Integrand size = 15, antiderivative size = 113 \[ \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {(4 a+5 b) x}{2 b^3}-\frac {(4 a-b) (a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} b^3}-\frac {\cos (x) \sin (x)}{2 b \left (a+(a+b) \tan ^2(x)\right )}+\frac {(a+b) (2 a+b) \tan (x)}{2 a b^2 \left (a+(a+b) \tan ^2(x)\right )} \]
1/2*(4*a+5*b)*x/b^3-1/2*(4*a-b)*(a+b)^(3/2)*arctan((a+b)^(1/2)*tan(x)/a^(1 /2))/a^(3/2)/b^3-1/2*cos(x)*sin(x)/b/(a+(a+b)*tan(x)^2)+1/2*(a+b)*(2*a+b)* tan(x)/a/b^2/(a+(a+b)*tan(x)^2)
Time = 0.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {2 (4 a+5 b) x-\frac {2 (4 a-b) (a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{a^{3/2}}+b \sin (2 x)+\frac {2 b (a+b)^2 \sin (2 x)}{a (2 a+b-b \cos (2 x))}}{4 b^3} \]
(2*(4*a + 5*b)*x - (2*(4*a - b)*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x])/ Sqrt[a]])/a^(3/2) + b*Sin[2*x] + (2*b*(a + b)^2*Sin[2*x])/(a*(2*a + b - b* Cos[2*x])))/(4*b^3)
Time = 0.36 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 3670, 316, 402, 27, 397, 216, 218}
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.
| \(\displaystyle \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (x)^6}{\left (a+b \sin (x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 3670 |
| \(\displaystyle \int \frac {1}{\left (\tan ^2(x)+1\right )^2 \left ((a+b) \tan ^2(x)+a\right )^2}d\tan (x)\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int \frac {-3 (a+b) \tan ^2(x)+a+2 b}{\left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )^2}d\tan (x)}{2 b}-\frac {\tan (x)}{2 b \left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {(a+b) (2 a+b) \tan (x)}{a b \left ((a+b) \tan ^2(x)+a\right )}-\frac {\int \frac {2 \left (2 a^2+2 b a-b^2-(a+b) (2 a+b) \tan ^2(x)\right )}{\left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}d\tan (x)}{2 a b}}{2 b}-\frac {\tan (x)}{2 b \left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {(a+b) (2 a+b) \tan (x)}{a b \left ((a+b) \tan ^2(x)+a\right )}-\frac {\int \frac {2 a^2+2 b a-b^2-(a+b) (2 a+b) \tan ^2(x)}{\left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}d\tan (x)}{a b}}{2 b}-\frac {\tan (x)}{2 b \left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {(a+b) (2 a+b) \tan (x)}{a b \left ((a+b) \tan ^2(x)+a\right )}-\frac {\frac {(4 a-b) (a+b)^2 \int \frac {1}{(a+b) \tan ^2(x)+a}d\tan (x)}{b}-\frac {a (4 a+5 b) \int \frac {1}{\tan ^2(x)+1}d\tan (x)}{b}}{a b}}{2 b}-\frac {\tan (x)}{2 b \left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {(a+b) (2 a+b) \tan (x)}{a b \left ((a+b) \tan ^2(x)+a\right )}-\frac {\frac {(4 a-b) (a+b)^2 \int \frac {1}{(a+b) \tan ^2(x)+a}d\tan (x)}{b}-\frac {a (4 a+5 b) \arctan (\tan (x))}{b}}{a b}}{2 b}-\frac {\tan (x)}{2 b \left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {(a+b) (2 a+b) \tan (x)}{a b \left ((a+b) \tan ^2(x)+a\right )}-\frac {\frac {(4 a-b) (a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} b}-\frac {a (4 a+5 b) \arctan (\tan (x))}{b}}{a b}}{2 b}-\frac {\tan (x)}{2 b \left (\tan ^2(x)+1\right ) \left ((a+b) \tan ^2(x)+a\right )}\) |
