Integrand size = 24, antiderivative size = 142 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=-\frac {\left (3 a^2+b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{16 b^5 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3208, 3232, 3202, 31} \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))}-\frac {\left (3 a^2+b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{16 b^5 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2} \]
[In]
[Out]
Rule 31
Rule 3202
Rule 3208
Rule 3232
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {\int \frac {-4 a+2 b \cos (d+e x)+2 a \sin (d+e x)}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^2} \, dx}{8 b^2} \\ & = -\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))}+\frac {\left (3 a^2+b^2\right ) \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx}{8 b^4} \\ & = -\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))}-\frac {\left (3 a^2+b^2\right ) \text {Subst}\left (\int \frac {1}{2 a+2 b x} \, dx,x,\cot \left (\frac {\pi }{4}+\frac {1}{2} (d+e x)\right )\right )}{8 b^4 e} \\ & = -\frac {\left (3 a^2+b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{16 b^5 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))} \\ \end{align*}
Time = 1.92 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.80 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=-\frac {-2 \left (3 a^2+b^2\right ) \log \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )+2 \left (3 a^2+b^2\right ) \log \left ((a+b) \cos \left (\frac {1}{2} (d+e x)\right )+(a-b) \sin \left (\frac {1}{2} (d+e x)\right )\right )+\frac {b^2}{\left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^2}+\frac {6 a b \sin \left (\frac {1}{2} (d+e x)\right )}{\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )}-\frac {b^2 \left (a^2+b^2\right )}{\left ((a+b) \cos \left (\frac {1}{2} (d+e x)\right )+(a-b) \sin \left (\frac {1}{2} (d+e x)\right )\right )^2}+\frac {6 a b \left (a^2+b^2\right ) \sin \left (\frac {1}{2} (d+e x)\right )}{(a+b) \left ((a+b) \cos \left (\frac {1}{2} (d+e x)\right )+(a-b) \sin \left (\frac {1}{2} (d+e x)\right )\right )}}{32 b^5 e} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(130)=260\).
Time = 1.95 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.86
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 b^{3} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {-3 a -b}{2 b^{4} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b^{5}}+\frac {\left (-3 a^{3}+3 a^{2} b -a \,b^{2}+b^{3}\right ) \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{2 b^{5} \left (a -b \right )}-\frac {-a^{4}-2 a^{2} b^{2}-b^{4}}{2 b^{3} \left (a -b \right )^{2} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{2}}-\frac {-3 a^{4}+4 a^{3} b -2 a^{2} b^{2}+4 a \,b^{3}+b^{4}}{2 b^{4} \left (a -b \right )^{2} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}}{8 e}\) | \(264\) |
default | \(\frac {-\frac {1}{2 b^{3} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {-3 a -b}{2 b^{4} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b^{5}}+\frac {\left (-3 a^{3}+3 a^{2} b -a \,b^{2}+b^{3}\right ) \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{2 b^{5} \left (a -b \right )}-\frac {-a^{4}-2 a^{2} b^{2}-b^{4}}{2 b^{3} \left (a -b \right )^{2} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{2}}-\frac {-3 a^{4}+4 a^{3} b -2 a^{2} b^{2}+4 a \,b^{3}+b^{4}}{2 b^{4} \left (a -b \right )^{2} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}}{8 e}\) | \(264\) |
