Integrand size = 29, antiderivative size = 212 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 d}-\frac {\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{2 a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^4(c+d x)}{4 a^3 d}+\frac {b \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{6 a d}+\frac {b^2 \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^7 d}-\frac {b^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^7 d} \]
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Time = 0.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 908} \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \csc ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^7 d}-\frac {b^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^7 d}+\frac {b \left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 d}-\frac {\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{2 a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^4(c+d x)}{4 a^3 d}-\frac {\csc ^6(c+d x)}{6 a d} \]
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Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^7 \left (b^2-x^2\right )^2}{x^7 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {b^2 \text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x^7 (a+x)} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^2 \text {Subst}\left (\int \left (\frac {b^4}{a x^7}-\frac {b^4}{a^2 x^6}+\frac {-2 a^2 b^2+b^4}{a^3 x^5}+\frac {2 a^2 b^2-b^4}{a^4 x^4}+\frac {\left (a^2-b^2\right )^2}{a^5 x^3}-\frac {\left (a^2-b^2\right )^2}{a^6 x^2}+\frac {\left (a^2-b^2\right )^2}{a^7 x}-\frac {\left (a^2-b^2\right )^2}{a^7 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b \left (a^2-b^2\right )^2 \csc (c+d x)}{a^6 d}-\frac {\left (a^2-b^2\right )^2 \csc ^2(c+d x)}{2 a^5 d}-\frac {b \left (2 a^2-b^2\right ) \csc ^3(c+d x)}{3 a^4 d}+\frac {\left (2 a^2-b^2\right ) \csc ^4(c+d x)}{4 a^3 d}+\frac {b \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{6 a d}+\frac {b^2 \left (a^2-b^2\right )^2 \log (\sin (c+d x))}{a^7 d}-\frac {b^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^7 d} \\ \end{align*}
Time = 2.59 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {60 a b \left (a^2-b^2\right )^2 \csc (c+d x)-30 a^2 \left (a^2-b^2\right )^2 \csc ^2(c+d x)+20 a^3 b \left (-2 a^2+b^2\right ) \csc ^3(c+d x)+15 a^4 \left (2 a^2-b^2\right ) \csc ^4(c+d x)+12 a^5 b \csc ^5(c+d x)-10 a^6 \csc ^6(c+d x)+60 \left (-a^2 b+b^3\right )^2 (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))}{60 a^7 d} \]
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Time = 0.70 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(-\frac {\csc ^{6}\left (d x +c \right )}{6 a d}+\frac {b \left (\csc ^{5}\left (d x +c \right )\right )}{5 a^{2} d}+\frac {\csc ^{4}\left (d x +c \right )}{2 a d}-\frac {\left (\csc ^{4}\left (d x +c \right )\right ) b^{2}}{4 d \,a^{3}}-\frac {2 b \left (\csc ^{3}\left (d x +c \right )\right )}{3 a^{2} d}+\frac {\left (\csc ^{3}\left (d x +c \right )\right ) b^{3}}{3 d \,a^{4}}+\frac {\left (\csc ^{2}\left (d x +c \right )\right ) b^{2}}{d \,a^{3}}-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) b^{4}}{2 d \,a^{5}}-\frac {\csc ^{2}\left (d x +c \right )}{2 a d}+\frac {b \csc \left (d x +c \right )}{a^{2} d}-\frac {2 b^{3} \csc \left (d x +c \right )}{d \,a^{4}}+\frac {b^{5} \csc \left (d x +c \right )}{d \,a^{6}}-\frac {b^{2} \ln \left (a \csc \left (d x +c \right )+b \right )}{d \,a^{3}}+\frac {2 b^{4} \ln \left (a \csc \left (d x +c \right )+b \right )}{d \,a^{5}}-\frac {b^{6} \ln \left (a \csc \left (d x +c \right )+b \right )}{d \,a^{7}}\) | \(272\) |
default | \(-\frac {\csc ^{6}\left (d x +c \right )}{6 a d}+\frac {b \left (\csc ^{5}\left (d x +c \right )\right )}{5 a^{2} d}+\frac {\csc ^{4}\left (d x +c \right )}{2 a d}-\frac {\left (\csc ^{4}\left (d x +c \right )\right ) b^{2}}{4 d \,a^{3}}-\frac {2 b \left (\csc ^{3}\left (d x +c \right )\right )}{3 a^{2} d}+\frac {\left (\csc ^{3}\left (d x +c \right )\right ) b^{3}}{3 d \,a^{4}}+\frac {\left (\csc ^{2}\left (d x +c \right )\right ) b^{2}}{d \,a^{3}}-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) b^{4}}{2 d \,a^{5}}-\frac {\csc ^{2}\left (d x +c \right )}{2 a d}+\frac {b \csc \left (d x +c \right )}{a^{2} d}-\frac {2 b^{3} \csc \left (d x +c \right )}{d \,a^{4}}+\frac {b^{5} \csc \left (d x +c \right )}{d \,a^{6}}-\frac {b^{2} \ln \left (a \csc \left (d x +c \right )+b \right )}{d \,a^{3}}+\frac {2 b^{4} \ln \left (a \csc \left (d x +c \right )+b \right )}{d \,a^{5}}-\frac {b^{6} \ln \left (a \csc \left (d x +c \right )+b \right )}{d \,a^{7}}\) | \(272\) |
