Integrand size = 24, antiderivative size = 95 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d} \]
-1/2*arctanh(b^(1/4)*sin(d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))/a^(3/4)/b^(3/4) /d+1/2*arctan(b^(1/4)*sin(d*x+c)/a^(1/4))*(a^(1/2)+b^(1/2))/a^(3/4)/b^(3/4 )/d
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )+i \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )-i \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )-\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )}{4 a^{3/4} b^{3/4} d} \]
((Sqrt[a] - Sqrt[b])*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]] + I*(Sqrt[a] + Sq rt[b])*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]] - I*(Sqrt[a] + Sqrt[b])*Log[a ^(1/4) + I*b^(1/4)*Sin[c + d*x]] - (Sqrt[a] - Sqrt[b])*Log[a^(1/4) + b^(1/ 4)*Sin[c + d*x]])/(4*a^(3/4)*b^(3/4)*d)
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 3702, 1481, 218, 221}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^3}{a-b \sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1-\sin ^2(c+d x)}{a-b \sin ^4(c+d x)}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 1481 |
| \(\displaystyle \frac {-\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \int \frac {1}{-b \sin ^2(c+d x)-\sqrt {a} \sqrt {b}}d\sin (c+d x)-\frac {1}{2} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b \sin ^2(c+d x)}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{3/4}}-\frac {1}{2} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b \sin ^2(c+d x)}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{3/4}}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{3/4}}}{d}\) |
(((1 + Sqrt[b]/Sqrt[a])*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(1/4) *b^(3/4)) - ((1 - Sqrt[b]/Sqrt[a])*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)] )/(2*a^(1/4)*b^(3/4)))/d
3.5.6.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ (-a)*c, 2]}, Simp[(e/2 + c*(d/(2*q))) Int[1/(-q + c*x^2), x], x] + Simp[( e/2 - c*(d/(2*q))) Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] & & NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x _)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(67)=134\).
Time = 0.72 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{d}\) | \(136\) |
| default | \(\frac {\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{d}\) | \(136\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 a^{3} b^{3} d^{4} \textit {\_Z}^{4}+64 a^{2} b^{2} d^{2} \textit {\_Z}^{2}-a^{2}+2 a b -b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (-\frac {128 i a^{3} b^{2} d^{3} \textit {\_R}^{3}}{a^{2}-b^{2}}+\left (-\frac {24 i d b \,a^{2}}{a^{2}-b^{2}}-\frac {8 i a \,b^{2} d}{a^{2}-b^{2}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{2}}{a^{2}-b^{2}}+\frac {b^{2}}{a^{2}-b^{2}}\right )\) | \(170\) |
1/d*(1/4*(1/b*a)^(1/4)/a*(ln((sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1/b*a )^(1/4)))+2*arctan(sin(d*x+c)/(1/b*a)^(1/4)))+1/4/b/(1/b*a)^(1/4)*(2*arcta n(sin(d*x+c)/(1/b*a)^(1/4))-ln((sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1/b *a)^(1/4)))))
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (67) = 134\).
Time = 0.37 (sec) , antiderivative size = 631, normalized size of antiderivative = 6.64 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {1}{4} \, \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}} \log \left (\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}}\right ) - \frac {1}{4} \, \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}} \log \left (\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}}\right ) - \frac {1}{4} \, \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}} \log \left (-\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}}\right ) + \frac {1}{4} \, \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}} \log \left (-\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}}\right ) \]
1/4*sqrt(-(a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d^2)) *log(1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a*b + b ^2)/(a^3*b^3*d^4)) - (a^2*b + a*b^2)*d)*sqrt(-(a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d^2))) - 1/4*sqrt((a*b*d^2*sqrt((a^2 + 2*a* b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b*d^2))*log(1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + (a^2*b + a*b^2 )*d)*sqrt((a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b*d^2)) ) - 1/4*sqrt(-(a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d ^2))*log(-1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a* b + b^2)/(a^3*b^3*d^4)) - (a^2*b + a*b^2)*d)*sqrt(-(a*b*d^2*sqrt((a^2 + 2* a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d^2))) + 1/4*sqrt((a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b*d^2))*log(-1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + (a^2*b + a*b^2)*d)*sqrt((a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b *d^2)))
Timed out. \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {2 \, {\left (\sqrt {a} + \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (\sqrt {a} - \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{4 \, d} \]
1/4*(2*(sqrt(a) + sqrt(b))*arctan(sqrt(b)*sin(d*x + c)/sqrt(sqrt(a)*sqrt(b )))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + (sqrt(a) - sqrt(b))*log((sqr t(b)*sin(d*x + c) - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*sin(d*x + c) + sqrt(sq rt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/d
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (67) = 134\).