-1/2*Tan[x]/(b*(1 + Tan[x]^2)*(a + (a + b)*Tan[x]^2)) + (-((-((a*(4*a + 5* b)*ArcTan[Tan[x]])/b) + ((4*a - b)*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x ])/Sqrt[a]])/(Sqrt[a]*b))/(a*b)) + ((a + b)*(2*a + b)*Tan[x])/(a*b*(a + (a + b)*Tan[x]^2)))/(2*b)
3.4.14.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Su bst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Time = 1.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\frac {b \tan \left (x \right )}{2+2 \left (\tan ^{2}\left (x \right )\right )}+\frac {\left (4 a +5 b \right ) \arctan \left (\tan \left (x \right )\right )}{2}}{b^{3}}-\frac {\left (a +b \right )^{2} \left (-\frac {b \tan \left (x \right )}{2 a \left (a \left (\tan ^{2}\left (x \right )\right )+\left (\tan ^{2}\left (x \right )\right ) b +a \right )}+\frac {\left (4 a -b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (x \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 a \sqrt {a \left (a +b \right )}}\right )}{b^{3}}\) | \(100\) |
| risch | \(\frac {2 a x}{b^{3}}+\frac {5 x}{2 b^{2}}-\frac {i {\mathrm e}^{2 i x}}{8 b^{2}}+\frac {i {\mathrm e}^{-2 i x}}{8 b^{2}}-\frac {i \left (2 a^{3} {\mathrm e}^{2 i x}+5 a^{2} b \,{\mathrm e}^{2 i x}+4 a \,b^{2} {\mathrm e}^{2 i x}+b^{3} {\mathrm e}^{2 i x}-a^{2} b -2 a \,b^{2}-b^{3}\right )}{a \,b^{3} \left (-b \,{\mathrm e}^{4 i x}+4 a \,{\mathrm e}^{2 i x}+2 b \,{\mathrm e}^{2 i x}-b \right )}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{b^{3}}+\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{4 a \,b^{2}}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{4 a^{2} b}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{b^{3}}-\frac {3 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{4 a \,b^{2}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{4 a^{2} b}\) | \(395\) |
1/b^3*(1/2*b*tan(x)/(1+tan(x)^2)+1/2*(4*a+5*b)*arctan(tan(x)))-(a+b)^2/b^3 *(-1/2/a*b*tan(x)/(a*tan(x)^2+tan(x)^2*b+a)+1/2*(4*a-b)/a/(a*(a+b))^(1/2)* arctan((a+b)*tan(x)/(a*(a+b))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (97) = 194\).
Time = 0.34 (sec) , antiderivative size = 491, normalized size of antiderivative = 4.35 \[ \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\left [\frac {4 \, {\left (4 \, a^{2} b + 5 \, a b^{2}\right )} x \cos \left (x\right )^{2} + {\left (4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3} - {\left (4 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (x\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \left (x\right )^{3} - {\left (a^{2} + a b\right )} \cos \left (x\right )\right )} \sqrt {-\frac {a + b}{a}} \sin \left (x\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (x\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (x\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) - 4 \, {\left (4 \, a^{3} + 9 \, a^{2} b + 5 \, a b^{2}\right )} x + 4 \, {\left (a b^{2} \cos \left (x\right )^{3} - {\left (2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{8 \, {\left (a b^{4} \cos \left (x\right )^{2} - a^{2} b^{3} - a b^{4}\right )}}, \frac {2 \, {\left (4 \, a^{2} b + 5 \, a b^{2}\right )} x \cos \left (x\right )^{2} - {\left (4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3} - {\left (4 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (x\right )^{2}\right )} \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a - b\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \left (x\right ) \sin \left (x\right )}\right ) - 2 \, {\left (4 \, a^{3} + 9 \, a^{2} b + 5 \, a b^{2}\right )} x + 2 \, {\left (a b^{2} \cos \left (x\right )^{3} - {\left (2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{4 \, {\left (a b^{4} \cos \left (x\right )^{2} - a^{2} b^{3} - a b^{4}\right )}}\right ] \]
