d^{2}}{128 a}\right ) \textit {\_R}^{2}+\frac {i d b \textit {\_R}}{4}-\frac {2 a}{b}-1\right )\right )}{64}\) | \(170\) |
| derivativedivides | \(\frac {-\frac {-\frac {\tan \left (d x +c \right )}{2 \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{2}}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\) | \(186\) |
| default | \(\frac {-\frac {-\frac {\tan \left (d x +c \right )}{2 \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{2}}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\) | \(186\) |
-1/2*x/b-1/8*I/b/d*exp(2*I*(d*x+c))+1/8*I/b/d*exp(-2*I*(d*x+c))-1/64*sum(_ R*ln(exp(2*I*(d*x+c))+(1/2048*I/a*b^4*d^3-1/2048*I/a^2*b^5*d^3)*_R^3+(-1/1 28*b^2*d^2+1/128/a*b^3*d^2)*_R^2+1/4*I*d*b*_R-2/b*a-1),_R=RootOf((a*b^6*d^ 4-b^7*d^4)*_Z^4+512*a^2*b^3*d^2*_Z^2+65536*a^3))
Leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (111) = 222\).
Time = 0.45 (sec) , antiderivative size = 1275, normalized size of antiderivative = 8.23 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
-1/8*(b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4 )) + a^2)/((a*b^3 - b^4)*d^2))*log(1/4*a^2*cos(d*x + c)^2 - 1/4*a^2 - 1/4* (2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/ ((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b ^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - a^2*b*d*cos(d*x + c) *sin(d*x + c))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7 )*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) - b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a ^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(1/4*a^ 2*cos(d*x + c)^2 - 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - 1/2*(( a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*si n(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^3 - b^4)*d^2*s qrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + b* d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2) /((a*b^3 - b^4)*d^2))*log(-1/4*a^2*cos(d*x + c)^2 + 1/4*a^2 - 1/4*(2*(a^2* b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^ 5 - 2*a*b^6 + b^7)*d^4)) + 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a *b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) - b*d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^...
Timed out. \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{6}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
1/4*(4*b*d*integrate(-4*(4*a*b*cos(6*d*x + 6*c)^2 + 4*a*b*cos(2*d*x + 2*c) ^2 + 4*a*b*sin(6*d*x + 6*c)^2 + 4*a*b*sin(2*d*x + 2*c)^2 - 4*(8*a^2 - 3*a* b)*cos(4*d*x + 4*c)^2 - a*b*cos(2*d*x + 2*c) - 4*(8*a^2 - 3*a*b)*sin(4*d*x + 4*c)^2 + 2*(8*a^2 - 7*a*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (a*b*cos (6*d*x + 6*c) - 2*a*b*cos(4*d*x + 4*c) + a*b*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + (8*a*b*cos(2*d*x + 2*c) - a*b + 2*(8*a^2 - 7*a*b)*cos(4*d*x + 4*c) )*cos(6*d*x + 6*c) + 2*(a*b + (8*a^2 - 7*a*b)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*b*sin(6*d*x + 6*c) - 2*a*b*sin(4*d*x + 4*c) + a*b*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*(4*a*b*sin(2*d*x + 2*c) + (8*a^2 - 7*a*b)*sin(4 *d*x + 4*c))*sin(6*d*x + 6*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)*s in(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*cos(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) + (...
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (111) = 222\).
Time = 0.78 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.48 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {d x + c}{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 )}} \sqrt {a b} a^{3} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a 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 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 )}} \sqrt {a b} a^{3} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a 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 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 {\tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} b}}{2 \, d} \]
-1/2*((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 + s qrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b + sqr t(a*b)*(a - b))*sqrt(a*b)*a^3*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sq rt(a*b)*a^2*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3)*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 - s qrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sq rt(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))*sqrt(a*b)*a^3*b - 6*sqrt(a ^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b - sqrt(a* b)*(a - b))*sqrt(a*b)*a*b^3)*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)) - tan(d*x + c)/((tan(d*x + c)^2 + 1)*b))/d
Time = 16.09 (sec) , antiderivative size = 1273, normalized size of antiderivative = 8.21 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
sin(2*c + 2*d*x)/(4*b*d) - (atan((a*b^7*sin(c + d*x)*(((a^3*b^7)^(1/2) - a ^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*4i - b^12*sin(c + d*x)*(((a^3*b^7)^(1/2 ) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(5/2)*3072i - b^10*sin(c + d*x)*(((a^3*b ^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*192i + a*b^9*sin(c + d*x)* (((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*192i + a^2*b^6*sin (c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*24i + a^ 3*b^5*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2) *4i + a^4*b^4*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7 ))^(1/2)*8i + a^2*b^8*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*448i + a^3*b^7*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/ (16*a*b^6 - 16*b^7))^(3/2)*320i + a^2*b^10*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(5/2)*3072i)/(a^2*b^5*cos(c + d*x) + a^3*b ^4*cos(c + d*x) - a^4*b^3*cos(c + d*x) - a^2*cos(c + d*x)*(a^3*b^7)^(1/2) + 2*a*b*cos(c + d*x)*(a^3*b^7)^(1/2)))*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a* b^6 - 16*b^7))^(1/2)*2i)/d - (atan((a*b^7*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*4i - b^12*sin(c + d*x)*(-((a^3*b^7)^ (1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(5/2)*3072i - b^10*sin(c + d*x)*(-(( a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*192i + a*b^9*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*192i + a^2*b ^6*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2...