risch | \(-\frac {i \left (3 a^{2} b \,{\mathrm e}^{3 i \left (e x +d \right )}+b^{3} {\mathrm e}^{3 i \left (e x +d \right )}+9 a^{3} {\mathrm e}^{2 i \left (e x +d \right )}+3 a \,b^{2} {\mathrm e}^{2 i \left (e x +d \right )}-3 i a^{3} {\mathrm e}^{3 i \left (e x +d \right )}-i a \,b^{2} {\mathrm e}^{3 i \left (e x +d \right )}+9 a^{2} b \,{\mathrm e}^{i \left (e x +d \right )}-b^{3} {\mathrm e}^{i \left (e x +d \right )}-3 a^{3}+3 a \,b^{2}+9 i a^{3} {\mathrm e}^{i \left (e x +d \right )}-i {\mathrm e}^{i \left (e x +d \right )} a \,b^{2}+6 i b \,a^{2}\right )}{8 \left (-i a \,{\mathrm e}^{2 i \left (e x +d \right )}+b \,{\mathrm e}^{2 i \left (e x +d \right )}+i a +2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2} b^{4} e}-\frac {3 \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a +b}{i b +a}\right ) a^{2}}{16 b^{5} e}-\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a +b}{i b +a}\right )}{16 b^{3} e}+\frac {3 \ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right ) a^{2}}{16 b^{5} e}+\frac {\ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{16 b^{3} e}\) | \(338\) |
norman | \(\frac {\frac {9 a^{5}+18 a^{4} b +12 a^{3} b^{2}+6 a^{2} b^{3}+a \,b^{4}}{16 b^{4} e \left (3 a^{2}-b^{2}\right )}-\frac {\left (9 a^{5}-9 a^{4} b +6 a^{3} b^{2}-6 a^{2} b^{3}+3 a \,b^{4}+b^{5}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{8 b^{4} e \left (3 a^{2}-b^{2}\right )}+\frac {\left (9 a^{5}+9 a^{4} b +6 a^{3} b^{2}+6 a^{2} b^{3}+3 a \,b^{4}-b^{5}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{8 b^{4} e \left (3 a^{2}-b^{2}\right )}-\frac {\left (9 a^{5}-18 a^{4} b +12 a^{3} b^{2}-6 a^{2} b^{3}+a \,b^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{16 b^{4} e \left (3 a^{2}-b^{2}\right )}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{2}}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{16 b^{5} e}-\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{16 b^{5} e}\) | \(375\) |
parallelrisch | \(\frac {-9 \left (a^{2}+\frac {b^{2}}{3}\right ) \left (\left (a^{2}-b^{2}\right ) \cos \left (2 e x +2 d \right )-4 a^{2} \sin \left (e x +d \right )-4 a b \cos \left (e x +d \right )-2 a b \sin \left (2 e x +2 d \right )-3 a^{2}-b^{2}\right ) \left (a^{2}-\frac {b^{2}}{3}\right ) \ln \left (a +b +\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+9 \left (a^{2}+\frac {b^{2}}{3}\right ) \left (\left (a^{2}-b^{2}\right ) \cos \left (2 e x +2 d \right )-4 a^{2} \sin \left (e x +d \right )-4 a b \cos \left (e x +d \right )-2 a b \sin \left (2 e x +2 d \right )-3 a^{2}-b^{2}\right ) \left (a^{2}-\frac {b^{2}}{3}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\left (-9 a^{4} b^{2}-3 a^{2} b^{4}\right ) \cos \left (2 e x +2 d \right )+\left (-9 a^{5} b -6 a^{3} b^{3}-3 a \,b^{5}\right ) \sin \left (2 e x +2 d \right )+\left (-18 a^{5} b -24 a^{3} b^{3}-2 a \,b^{5}\right ) \cos \left (e x +d \right )+\left (-18 a^{4} b^{2}-12 a^{2} b^{4}+2 b^{6}\right ) \sin \left (e x +d \right )-27 a^{4} b^{2}-9 a^{2} b^{4}}{48 b^{5} \left (\left (a^{2}-b^{2}\right ) \cos \left (2 e x +2 d \right )-4 a^{2} \sin \left (e x +d \right )-4 a b \cos \left (e x +d \right )-2 a b \sin \left (2 e x +2 d \right )-3 a^{2}-b^{2}\right ) e \left (a^{2}-\frac {b^{2}}{3}\right )}\) | \(417\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (130) = 260\).
