Integrand size = 31, antiderivative size = 238 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {(8 A-21 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}+\frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3} \]
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Time = 0.78 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4104, 3872, 3852, 8, 3853, 3855} \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {(8 A-21 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}+\frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}+\frac {(52 A-129 B) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {4 (83 A-216 B) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(8 A-21 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac {(A-B) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(A-2 B) \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4104
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec ^5(c+d x) (5 a (A-B)-a (2 A-9 B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = \frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^4(c+d x) \left (28 a^2 (A-2 B)-a^2 (24 A-73 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^3(c+d x) \left (3 a^3 (52 A-129 B)-a^3 (176 A-477 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = \frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \sec ^2(c+d x) \left (8 a^4 (83 A-216 B)-105 a^4 (8 A-21 B) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = \frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(8 (83 A-216 B)) \int \sec ^2(c+d x) \, dx}{105 a^4}-\frac {(8 A-21 B) \int \sec ^3(c+d x) \, dx}{a^4} \\ & = -\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(8 A-21 B) \int \sec (c+d x) \, dx}{2 a^4}-\frac {(8 (83 A-216 B)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = -\frac {(8 A-21 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}+\frac {8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(A-2 B) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {4 (83 A-216 B) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Time = 4.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec ^4(c+d x) \left (-13440 (8 A-21 B) \text {arctanh}(\sin (c+d x)) \cos ^8\left (\frac {1}{2} (c+d x)\right )+(22888 A-58161 B+8 (4994 A-12813 B) \cos (c+d x)+60 (456 A-1177 B) \cos (2 (c+d x))+13864 A \cos (3 (c+d x))-35928 B \cos (3 (c+d x))+4472 A \cos (4 (c+d x))-11619 B \cos (4 (c+d x))+664 A \cos (5 (c+d x))-1728 B \cos (5 (c+d x))) \sec (c+d x) \tan (c+d x)\right )}{1680 a^4 d (1+\sec (c+d x))^4} \]
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Time = 1.10 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {3360 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {21 B}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3360 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {21 B}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+559 \left (\frac {15 \left (228 A -\frac {1177 B}{2}\right ) \cos \left (2 d x +2 c \right )}{559}+\frac {\left (1733 A -4491 B \right ) \cos \left (3 d x +3 c \right )}{559}+\left (A -\frac {11619 B}{4472}\right ) \cos \left (4 d x +4 c \right )+\frac {\left (83 A -216 B \right ) \cos \left (5 d x +5 c \right )}{559}+\frac {\left (4994 A -12813 B \right ) \cos \left (d x +c \right )}{559}+\frac {2861 A}{559}-\frac {58161 B}{4472}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{840 d \,a^{4} \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(196\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-84 B +32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 B -32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(234\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-84 B +32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 B +8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 B -32 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 B}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(234\) |
risch | \(\frac {i \left (840 A \,{\mathrm e}^{10 i \left (d x +c \right )}-2205 \,{\mathrm e}^{10 i \left (d x +c \right )} B +5880 A \,{\mathrm e}^{9 i \left (d x +c \right )}-15435 B \,{\mathrm e}^{9 i \left (d x +c \right )}+19040 A \,{\mathrm e}^{8 i \left (d x +c \right )}-49980 B \,{\mathrm e}^{8 i \left (d x +c \right )}+39200 A \,{\mathrm e}^{7 i \left (d x +c \right )}-102900 B \,{\mathrm e}^{7 i \left (d x +c \right )}+59248 A \,{\mathrm e}^{6 i \left (d x +c \right )}-155526 B \,{\mathrm e}^{6 i \left (d x +c \right )}+70896 A \,{\mathrm e}^{5 i \left (d x +c \right )}-183162 B \,{\mathrm e}^{5 i \left (d x +c \right )}+64384 A \,{\mathrm e}^{4 i \left (d x +c \right )}-166668 B \,{\mathrm e}^{4 i \left (d x +c \right )}+46032 A \,{\mathrm e}^{3 i \left (d x +c \right )}-119364 B \,{\mathrm e}^{3 i \left (d x +c \right )}+24664 A \,{\mathrm e}^{2 i \left (d x +c \right )}-64053 B \,{\mathrm e}^{2 i \left (d x +c \right )}+8456 \,{\mathrm e}^{i \left (d x +c \right )} A -21987 B \,{\mathrm e}^{i \left (d x +c \right )}+1328 A -3456 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{4} d}-\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{4} d}+\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 a^{4} d}\) | \(372\) |
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Time = 0.29 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.50 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {105 \, {\left ({\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (8 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (83 \, A - 216 \, B\right )} \cos \left (d x + c\right )^{5} + {\left (4472 \, A - 11619 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (1318 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (592 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right ) + 105 \, B\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
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\[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{7}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Time = 0.24 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.76 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {3 \, B {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.12 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {420 \, {\left (8 \, A - 21 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (8 \, A - 21 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 189 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11655 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 13.98 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.14 \[ \int \frac {\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^4}+\frac {4\,A-6\,B}{8\,a^4}+\frac {5\,A-15\,B}{24\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{4\,a^4}-\frac {5\,B}{2\,a^4}+\frac {3\,\left (4\,A-6\,B\right )}{4\,a^4}+\frac {3\,\left (5\,A-15\,B\right )}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,A-9\,B\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A-7\,B\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A-B\right )}{40\,a^4}+\frac {4\,A-6\,B}{40\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (8\,A-21\,B\right )}{a^4\,d} \]
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