Integrand size = 29, antiderivative size = 178 \[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a (a+3 b) \log (1-\sin (c+d x))}{16 (a+b)^3 d}-\frac {a (a-3 b) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac {a^2 b^3 \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 d}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d}+\frac {a \sec ^2(c+d x) \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \]
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Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2916, 12, 1661, 837, 815} \[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right )}+\frac {a \sec ^2(c+d x) \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{8 d \left (a^2-b^2\right )^2}-\frac {a^2 b^3 \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}+\frac {a (a+3 b) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac {a (a-3 b) \log (\sin (c+d x)+1)}{16 d (a-b)^3} \]
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Rule 12
Rule 815
Rule 837
Rule 1661
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {x^2}{b^2 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^3 \text {Subst}\left (\int \frac {x^2}{(a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {\sec ^4(c+d x) \left (\frac {b}{a^2-b^2}-\frac {a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac {b \text {Subst}\left (\int \frac {-\frac {a^2 b^2}{a^2-b^2}+\frac {3 a b^2 x}{a^2-b^2}}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d} \\ & = -\frac {\sec ^4(c+d x) \left (\frac {b}{a^2-b^2}-\frac {a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac {a \sec ^2(c+d x) \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}-\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^2 \left (a^2-5 b^2\right )}{a^2-b^2}+\frac {a b^2 \left (a^2+3 b^2\right ) x}{a^2-b^2}}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 b \left (a^2-b^2\right ) d} \\ & = -\frac {\sec ^4(c+d x) \left (\frac {b}{a^2-b^2}-\frac {a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac {a \sec ^2(c+d x) \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}-\frac {\text {Subst}\left (\int \left (\frac {a (a-b) b (a+3 b)}{2 (a+b)^2 (b-x)}+\frac {8 a^2 b^4}{(a-b)^2 (a+b)^2 (a+x)}+\frac {a (a-3 b) b (a+b)}{2 (a-b)^2 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 b \left (a^2-b^2\right ) d} \\ & = \frac {a (a+3 b) \log (1-\sin (c+d x))}{16 (a+b)^3 d}-\frac {a (a-3 b) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac {a^2 b^3 \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 d}-\frac {\sec ^4(c+d x) \left (\frac {b}{a^2-b^2}-\frac {a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac {a \sec ^2(c+d x) \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.92 \[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {a (a+3 b) \log (1-\sin (c+d x))}{(a+b)^3}-\frac {a (a-3 b) \log (1+\sin (c+d x))}{(a-b)^3}-\frac {16 a^2 b^3 \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac {1}{(a+b) (-1+\sin (c+d x))^2}+\frac {a-b}{(a+b)^2 (-1+\sin (c+d x))}-\frac {1}{(a-b) (1+\sin (c+d x))^2}+\frac {a+b}{(a-b)^2 (1+\sin (c+d x))}}{16 d} \]
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Time = 0.92 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {-a +b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {a \left (a +3 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-a -b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}-\frac {a \left (a -3 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}-\frac {a^{2} b^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}}{d}\) | \(173\) |
default | \(\frac {\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {-a +b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {a \left (a +3 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-a -b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}-\frac {a \left (a -3 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}-\frac {a^{2} b^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}}{d}\) | \(173\) |
parallelrisch | \(\frac {-8 b^{3} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )+\left (a +3 b \right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right )^{3} a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\left (a -3 b \right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {7 \left (\frac {4 \left (a^{2} b -b^{3}\right ) \cos \left (2 d x +2 c \right )}{7}+\frac {\left (-a^{2} b -b^{3}\right ) \cos \left (4 d x +4 c \right )}{7}+\frac {\left (-a^{3}-3 a \,b^{2}\right ) \sin \left (3 d x +3 c \right )}{7}+\left (a^{3}-\frac {11}{7} a \,b^{2}\right ) \sin \left (d x +c \right )-\frac {3 a^{2} b}{7}+\frac {5 b^{3}}{7}\right ) \left (a -b \right )}{2}\right ) \left (a +b \right )}{2 \left (a -b \right )^{3} \left (a +b \right )^{3} d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(304\) |