Time = 0.26 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.96 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=\frac {12 \, a^{2} b^{2} \cos \left (e x + d\right )^{2} - 6 \, a^{2} b^{2} + 2 \, {\left (3 \, a^{3} b - a b^{3}\right )} \cos \left (e x + d\right ) - {\left (6 \, a^{4} + 2 \, a^{2} b^{2} - {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{4} + a^{2} b^{2} + {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (2 \, a b \cos \left (e x + d\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \sin \left (e x + d\right )\right ) + {\left (6 \, a^{4} + 2 \, a^{2} b^{2} - {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{4} + a^{2} b^{2} + {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (\sin \left (e x + d\right ) + 1\right ) - 2 \, {\left (3 \, a^{2} b^{2} - b^{4} - 3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{32 \, {\left (2 \, a b^{6} e \cos \left (e x + d\right ) + 2 \, a^{2} b^{5} e - {\left (a^{2} b^{5} - b^{7}\right )} e \cos \left (e x + d\right )^{2} + 2 \, {\left (a b^{6} e \cos \left (e x + d\right ) + a^{2} b^{5} e\right )} \sin \left (e x + d\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (130) = 260\).
Time = 0.23 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.47 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=\frac {\frac {2 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2} - a b^{4} + \frac {{\left (9 \, a^{5} - 9 \, a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 5 \, a b^{4} + b^{5}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {{\left (9 \, a^{5} - 18 \, a^{4} b + 12 \, a^{3} b^{2} - 6 \, a^{2} b^{3} + a b^{4}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {{\left (3 \, a^{5} - 9 \, a^{4} b + 10 \, a^{3} b^{2} - 6 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}\right )}}{a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8} + \frac {4 \, {\left (a^{4} b^{4} - a^{3} b^{5} - a^{2} b^{6} + a b^{7}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {2 \, {\left (3 \, a^{4} b^{4} - 6 \, a^{3} b^{5} + 2 \, a^{2} b^{6} + 2 \, a b^{7} - b^{8}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {4 \, {\left (a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {{\left (a^{4} b^{4} - 4 \, a^{3} b^{5} + 6 \, a^{2} b^{6} - 4 \, a b^{7} + b^{8}\right )} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}}} - \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (-a - b - \frac {{\left (a - b\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{b^{5}} + \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{b^{5}}}{16 \, e} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (130) = 260\).
Time = 0.30 (sec) , antiderivative size = 458, normalized size of antiderivative = 3.23 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=\frac {\frac {2 \, {\left (3 \, a^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 9 \, a^{4} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 10 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 6 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + a b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + b^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 9 \, a^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - 18 \, a^{4} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 12 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - 6 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + a b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 9 \, a^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 9 \, a^{4} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 2 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 5 \, a b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + b^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 3 \, a^{5} - 4 \, a^{3} b^{2} - a b^{4}\right )}}{{\left (a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a + b\right )}^{2}} + \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 2 \, a - 2 \, {\left | b \right |} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 2 \, a + 2 \, {\left | b \right |} \right |}}\right )}{b^{4} {\left | b \right |}}}{16 \, e} \]
[In]
[Out]
Time = 30.88 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.54 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx=-\frac {\frac {-3\,a^5+4\,a^3\,b^2+a\,b^4}{2\,b^4\,{\left (a-b\right )}^2}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (-9\,a^5+9\,a^4\,b+2\,a^3\,b^2-2\,a^2\,b^3+5\,a\,b^4-b^5\right )}{2\,b^4\,{\left (a-b\right )}^2}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (-3\,a^4+6\,a^3\,b-4\,a^2\,b^2+2\,a\,b^3+b^4\right )}{2\,b^4\,\left (a-b\right )}-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (9\,a^5-18\,a^4\,b+12\,a^3\,b^2-6\,a^2\,b^3+a\,b^4\right )}{2\,b^4\,{\left (a-b\right )}^2}}{e\,\left (8\,a\,b+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (24\,a^2-8\,b^2\right )-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (16\,a\,b-16\,a^2\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (4\,a^2-8\,a\,b+4\,b^2\right )+4\,a^2+4\,b^2+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (16\,a^2+16\,b\,a\right )\right )}-\frac {\mathrm {atanh}\left (\frac {2\,a+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )}{2\,b}\right )\,\left (3\,a^2+b^2\right )}{8\,b^5\,e} \]
[In]
[Out]