Time = 0.86 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.95 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3}} + \frac {2 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3}} + \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{3}} - \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{3}}}{8 \, d} \]
1/8*(2*sqrt(2)*((-a*b^3)^(1/4)*b^2 - (-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(s qrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^3) + 2*sqrt(2)*(( -a*b^3)^(1/4)*b^2 - (-a*b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1 /4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^3) + sqrt(2)*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^3) - sqrt(2)*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*log(s in(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^3))/d
Time = 14.79 (sec) , antiderivative size = 489, normalized size of antiderivative = 5.15 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {8\,b^3\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {1}{8\,a\,b}-\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b+\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2+\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}+\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {1}{8\,a\,b}-\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b+\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2+\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}\right )\,\sqrt {-\frac {a\,\sqrt {a^3\,b^3}+b\,\sqrt {a^3\,b^3}+2\,a^2\,b^2}{16\,a^3\,b^3}}}{d}-\frac {2\,\mathrm {atanh}\left (\frac {8\,b^3\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {1}{8\,a\,b}+\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b-\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2-\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}+\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {1}{8\,a\,b}+\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b-\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2-\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}\right )\,\sqrt {\frac {a\,\sqrt {a^3\,b^3}+b\,\sqrt {a^3\,b^3}-2\,a^2\,b^2}{16\,a^3\,b^3}}}{d} \]
- (2*atanh((8*b^3*sin(c + d*x)*(- 1/(8*a*b) - (a^3*b^3)^(1/2)/(16*a^2*b^3) - (a^3*b^3)^(1/2)/(16*a^3*b^2))^(1/2))/(2*a*b + (2*(a^3*b^3)^(1/2))/a + 2 *b^2 + (2*b*(a^3*b^3)^(1/2))/a^2) + (8*a*b^2*sin(c + d*x)*(- 1/(8*a*b) - ( a^3*b^3)^(1/2)/(16*a^2*b^3) - (a^3*b^3)^(1/2)/(16*a^3*b^2))^(1/2))/(2*a*b + (2*(a^3*b^3)^(1/2))/a + 2*b^2 + (2*b*(a^3*b^3)^(1/2))/a^2))*(-(a*(a^3*b^ 3)^(1/2) + b*(a^3*b^3)^(1/2) + 2*a^2*b^2)/(16*a^3*b^3))^(1/2))/d - (2*atan h((8*b^3*sin(c + d*x)*((a^3*b^3)^(1/2)/(16*a^2*b^3) - 1/(8*a*b) + (a^3*b^3 )^(1/2)/(16*a^3*b^2))^(1/2))/(2*a*b - (2*(a^3*b^3)^(1/2))/a + 2*b^2 - (2*b *(a^3*b^3)^(1/2))/a^2) + (8*a*b^2*sin(c + d*x)*((a^3*b^3)^(1/2)/(16*a^2*b^ 3) - 1/(8*a*b) + (a^3*b^3)^(1/2)/(16*a^3*b^2))^(1/2))/(2*a*b - (2*(a^3*b^3 )^(1/2))/a + 2*b^2 - (2*b*(a^3*b^3)^(1/2))/a^2))*((a*(a^3*b^3)^(1/2) + b*( a^3*b^3)^(1/2) - 2*a^2*b^2)/(16*a^3*b^3))^(1/2))/d