Integrand size = 33, antiderivative size = 198 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {(2 A+13 C) \text {arctanh}(\sin (c+d x))}{2 a^3 d}-\frac {2 (11 A+76 C) \tan (c+d x)}{15 a^3 d}+\frac {(2 A+13 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \]
[Out]
Time = 0.58 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4170, 4104, 3872, 3852, 8, 3853, 3855} \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {(2 A+13 C) \text {arctanh}(\sin (c+d x))}{2 a^3 d}-\frac {2 (11 A+76 C) \tan (c+d x)}{15 a^3 d}-\frac {(11 A+76 C) \tan (c+d x) \sec ^2(c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac {(2 A+13 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac {(A+11 C) \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2} \]
[In]
[Out]
Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4104
Rule 4170
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^4(c+d x) (-a (A-4 C)-a (2 A+7 C) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {\int \frac {\sec ^3(c+d x) \left (3 a^2 (A+11 C)-a^2 (8 A+43 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \sec ^2(c+d x) \left (2 a^3 (11 A+76 C)-15 a^3 (2 A+13 C) \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(2 A+13 C) \int \sec ^3(c+d x) \, dx}{a^3}-\frac {(2 (11 A+76 C)) \int \sec ^2(c+d x) \, dx}{15 a^3} \\ & = \frac {(2 A+13 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(2 A+13 C) \int \sec (c+d x) \, dx}{2 a^3}+\frac {(2 (11 A+76 C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 a^3 d} \\ & = \frac {(2 A+13 C) \text {arctanh}(\sin (c+d x))}{2 a^3 d}-\frac {2 (11 A+76 C) \tan (c+d x)}{15 a^3 d}+\frac {(2 A+13 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(632\) vs. \(2(198)=396\).
Time = 3.95 (sec) , antiderivative size = 632, normalized size of antiderivative = 3.19 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (1920 (2 A+13 C) \cos ^5\left (\frac {1}{2} (c+d x)\right ) \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 )+\sec \left (\frac {c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (-5 (98 A+247 C) \sin \left (\frac {d x}{2}\right )+5 (106 A+761 C) \sin \left (\frac {3 d x}{2}\right )-654 A \sin \left (c-\frac {d x}{2}\right )-4329 C \sin \left (c-\frac {d x}{2}\right )+654 A \sin \left (c+\frac {d x}{2}\right )+1989 C \sin \left (c+\frac {d x}{2}\right )-490 A \sin \left (2 c+\frac {d x}{2}\right )-3575 C \sin \left (2 c+\frac {d x}{2}\right )-350 A \sin \left (c+\frac {3 d x}{2}\right )-475 C \sin \left (c+\frac {3 d x}{2}\right )+530 A \sin \left (2 c+\frac {3 d x}{2}\right )+2005 C \sin \left (2 c+\frac {3 d x}{2}\right )-350 A \sin \left (3 c+\frac {3 d x}{2}\right )-2275 C \sin \left (3 c+\frac {3 d x}{2}\right )+378 A \sin \left (c+\frac {5 d x}{2}\right )+2673 C \sin \left (c+\frac {5 d x}{2}\right )-150 A \sin \left (2 c+\frac {5 d x}{2}\right )+105 C \sin \left (2 c+\frac {5 d x}{2}\right )+378 A \sin \left (3 c+\frac {5 d x}{2}\right )+1593 C \sin \left (3 c+\frac {5 d x}{2}\right )-150 A \sin \left (4 c+\frac {5 d x}{2}\right )-975 C \sin \left (4 c+\frac {5 d x}{2}\right )+190 A \sin \left (2 c+\frac {7 d x}{2}\right )+1325 C \sin \left (2 c+\frac {7 d x}{2}\right )-30 A \sin \left (3 c+\frac {7 d x}{2}\right )+255 C \sin \left (3 c+\frac {7 d x}{2}\right )+190 A \sin \left (4 c+\frac {7 d x}{2}\right )+875 C \sin \left (4 c+\frac {7 d x}{2}\right )-30 A \sin \left (5 c+\frac {7 d x}{2}\right )-195 C \sin \left (5 c+\frac {7 d x}{2}\right )+44 A \sin \left (3 c+\frac {9 d x}{2}\right )+304 C \sin \left (3 c+\frac {9 d x}{2}\right )+90 C \sin \left (4 c+\frac {9 d x}{2}\right )+44 A \sin \left (5 c+\frac {9 d x}{2}\right )+214 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )\right )}{240 a^3 d (A+2 C+A \cos (2 (c+d x))) (1+\sec (c+d x))^3} \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(\frac {-240 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (\frac {13 C}{2}+A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+240 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (\frac {13 C}{2}+A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\left (\frac {108 A}{11}+\frac {783 C}{11}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {51 A}{11}+\frac {717 C}{22}\right ) \cos \left (3 d x +3 c \right )+\left (A +\frac {76 C}{11}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {153 A}{11}+\frac {2331 C}{22}\right ) \cos \left (d x +c \right )+\frac {97 A}{11}+\frac {677 C}{11}\right )}{240 d \,a^{3} \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(175\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (-26 C -4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {2 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {14 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (26 C +4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {14 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{4 d \,a^{3}}\) | \(194\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (-26 C -4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {2 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {14 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (26 C +4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {14 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{4 d \,a^{3}}\) | \(194\) |
norman | \(\frac {-\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{20 a d}-\frac {\left (A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 a d}-\frac {\left (7 A +59 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 a d}+\frac {\left (51 C +7 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (475 C +71 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 a d}-\frac {\left (721 C +101 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}-\frac {\left (1165 C +173 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{12 a d}+\frac {\left (6613 C +953 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5} a^{2}}-\frac {\left (2 A +13 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{3} d}+\frac {\left (2 A +13 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{3} d}\) | \(276\) |
risch | \(-\frac {i \left (30 A \,{\mathrm e}^{8 i \left (d x +c \right )}+195 C \,{\mathrm e}^{8 i \left (d x +c \right )}+150 A \,{\mathrm e}^{7 i \left (d x +c \right )}+975 C \,{\mathrm e}^{7 i \left (d x +c \right )}+350 A \,{\mathrm e}^{6 i \left (d x +c \right )}+2275 C \,{\mathrm e}^{6 i \left (d x +c \right )}+490 A \,{\mathrm e}^{5 i \left (d x +c \right )}+3575 C \,{\mathrm e}^{5 i \left (d x +c \right )}+654 A \,{\mathrm e}^{4 i \left (d x +c \right )}+4329 C \,{\mathrm e}^{4 i \left (d x +c \right )}+530 A \,{\mathrm e}^{3 i \left (d x +c \right )}+3805 C \,{\mathrm e}^{3 i \left (d x +c \right )}+378 A \,{\mathrm e}^{2 i \left (d x +c \right )}+2673 C \,{\mathrm e}^{2 i \left (d x +c \right )}+190 A \,{\mathrm e}^{i \left (d x +c \right )}+1325 C \,{\mathrm e}^{i \left (d x +c \right )}+44 A +304 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{3} d}-\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 a^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{3} d}+\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 a^{3} d}\) | \(323\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.46 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {15 \, {\left ({\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (11 \, A + 76 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (34 \, A + 239 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (64 \, A + 479 \, C\right )} \cos \left (d x + c\right )^{2} + 45 \, C \cos \left (d x + c\right ) - 15 \, C\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.67 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {C {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {390 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {390 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )} + A {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )}}{60 \, d} \]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {30 \, {\left (2 \, A + 13 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {30 \, {\left (2 \, A + 13 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac {60 \, {\left (7 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 465 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
[In]
[Out]
Time = 15.21 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A+\frac {13\,C}{2}\right )}{a^3\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A+C\right )}{2\,a^3}+\frac {3\,\left (A+5\,C\right )}{4\,a^3}-\frac {2\,A-10\,C}{4\,a^3}\right )}{d}-\frac {5\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-7\,C\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{4\,a^3}+\frac {A+5\,C}{12\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A+C\right )}{20\,a^3\,d} \]
[In]
[Out]