parallelrisch | \(\frac {-1920 b^{2} \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+1920 b^{2} \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}-\frac {12 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b}{5}+\left (-6 a^{5}+6 a^{3} b^{2}\right ) \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 a^{4} b -16 a^{2} b^{3}\right ) \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{5}-72 a^{3} b^{2}+48 a \,b^{4}\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-120 a^{4} b +336 a^{2} b^{3}-192 b^{5}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}-\frac {12 a^{4} b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\left (-6 a^{5}+6 a^{3} b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 a^{4} b -16 a^{2} b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{5}-72 a^{3} b^{2}+48 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-120 a^{4} b +336 a^{2} b^{3}-192 b^{5}\right )\right ) a}{1920 a^{7} d}\) | \(377\) |
norman | \(\frac {-\frac {1}{384 a d}-\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}+\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{160 a^{2} d}+\frac {b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 a^{2} d}+\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{3} d}+\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{3} d}-\frac {\left (5 a^{4}-24 a^{2} b^{2}+16 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{5} d}-\frac {\left (5 a^{4}-24 a^{2} b^{2}+16 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{5} d}-\frac {b \left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a^{4} d}-\frac {b \left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a^{4} d}+\frac {b \left (5 a^{4}-14 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{6} d}+\frac {b \left (5 a^{4}-14 a^{2} b^{2}+8 b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{6} d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{7} d}-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a^{7} d}\) | \(441\) |
risch | \(\frac {2 i \left (-150 b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+15 b^{5} {\mathrm e}^{11 i \left (d x +c \right )}+75 b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-75 b^{5} {\mathrm e}^{9 i \left (d x +c \right )}+150 b^{5} {\mathrm e}^{7 i \left (d x +c \right )}-15 b^{5} {\mathrm e}^{i \left (d x +c \right )}-15 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-90 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+120 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-90 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+30 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-15 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-90 i a^{3} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+60 i a \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+30 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+60 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-15 a^{4} b \,{\mathrm e}^{i \left (d x +c \right )}+240 a^{2} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-15 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}-15 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-50 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}+15 a^{4} b \,{\mathrm e}^{11 i \left (d x +c \right )}-30 a^{2} b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-130 a^{2} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-78 a^{4} b \,{\mathrm e}^{5 i \left (d x +c \right )}+35 a^{4} b \,{\mathrm e}^{3 i \left (d x +c \right )}-35 a^{4} b \,{\mathrm e}^{9 i \left (d x +c \right )}+78 a^{4} b \,{\mathrm e}^{7 i \left (d x +c \right )}+130 a^{2} b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-240 a^{2} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+30 a^{2} b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \,a^{6} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{3} d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right ) b^{4}}{a^{5} d}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{7} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{3} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{4}}{a^{5} d}+\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{7} d}\) | \(700\) |
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Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (202) = 404\).
