Integrand size = 28, antiderivative size = 169 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {1}{2} a \left (a^4+10 a^2 b^2-15 b^4\right ) x-\frac {2 b^3 \left (5 a^2-b^2\right ) \log (\sin (c+d x))}{d}+\frac {2 b^3 \left (5 a^2-b^2\right ) \log (\tan (c+d x))}{d}+\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac {5 a b^4 \tan (c+d x)}{d}+\frac {b^5 \tan ^2(c+d x)}{2 d} \]
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Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3167, 1819, 1816, 649, 209, 266} \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {2 b^3 \left (5 a^2-b^2\right ) \log (\sin (c+d x))}{d}+\frac {2 b^3 \left (5 a^2-b^2\right ) \log (\tan (c+d x))}{d}+\frac {\sin ^2(c+d x) \left (a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)+b \left (5 a^4-10 a^2 b^2+b^4\right )\right )}{2 d}+\frac {1}{2} a x \left (a^4+10 a^2 b^2-15 b^4\right )+\frac {5 a b^4 \tan (c+d x)}{d}+\frac {b^5 \tan ^2(c+d x)}{2 d} \]
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Rule 209
Rule 266
Rule 649
Rule 1816
Rule 1819
Rule 3167
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(b+a x)^5}{x^3 \left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac {\text {Subst}\left (\int \frac {-2 b^5-10 a b^4 x-2 b^3 \left (10 a^2-b^2\right ) x^2-a \left (a^4+10 a^2 b^2-5 b^4\right ) x^3}{x^3 \left (1+x^2\right )} \, dx,x,\cot (c+d x)\right )}{2 d} \\ & = \frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac {\text {Subst}\left (\int \left (-\frac {2 b^5}{x^3}-\frac {10 a b^4}{x^2}-\frac {4 \left (5 a^2 b^3-b^5\right )}{x}+\frac {-a \left (a^4+10 a^2 b^2-15 b^4\right )+4 b^3 \left (5 a^2-b^2\right ) x}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 d} \\ & = \frac {2 b^3 \left (5 a^2-b^2\right ) \log (\tan (c+d x))}{d}+\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac {5 a b^4 \tan (c+d x)}{d}+\frac {b^5 \tan ^2(c+d x)}{2 d}+\frac {\text {Subst}\left (\int \frac {-a \left (a^4+10 a^2 b^2-15 b^4\right )+4 b^3 \left (5 a^2-b^2\right ) x}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d} \\ & = \frac {2 b^3 \left (5 a^2-b^2\right ) \log (\tan (c+d x))}{d}+\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac {5 a b^4 \tan (c+d x)}{d}+\frac {b^5 \tan ^2(c+d x)}{2 d}+\frac {\left (2 b^3 \left (5 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (a \left (a^4+10 a^2 b^2-15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d} \\ & = \frac {1}{2} a \left (a^4+10 a^2 b^2-15 b^4\right ) x-\frac {2 b^3 \left (5 a^2-b^2\right ) \log (\sin (c+d x))}{d}+\frac {2 b^3 \left (5 a^2-b^2\right ) \log (\tan (c+d x))}{d}+\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac {5 a b^4 \tan (c+d x)}{d}+\frac {b^5 \tan ^2(c+d x)}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(571\) vs. \(2(169)=338\).