norman | \(\frac {\frac {2 b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 b^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}-\frac {4 a^{2} b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {a \left (7 a^{2}-3 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}+\frac {a \left (7 a^{2}-3 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) d}+\frac {a \left (a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {a \left (a^{2}-5 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {a \left (a +3 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {a^{2} b^{3} \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {\left (a -3 b \right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\) | \(458\) |
risch | \(-\frac {i a^{2} x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {3 i a b c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {3 i a b x}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {i a^{2} c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 i a^{2} b^{3} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {2 i a^{2} b^{3} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {i a^{2} x}{8 a^{3}-24 a^{2} b +24 a \,b^{2}-8 b^{3}}+\frac {i a^{2} c}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}-\frac {3 i a b c}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}-\frac {3 i a b x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-7 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+11 i b^{2} a \,{\mathrm e}^{5 i \left (d x +c \right )}+8 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+7 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-11 i b^{2} a \,{\mathrm e}^{3 i \left (d x +c \right )}+16 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-i a^{3} {\mathrm e}^{i \left (d x +c \right )}-3 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+8 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {a^{2} b^{3} \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 )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(759\) |
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Time = 0.42 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.45 \[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {16 \, a^{2} b^{3} \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (a^{5} - 6 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{5} - 6 \, a^{3} b^{2} + 8 \, a^{2} b^{3} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, a^{4} b - 8 \, a^{2} b^{3} + 4 \, b^{5} - 8 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (2 \, a^{5} - 4 \, a^{3} b^{2} + 2 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{4}} \]
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\[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\sin ^{2}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.49 \[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {16 \, a^{2} b^{3} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {{\left (a^{2} - 3 \, a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (a^{2} + 3 \, a b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (4 \, a^{2} b \sin \left (d x + c\right )^{2} - {\left (a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{3} - 2 \, a^{2} b - 2 \, b^{3} - {\left (a^{3} - 5 \, a b^{2}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}}{16 \, d} \]
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Time = 0.49 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.83 \[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {16 \, a^{2} b^{4} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} + \frac {{\left (a^{2} - 3 \, a b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (a^{2} + 3 \, a b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (6 \, a^{2} b^{3} \sin \left (d x + c\right )^{4} - a^{5} \sin \left (d x + c\right )^{3} - 2 \, a^{3} b^{2} \sin \left (d x + c\right )^{3} + 3 \, a b^{4} \sin \left (d x + c\right )^{3} + 4 \, a^{4} b \sin \left (d x + c\right )^{2} - 16 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} - a^{5} \sin \left (d x + c\right ) + 6 \, a^{3} b^{2} \sin \left (d x + c\right ) - 5 \, a b^{4} \sin \left (d x + c\right ) - 2 \, a^{4} b + 6 \, a^{2} b^{3} + 2 \, b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
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Time = 12.81 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.80 \[ \int \frac {\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (\frac {b^2}{4\,{\left (a-b\right )}^3}+\frac {b}{8\,{\left (a-b\right )}^2}-\frac {1}{8\,\left (a-b\right )}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (\frac {b}{8\,{\left (a+b\right )}^2}+\frac {1}{8\,\left (a+b\right )}-\frac {b^2}{4\,{\left (a+b\right )}^3}\right )}{d}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a\,b^2-7\,a^3\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{a^4-2\,a^2\,b^2+b^4}-\frac {2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a^4-2\,a^2\,b^2+b^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (5\,a\,b^2-a^3\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a\,b^2-7\,a^3\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {4\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{a^4-2\,a^2\,b^2+b^4}-\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^2-5\,b^2\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^2\,b^3\,\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{d\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )} \]
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