Integrand size = 31, antiderivative size = 159 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {7}{16} a^3 (2 A+B) x-\frac {7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}+\frac {7 a^3 (2 A+B) \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a (2 A+B) \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 (2 A+B) \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}-\frac {7 (2 A+B) \cos ^3(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{40 d}+\frac {7 a^3 (2 A+B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {7}{16} a^3 x (2 A+B)-\frac {a (2 A+B) \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{10 d}-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 2748
Rule 2757
Rule 2939
Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}+\frac {1}{2} (2 A+B) \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = -\frac {a (2 A+B) \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}+\frac {1}{10} (7 a (2 A+B)) \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {a (2 A+B) \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 (2 A+B) \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d}+\frac {1}{8} \left (7 a^2 (2 A+B)\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}-\frac {a (2 A+B) \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 (2 A+B) \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d}+\frac {1}{8} \left (7 a^3 (2 A+B)\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}+\frac {7 a^3 (2 A+B) \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a (2 A+B) \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 (2 A+B) \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d}+\frac {1}{16} \left (7 a^3 (2 A+B)\right ) \int 1 \, dx \\ & = \frac {7}{16} a^3 (2 A+B) x-\frac {7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}+\frac {7 a^3 (2 A+B) \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a (2 A+B) \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 (2 A+B) \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d} \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 \cos (c+d x) \left (284 A+212 B+\frac {420 (2 A+B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+16 (17 A+11 B) \cos (2 (c+d x))-12 (A+3 B) \cos (4 (c+d x))-330 A \sin (c+d x)-95 B \sin (c+d x)+90 A \sin (3 (c+d x))+110 B \sin (3 (c+d x))-5 B \sin (5 (c+d x))\right )}{480 d} \]
[In]
[Out]
Time = 0.82 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\frac {\left (\frac {\left (-13 A -7 B \right ) \cos \left (3 d x +3 c \right )}{12}+\frac {\left (A +3 B \right ) \cos \left (5 d x +5 c \right )}{20}+\left (A -\frac {B}{16}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (-3 A -\frac {7 B}{2}\right ) \sin \left (4 d x +4 c \right )}{8}+\frac {B \sin \left (6 d x +6 c \right )}{48}+\frac {\left (-7 A -5 B \right ) \cos \left (d x +c \right )}{2}+\frac {7 d x A}{2}+\frac {7 d x B}{4}-\frac {68 A}{15}-\frac {44 B}{15}\right ) a^{3}}{4 d}\) | \(120\) |
risch | \(\frac {7 a^{3} x A}{8}+\frac {7 a^{3} B x}{16}-\frac {7 A \,a^{3} \cos \left (d x +c \right )}{8 d}-\frac {5 a^{3} \cos \left (d x +c \right ) B}{8 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{3}}{192 d}+\frac {a^{3} \cos \left (5 d x +5 c \right ) A}{80 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right ) B}{80 d}-\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{3}}{32 d}-\frac {7 \sin \left (4 d x +4 c \right ) B \,a^{3}}{64 d}-\frac {13 a^{3} \cos \left (3 d x +3 c \right ) A}{48 d}-\frac {7 a^{3} \cos \left (3 d x +3 c \right ) B}{48 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{4 d}-\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{64 d}\) | \(208\) |
derivativedivides | \(\frac {A \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 A \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )-A \,a^{3} \left (\cos ^{3}\left (d x +c \right )\right )+3 B \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {B \,a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(279\) |
default | \(\frac {A \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 A \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )-A \,a^{3} \left (\cos ^{3}\left (d x +c \right )\right )+3 B \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {B \,a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(279\) |
norman | \(\frac {\left (\frac {7}{8} A \,a^{3}+\frac {7}{16} B \,a^{3}\right ) x +\left (\frac {7}{8} A \,a^{3}+\frac {7}{16} B \,a^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{4} A \,a^{3}+\frac {21}{8} B \,a^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{4} A \,a^{3}+\frac {21}{8} B \,a^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35}{2} A \,a^{3}+\frac {35}{4} B \,a^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{8} A \,a^{3}+\frac {105}{16} B \,a^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{8} A \,a^{3}+\frac {105}{16} B \,a^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {34 A \,a^{3}+22 B \,a^{3}}{15 d}-\frac {2 \left (3 A \,a^{3}+B \,a^{3}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (12 A \,a^{3}+4 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 \left (17 A \,a^{3}+11 B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 \left (19 A \,a^{3}+17 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (22 A \,a^{3}+18 B \,a^{3}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a^{3} \left (2 A -7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{3} \left (2 A -7 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (26 A +37 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a^{3} \left (26 A +37 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{3} \left (162 A +73 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a^{3} \left (162 A +73 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(505\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.70 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {48 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} - 320 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + 105 \, {\left (2 \, A + B\right )} a^{3} d x + 5 \, {\left (8 \, B a^{3} \cos \left (d x + c\right )^{5} - 2 \, {\left (18 \, A + 25 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 21 \, {\left (2 \, A + B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (151) = 302\).
Time = 0.39 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.70 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {3 A a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {A a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {2 A a^{3} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {A a^{3} \cos ^{3}{\left (c + d x \right )}}{d} + \frac {B a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 B a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 B a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {B a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 B a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {3 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {B a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {B a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 B a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {2 B a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {B a^{3} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right )^{3} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.25 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {960 \, A a^{3} \cos \left (d x + c\right )^{3} + 320 \, B a^{3} \cos \left (d x + c\right )^{3} - 64 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} A a^{3} - 90 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 192 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} B a^{3} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3} - 90 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3}}{960 \, d} \]
[In]
[Out]
none
Time = 0.56 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.04 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {B a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {7}{16} \, {\left (2 \, A a^{3} + B a^{3}\right )} x + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {{\left (13 \, A a^{3} + 7 \, B a^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (7 \, A a^{3} + 5 \, B a^{3}\right )} \cos \left (d x + c\right )}{8 \, d} - \frac {{\left (6 \, A a^{3} + 7 \, B a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, A a^{3} - B a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
[In]
[Out]
Time = 11.03 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.84 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {7\,a^3\,\mathrm {atan}\left (\frac {7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A+B\right )}{8\,\left (\frac {7\,A\,a^3}{4}+\frac {7\,B\,a^3}{8}\right )}\right )\,\left (2\,A+B\right )}{8\,d}-\frac {\frac {34\,A\,a^3}{15}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A\,a^3}{4}-\frac {7\,B\,a^3}{8}\right )+\frac {22\,B\,a^3}{15}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (6\,A\,a^3+2\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (12\,A\,a^3+4\,B\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {A\,a^3}{4}-\frac {7\,B\,a^3}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (22\,A\,a^3+18\,B\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {13\,A\,a^3}{2}+\frac {37\,B\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {13\,A\,a^3}{2}+\frac {37\,B\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {38\,A\,a^3}{5}+\frac {34\,B\,a^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {68\,A\,a^3}{3}+\frac {44\,B\,a^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {27\,A\,a^3}{4}+\frac {73\,B\,a^3}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {27\,A\,a^3}{4}+\frac {73\,B\,a^3}{24}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^3\,\left (2\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \]
[In]
[Out]