Time = 6.36 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.38 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {b^3 \left (\frac {\cos ^2(c+d x) (a+b \tan (c+d x))^6 \left (b^2+a b \tan (c+d x)\right )}{2 b^4 \left (a^2+b^2\right )}-\frac {\left (-6 a^2+4 b^2\right ) \left (\frac {1}{2} \left (5 a^4-10 a^2 b^2+b^4+\frac {a^5-10 a^3 b^2+5 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )+\frac {1}{2} \left (5 a^4-10 a^2 b^2+b^4-\frac {a^5-10 a^3 b^2+5 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+5 a b \left (2 a^2-b^2\right ) \tan (c+d x)+\frac {1}{2} b^2 \left (10 a^2-b^2\right ) \tan ^2(c+d x)+\frac {5}{3} a b^3 \tan ^3(c+d x)+\frac {1}{4} b^4 \tan ^4(c+d x)\right )+5 a \left (\frac {1}{2} \left (6 a^5-20 a^3 b^2+6 a b^4+\frac {a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )+\frac {1}{2} \left (6 a^5-20 a^3 b^2+6 a b^4-\frac {a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+b \left (15 a^4-15 a^2 b^2+b^4\right ) \tan (c+d x)+a b^2 \left (10 a^2-3 b^2\right ) \tan ^2(c+d x)+\frac {1}{3} b^3 \left (15 a^2-b^2\right ) \tan ^3(c+d x)+\frac {3}{2} a b^4 \tan ^4(c+d x)+\frac {1}{5} b^5 \tan ^5(c+d x)\right )}{2 b^2 \left (a^2+b^2\right )}\right )}{d} \]
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Time = 1.58 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {a^{5} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {5 \cos \left (d x +c \right )^{2} a^{4} b}{2}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(209\) |
default | \(\frac {a^{5} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {5 \cos \left (d x +c \right )^{2} a^{4} b}{2}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(209\) |
parts | \(\frac {a^{5} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{4}}{2}+\sin \left (d x +c \right )^{2}+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}+\frac {10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}-\frac {5 a^{4} b}{2 d \sec \left (d x +c \right )^{2}}\) | \(223\) |
parallelrisch | \(\frac {80 b^{3} \left (a^{2}-\frac {b^{2}}{5}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-80 b^{3} \left (a^{2}-\frac {b^{2}}{5}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-80 b^{3} \left (a^{2}-\frac {b^{2}}{5}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (4 a^{5} d x +40 a^{3} b^{2} d x -60 a \,b^{4} d x -4 b^{5}\right ) \cos \left (2 d x +2 c \right )+\left (-5 a^{4} b +10 a^{2} b^{3}-b^{5}\right ) \cos \left (4 d x +4 c \right )+2 a \left (a^{2}-5 b^{2}\right )^{2} \sin \left (2 d x +2 c \right )+a \left (a^{4}-10 a^{2} b^{2}+5 b^{4}\right ) \sin \left (4 d x +4 c \right )+4 a^{5} d x +40 a^{3} b^{2} d x -60 a \,b^{4} d x +5 a^{4} b -10 a^{2} b^{3}+5 b^{5}}{8 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(297\) |
risch | \(-\frac {4 i b^{5} c}{d}-\frac {5 i {\mathrm e}^{2 i \left (d x +c \right )} a \,b^{4}}{8 d}+\frac {a^{5} x}{2}+5 a^{3} b^{2} x -\frac {15 a \,b^{4} x}{2}-\frac {5 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{4} b}{8 d}+\frac {5 \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2} b^{3}}{4 d}-\frac {{\mathrm e}^{2 i \left (d x +c \right )} b^{5}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{5}}{8 d}+\frac {20 i b^{3} a^{2} c}{d}+\frac {5 i {\mathrm e}^{2 i \left (d x +c \right )} a^{3} b^{2}}{4 d}-\frac {5 \,{\mathrm e}^{-2 i \left (d x +c \right )} a^{4} b}{8 d}+\frac {5 \,{\mathrm e}^{-2 i \left (d x +c \right )} a^{2} b^{3}}{4 d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )} b^{5}}{8 d}+10 i x \,a^{2} b^{3}-2 i x \,b^{5}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{5}}{8 d}+\frac {5 i {\mathrm e}^{-2 i \left (d x +c \right )} a \,b^{4}}{8 d}-\frac {5 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{3} b^{2}}{4 d}+\frac {2 b^{4} \left (5 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+5 i a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {10 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}+\frac {2 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(383\) |