[1/8*(4*(4*a^2*b + 5*a*b^2)*x*cos(x)^2 + (4*a^3 + 7*a^2*b + 2*a*b^2 - b^3 - (4*a^2*b + 3*a*b^2 - b^3)*cos(x)^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 - 4*((2*a^2 + a*b)*cos (x)^3 - (a^2 + a*b)*cos(x))*sqrt(-(a + b)/a)*sin(x) + a^2 + 2*a*b + b^2)/( b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - 4*(4*a^3 + 9 *a^2*b + 5*a*b^2)*x + 4*(a*b^2*cos(x)^3 - (2*a^2*b + 3*a*b^2 + b^3)*cos(x) )*sin(x))/(a*b^4*cos(x)^2 - a^2*b^3 - a*b^4), 1/4*(2*(4*a^2*b + 5*a*b^2)*x *cos(x)^2 - (4*a^3 + 7*a^2*b + 2*a*b^2 - b^3 - (4*a^2*b + 3*a*b^2 - b^3)*c os(x)^2)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(x)*sin(x))) - 2*(4*a^3 + 9*a^2*b + 5*a*b^2)*x + 2*(a*b ^2*cos(x)^3 - (2*a^2*b + 3*a*b^2 + b^3)*cos(x))*sin(x))/(a*b^4*cos(x)^2 - a^2*b^3 - a*b^4)]
Timed out. \[ \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\text {Timed out} \]
Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {{\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \tan \left (x\right )^{3} + {\left (2 \, a^{2} + 2 \, a b + b^{2}\right )} \tan \left (x\right )}{2 \, {\left ({\left (a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{4} + a^{2} b^{2} + {\left (2 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{2}\right )}} + \frac {{\left (4 \, a + 5 \, b\right )} x}{2 \, b^{3}} - \frac {{\left (4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{2 \, \sqrt {{\left (a + b\right )} a} a b^{3}} \]
1/2*((2*a^2 + 3*a*b + b^2)*tan(x)^3 + (2*a^2 + 2*a*b + b^2)*tan(x))/((a^2* b^2 + a*b^3)*tan(x)^4 + a^2*b^2 + (2*a^2*b^2 + a*b^3)*tan(x)^2) + 1/2*(4*a + 5*b)*x/b^3 - 1/2*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*arctan((a + b)*tan(x )/sqrt((a + b)*a))/(sqrt((a + b)*a)*a*b^3)
Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.55 \[ \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {{\left (4 \, a + 5 \, b\right )} x}{2 \, b^{3}} - \frac {{\left (4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3}\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (x\right ) + b \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{2 \, \sqrt {a^{2} + a b} a b^{3}} + \frac {2 \, a^{2} \tan \left (x\right )^{3} + 3 \, a b \tan \left (x\right )^{3} + b^{2} \tan \left (x\right )^{3} + 2 \, a^{2} \tan \left (x\right ) + 2 \, a b \tan \left (x\right ) + b^{2} \tan \left (x\right )}{2 \, {\left (a \tan \left (x\right )^{4} + b \tan \left (x\right )^{4} + 2 \, a \tan \left (x\right )^{2} + b \tan \left (x\right )^{2} + a\right )} a b^{2}} \]
1/2*(4*a + 5*b)*x/b^3 - 1/2*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*(pi*floor(x/ pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))/ (sqrt(a^2 + a*b)*a*b^3) + 1/2*(2*a^2*tan(x)^3 + 3*a*b*tan(x)^3 + b^2*tan(x )^3 + 2*a^2*tan(x) + 2*a*b*tan(x) + b^2*tan(x))/((a*tan(x)^4 + b*tan(x)^4 + 2*a*tan(x)^2 + b*tan(x)^2 + a)*a*b^2)