Time = 0.42 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.19 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {10 \, a^{6} - 45 \, a^{4} b^{2} + 30 \, a^{2} b^{4} + 30 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} - 15 \, {\left (2 \, a^{6} - 7 \, a^{4} b^{2} + 4 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} + 2 \, a^{2} b^{4} - b^{6} - 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 60 \, {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} + 2 \, a^{2} b^{4} - b^{6} - 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (8 \, a^{5} b - 25 \, a^{3} b^{3} + 15 \, a b^{5} + 15 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{4} - 5 \, {\left (4 \, a^{5} b - 11 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{7} d \cos \left (d x + c\right )^{6} - 3 \, a^{7} d \cos \left (d x + c\right )^{4} + 3 \, a^{7} d \cos \left (d x + c\right )^{2} - a^{7} d\right )}} \]
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Timed out. \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {60 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{7}} - \frac {12 \, a^{4} b \sin \left (d x + c\right ) + 60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{5} - 10 \, a^{5} - 30 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{4} - 20 \, {\left (2 \, a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )^{3} + 15 \, {\left (2 \, a^{5} - a^{3} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{6} \sin \left (d x + c\right )^{6}}}{60 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.42 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {60 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b} - \frac {147 \, a^{4} b^{2} \sin \left (d x + c\right )^{6} - 294 \, a^{2} b^{4} \sin \left (d x + c\right )^{6} + 147 \, b^{6} \sin \left (d x + c\right )^{6} - 60 \, a^{5} b \sin \left (d x + c\right )^{5} + 120 \, a^{3} b^{3} \sin \left (d x + c\right )^{5} - 60 \, a b^{5} \sin \left (d x + c\right )^{5} + 30 \, a^{6} \sin \left (d x + c\right )^{4} - 60 \, a^{4} b^{2} \sin \left (d x + c\right )^{4} + 30 \, a^{2} b^{4} \sin \left (d x + c\right )^{4} + 40 \, a^{5} b \sin \left (d x + c\right )^{3} - 20 \, a^{3} b^{3} \sin \left (d x + c\right )^{3} - 30 \, a^{6} \sin \left (d x + c\right )^{2} + 15 \, a^{4} b^{2} \sin \left (d x + c\right )^{2} - 12 \, a^{5} b \sin \left (d x + c\right ) + 10 \, a^{6}}{a^{7} \sin \left (d x + c\right )^{6}}}{60 \, d} \]
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Time = 11.97 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.42 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {1}{64\,a}-\frac {b^2}{64\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {b}{96\,a^2}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{3\,a}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {b}{32\,a^2}-\frac {2\,b\,\left (\frac {b^2}{16\,a^3}-\frac {5}{64\,a}+\frac {2\,b\,\left (\frac {b}{32\,a^2}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{a}\right )}{a}\right )}{a}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{a}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {b^2}{32\,a^3}-\frac {5}{128\,a}+\frac {b\,\left (\frac {b}{32\,a^2}+\frac {2\,b\,\left (\frac {1}{16\,a}-\frac {b^2}{16\,a^3}\right )}{a}\right )}{a}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4\,b^2-2\,a^2\,b^4+b^6\right )}{a^7\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^2\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4\,b^2-2\,a^2\,b^4+b^6\right )}{a^7\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {10\,a^4\,b}{3}-\frac {8\,a^2\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {5\,a^5}{2}-12\,a^3\,b^2+8\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (20\,a^4\,b-56\,a^2\,b^3+32\,b^5\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^5-a^3\,b^2\right )+\frac {a^5}{6}-\frac {2\,a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}}{64\,a^6\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]
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