norman | \(\frac {\left (\frac {1}{2} a^{5}+5 a^{3} b^{2}-\frac {15}{2} a \,b^{4}\right ) x +\left (-\frac {5}{2} a^{5}-25 a^{3} b^{2}+\frac {75}{2} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {5}{2} a^{5}-25 a^{3} b^{2}+\frac {75}{2} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {1}{2} a^{5}+5 a^{3} b^{2}-\frac {15}{2} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {1}{2} a^{5}+5 a^{3} b^{2}-\frac {15}{2} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {1}{2} a^{5}+5 a^{3} b^{2}-\frac {15}{2} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {3}{2} a^{5}+15 a^{3} b^{2}-\frac {45}{2} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {3}{2} a^{5}+15 a^{3} b^{2}-\frac {45}{2} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\frac {\left (10 a^{4} b -20 a^{2} b^{3}+4 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {\left (10 a^{4} b -20 a^{2} b^{3}+4 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {\left (10 a^{4} b -20 a^{2} b^{3}+12 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {\left (10 a^{4} b -20 a^{2} b^{3}+12 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}+\frac {a \left (a^{4}-10 a^{2} b^{2}+15 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (3 a^{4}-30 a^{2} b^{2}-35 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {\left (20 a^{4} b -40 a^{2} b^{3}-16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {\left (20 a^{4} b -40 a^{2} b^{3}-16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {40 a \,b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {40 a \,b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}-\frac {a \left (a^{4}-10 a^{2} b^{2}+15 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}-\frac {a \left (3 a^{4}-30 a^{2} b^{2}-35 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}-\frac {2 b^{3} \left (5 a^{2}-b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 b^{3} \left (5 a^{2}-b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 b^{3} \left (5 a^{2}-b^{2}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(770\) |
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Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {2 \, b^{5} - 2 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 8 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4}\right )} d x\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (10 \, a b^{4} \cos \left (d x + c\right ) + {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.06 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {10 \, a^{4} b \sin \left (d x + c\right )^{2} + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} + 10 \, {\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} b^{2} - 20 \, {\left (\sin \left (d x + c\right )^{2} + \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} a^{2} b^{3} - 10 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a b^{4} + 2 \, {\left (\sin \left (d x + c\right )^{2} - \frac {1}{\sin \left (d x + c\right )^{2} - 1} + 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} b^{5}}{4 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {b^{5} \tan \left (d x + c\right )^{2} + 10 \, a b^{4} \tan \left (d x + c\right ) + {\left (a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4}\right )} {\left (d x + c\right )} + 2 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac {10 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} - 2 \, b^{5} \tan \left (d x + c\right )^{2} - a^{5} \tan \left (d x + c\right ) + 10 \, a^{3} b^{2} \tan \left (d x + c\right ) - 5 \, a b^{4} \tan \left (d x + c\right ) + 5 \, a^{4} b - b^{5}}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
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Time = 25.01 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.09 \[ \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {2\,\left (b^5\,\ln \left (\frac {\cos \left (c+d\,x\right )}{\cos \left (c+d\,x\right )+1}\right )-b^5\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )+\frac {a^5\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}-5\,a^2\,b^3\,\ln \left (\frac {\cos \left (c+d\,x\right )}{\cos \left (c+d\,x\right )+1}\right )-\frac {15\,a\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+5\,a^2\,b^3\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )+5\,a^3\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\right )}{d}+\frac {\frac {5\,a^4\,b}{16}+\frac {9\,b^5}{16}-\frac {5\,a^2\,b^3}{8}-\frac {b^5\,\cos \left (4\,c+4\,d\,x\right )}{16}+\frac {a^5\,\sin \left (2\,c+2\,d\,x\right )}{8}+\frac {a^5\,\sin \left (4\,c+4\,d\,x\right )}{16}-\frac {5\,a^4\,b\,\cos \left (4\,c+4\,d\,x\right )}{16}+\frac {25\,a\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{8}+\frac {5\,a\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{16}+\frac {5\,a^2\,b^3\,\cos \left (4\,c+4\,d\,x\right )}{8}-\frac {5\,a^3\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4}-\frac {5\,a^3\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{8}}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \]
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