Time = 14.92 (sec) , antiderivative size = 463, normalized size of antiderivative = 4.10 \[ \int \frac {\cos ^6(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {\frac {\mathrm {tan}\left (x\right )\,\left (2\,a^2+2\,a\,b+b^2\right )}{2\,a\,b^2}+\frac {{\mathrm {tan}\left (x\right )}^3\,\left (a+b\right )\,\left (2\,a+b\right )}{2\,a\,b^2}}{\left (a+b\right )\,{\mathrm {tan}\left (x\right )}^4+\left (2\,a+b\right )\,{\mathrm {tan}\left (x\right )}^2+a}-\frac {\ln \left (a^2\,b-\mathrm {tan}\left (x\right )\,\sqrt {-a^3\,{\left (a+b\right )}^3}+a^3\right )\,\sqrt {-a^3\,{\left (a+b\right )}^3}\,\left (4\,a-b\right )}{4\,a^3\,b^3}+\frac {\ln \left (\mathrm {tan}\left (x\right )\,\sqrt {-a^3\,{\left (a+b\right )}^3}+a^2\,b+a^3\right )\,\left (a-\frac {b}{4}\right )\,\sqrt {-a^3\,{\left (a+b\right )}^3}}{a^3\,b^3}-\frac {\mathrm {atan}\left (\frac {41\,\mathrm {tan}\left (x\right )}{2\,\left (\frac {131\,a}{4\,b}+\frac {11\,b}{4\,a}-\frac {5\,b^2}{4\,a^2}+\frac {85\,a^2}{4\,b^2}+\frac {5\,a^3}{b^3}+\frac {41}{2}\right )}+\frac {11\,\mathrm {tan}\left (x\right )}{4\,\left (\frac {41\,a}{2\,b}-\frac {5\,b}{4\,a}+\frac {131\,a^2}{4\,b^2}+\frac {85\,a^3}{4\,b^3}+\frac {5\,a^4}{b^4}+\frac {11}{4}\right )}+\frac {131\,a\,\mathrm {tan}\left (x\right )}{4\,\left (\frac {131\,a}{4}+\frac {41\,b}{2}+\frac {11\,b^2}{4\,a}+\frac {85\,a^2}{4\,b}-\frac {5\,b^3}{4\,a^2}+\frac {5\,a^3}{b^2}\right )}-\frac {5\,b\,\mathrm {tan}\left (x\right )}{4\,\left (\frac {11\,a}{4}-\frac {5\,b}{4}+\frac {41\,a^2}{2\,b}+\frac {131\,a^3}{4\,b^2}+\frac {85\,a^4}{4\,b^3}+\frac {5\,a^5}{b^4}\right )}+\frac {85\,a^2\,\mathrm {tan}\left (x\right )}{4\,\left (\frac {131\,a\,b}{4}+\frac {85\,a^2}{4}+\frac {41\,b^2}{2}+\frac {11\,b^3}{4\,a}+\frac {5\,a^3}{b}-\frac {5\,b^4}{4\,a^2}\right )}+\frac {5\,a^3\,\mathrm {tan}\left (x\right )}{\frac {131\,a\,b^2}{4}+\frac {85\,a^2\,b}{4}+5\,a^3+\frac {41\,b^3}{2}+\frac {11\,b^4}{4\,a}-\frac {5\,b^5}{4\,a^2}}\right )\,\left (a\,1{}\mathrm {i}+\frac {b\,5{}\mathrm {i}}{4}\right )\,2{}\mathrm {i}}{b^3} \]
((tan(x)*(2*a*b + 2*a^2 + b^2))/(2*a*b^2) + (tan(x)^3*(a + b)*(2*a + b))/( 2*a*b^2))/(a + tan(x)^2*(2*a + b) + tan(x)^4*(a + b)) - (atan((41*tan(x))/ (2*((131*a)/(4*b) + (11*b)/(4*a) - (5*b^2)/(4*a^2) + (85*a^2)/(4*b^2) + (5 *a^3)/b^3 + 41/2)) + (11*tan(x))/(4*((41*a)/(2*b) - (5*b)/(4*a) + (131*a^2 )/(4*b^2) + (85*a^3)/(4*b^3) + (5*a^4)/b^4 + 11/4)) + (131*a*tan(x))/(4*(( 131*a)/4 + (41*b)/2 + (11*b^2)/(4*a) + (85*a^2)/(4*b) - (5*b^3)/(4*a^2) + (5*a^3)/b^2)) - (5*b*tan(x))/(4*((11*a)/4 - (5*b)/4 + (41*a^2)/(2*b) + (13 1*a^3)/(4*b^2) + (85*a^4)/(4*b^3) + (5*a^5)/b^4)) + (85*a^2*tan(x))/(4*((1 31*a*b)/4 + (85*a^2)/4 + (41*b^2)/2 + (11*b^3)/(4*a) + (5*a^3)/b - (5*b^4) /(4*a^2))) + (5*a^3*tan(x))/((131*a*b^2)/4 + (85*a^2*b)/4 + 5*a^3 + (41*b^ 3)/2 + (11*b^4)/(4*a) - (5*b^5)/(4*a^2)))*(a*1i + (b*5i)/4)*2i)/b^3 - (log (a^2*b - tan(x)*(-a^3*(a + b)^3)^(1/2) + a^3)*(-a^3*(a + b)^3)^(1/2)*(4*a - b))/(4*a^3*b^3) + (log(tan(x)*(-a^3*(a + b)^3)^(1/2) + a^2*b + a^3)*(a - b/4)*(-a^3*(a + b)^3)^(1/2))/(a^3*b^3)