Integrand size = 28, antiderivative size = 158 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 b \sec ^3(c+d x)}{d}-\frac {b^3 \sec ^3(c+d x)}{3 d}+\frac {b^3 \sec ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {3 a b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 3853, 3855, 2686, 30, 2691, 14} \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {a^2 b \sec ^3(c+d x)}{d}-\frac {3 a b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {3 a b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {3 a b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b^3 \sec ^5(c+d x)}{5 d}-\frac {b^3 \sec ^3(c+d x)}{3 d} \]
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Rule 14
Rule 30
Rule 2686
Rule 2691
Rule 3169
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \sec ^3(c+d x)+3 a^2 b \sec ^3(c+d x) \tan (c+d x)+3 a b^2 \sec ^3(c+d x) \tan ^2(c+d x)+b^3 \sec ^3(c+d x) \tan ^3(c+d x)\right ) \, dx \\ & = a^3 \int \sec ^3(c+d x) \, dx+\left (3 a^2 b\right ) \int \sec ^3(c+d x) \tan (c+d x) \, dx+\left (3 a b^2\right ) \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+b^3 \int \sec ^3(c+d x) \tan ^3(c+d x) \, dx \\ & = \frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {3 a b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{2} a^3 \int \sec (c+d x) \, dx-\frac {1}{4} \left (3 a b^2\right ) \int \sec ^3(c+d x) \, dx+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}+\frac {b^3 \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a^2 b \sec ^3(c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {3 a b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (3 a b^2\right ) \int \sec (c+d x) \, dx+\frac {b^3 \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 b \sec ^3(c+d x)}{d}-\frac {b^3 \sec ^3(c+d x)}{3 d}+\frac {b^3 \sec ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {3 a b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 a b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(464\) vs. \(2(158)=316\).
Time = 1.32 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.94 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {\sec ^5(c+d x) \left (960 a^2 b+64 b^3+320 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-300 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+225 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-60 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-150 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+300 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-225 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-45 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 a^3 \sin (2 (c+d x))+540 a b^2 \sin (2 (c+d x))+120 a^3 \sin (4 (c+d x))-90 a b^2 \sin (4 (c+d x))\right )}{1920 d} \]
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Time = 1.62 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.97
method | result | size |
parts | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{3} \left (\frac {\sec \left (d x +c \right )^{5}}{5}-\frac {\sec \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {a^{2} b \sec \left (d x +c \right )^{3}}{d}\) | \(154\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {a^{2} b}{\cos \left (d x +c \right )^{3}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{15}\right )}{d}\) | \(198\) |
default | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {a^{2} b}{\cos \left (d x +c \right )^{3}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{15}\right )}{d}\) | \(198\) |
parallelrisch | \(\frac {-600 \left (a^{2}-\frac {3 b^{2}}{4}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+600 \left (a^{2}-\frac {3 b^{2}}{4}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (600 a^{2} b -80 b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (120 a^{2} b -16 b^{3}\right ) \cos \left (5 d x +5 c \right )+\left (960 a^{2} b -320 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (240 a^{3}+540 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (120 a^{3}-90 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (1200 a^{2} b -160 b^{3}\right ) \cos \left (d x +c \right )+960 a^{2} b +64 b^{3}}{120 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(284\) |
risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )} \left (60 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-45 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+120 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+270 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-480 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+160 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-960 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-64 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-120 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-270 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-480 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+160 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-60 i a^{3}+45 i a \,b^{2}\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {a^{3} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d}-\frac {3 a \,b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{8 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {3 a \,b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(335\) |
norman | \(\frac {-\frac {30 a^{2} b -4 b^{3}}{15 d}-\frac {6 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}-\frac {2 \left (3 a^{2} b +2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {2 \left (15 a^{2} b -20 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d}-\frac {2 \left (15 a^{2} b +4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 d}-\frac {2 \left (15 a^{2} b +34 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 d}-\frac {2 \left (15 a^{2} b +88 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d}+\frac {3 a \left (4 a^{2}-21 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {3 a \left (4 a^{2}-21 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}+\frac {3 a \left (4 a^{2}-13 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}-\frac {3 a \left (4 a^{2}-13 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {a \left (4 a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a \left (4 a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}-\frac {a \left (4 a^{2}+27 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}+\frac {a \left (4 a^{2}+27 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}+\frac {2 b \left (15 a^{2}-26 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {a \left (4 a^{2}-3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a \left (4 a^{2}-3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(519\) |
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.93 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, b^{3} + 80 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (6 \, a b^{2} \cos \left (d x + c\right ) + {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {45 \, a b^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {240 \, a^{2} b}{\cos \left (d x + c\right )^{3}} - \frac {16 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} b^{3}}{\cos \left (d x + c\right )^{5}}}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (144) = 288\).
Time = 0.38 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.11 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 270 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 480 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 270 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, a^{2} b + 16 \, b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
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Time = 26.53 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.85 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (a^3+\frac {3\,a\,b^2}{4}\right )-2\,a^2\,b-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {9\,a\,b^2}{2}-2\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {9\,a\,b^2}{2}-2\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^2\,b-\frac {4\,b^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (8\,a^2\,b+\frac {4\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (12\,a^2\,b-4\,b^3\right )+\frac {4\,b^3}{15}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^3+\frac {3\,a\,b^2}{4}\right )-6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a\,b^2}{4}-a^3\right )}{d} \]
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