Optimal. Leaf size=256 \[ -\frac {(44 A-21 B) x}{2 a^4}+\frac {8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac {(44 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 a^4 d (1+\sec (c+d x))}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.47, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4105, 3872,
2713, 2715, 8} \begin {gather*} -\frac {8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d}+\frac {8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac {(44 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac {(44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {x (44 A-21 B)}{2 a^4}-\frac {(16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4105
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx &=-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\cos ^3(c+d x) (a (10 A-3 B)-6 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos ^3(c+d x) \left (14 a^2 (7 A-3 B)-5 a^2 (16 A-9 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos ^3(c+d x) \left (9 a^3 (92 A-43 B)-4 a^3 (178 A-87 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \cos ^3(c+d x) \left (24 a^4 (227 A-108 B)-105 a^4 (44 A-21 B) \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac {(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(8 (227 A-108 B)) \int \cos ^3(c+d x) \, dx}{35 a^4}-\frac {(44 A-21 B) \int \cos ^2(c+d x) \, dx}{a^4}\\ &=-\frac {(44 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(44 A-21 B) \int 1 \, dx}{2 a^4}-\frac {(8 (227 A-108 B)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 a^4 d}\\ &=-\frac {(44 A-21 B) x}{2 a^4}+\frac {8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac {(44 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(611\) vs. \(2(256)=512\).
time = 1.58, size = 611, normalized size = 2.39 \begin {gather*} \frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-14700 (44 A-21 B) d x \cos \left (\frac {d x}{2}\right )-14700 (44 A-21 B) d x \cos \left (c+\frac {d x}{2}\right )-388080 A d x \cos \left (c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (c+\frac {3 d x}{2}\right )-388080 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-129360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-129360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-18480 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (3 c+\frac {7 d x}{2}\right )-18480 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (4 c+\frac {7 d x}{2}\right )+1010660 A \sin \left (\frac {d x}{2}\right )-539490 B \sin \left (\frac {d x}{2}\right )-687260 A \sin \left (c+\frac {d x}{2}\right )+386190 B \sin \left (c+\frac {d x}{2}\right )+814107 A \sin \left (c+\frac {3 d x}{2}\right )-422478 B \sin \left (c+\frac {3 d x}{2}\right )-204645 A \sin \left (2 c+\frac {3 d x}{2}\right )+132930 B \sin \left (2 c+\frac {3 d x}{2}\right )+357609 A \sin \left (2 c+\frac {5 d x}{2}\right )-181461 B \sin \left (2 c+\frac {5 d x}{2}\right )+18025 A \sin \left (3 c+\frac {5 d x}{2}\right )+3675 B \sin \left (3 c+\frac {5 d x}{2}\right )+72522 A \sin \left (3 c+\frac {7 d x}{2}\right )-36003 B \sin \left (3 c+\frac {7 d x}{2}\right )+24010 A \sin \left (4 c+\frac {7 d x}{2}\right )-9555 B \sin \left (4 c+\frac {7 d x}{2}\right )+2310 A \sin \left (4 c+\frac {9 d x}{2}\right )-945 B \sin \left (4 c+\frac {9 d x}{2}\right )+2310 A \sin \left (5 c+\frac {9 d x}{2}\right )-945 B \sin \left (5 c+\frac {9 d x}{2}\right )-175 A \sin \left (5 c+\frac {11 d x}{2}\right )+105 B \sin \left (5 c+\frac {11 d x}{2}\right )-175 A \sin \left (6 c+\frac {11 d x}{2}\right )+105 B \sin \left (6 c+\frac {11 d x}{2}\right )+35 A \sin \left (6 c+\frac {13 d x}{2}\right )+35 A \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{6720 a^4 d (1+\cos (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.37, size = 210, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{5}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {59 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+13 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+209 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-13 A +\frac {9 B}{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {62 A}{3}+8 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-9 A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-8 \left (44 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(210\) |
default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{5}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {59 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+13 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+209 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-13 A +\frac {9 B}{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {62 A}{3}+8 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-9 A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-8 \left (44 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(210\) |
norman | \(\frac {-\frac {\left (44 A -21 B \right ) x}{2 a}-\frac {\left (A -B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\left (31 A -24 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}-\frac {3 \left (44 A -21 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3 \left (44 A -21 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {\left (44 A -21 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {7 \left (67 A -32 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {\left (353 A -167 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {3 \left (1297 A -613 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}+\frac {\left (1369 A -676 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 a d}-\frac {\left (1417 A -843 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{840 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} a^{3}}\) | \(284\) |
risch | \(-\frac {22 A x}{a^{4}}+\frac {21 x B}{2 a^{4}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{2 a^{4} d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B}{8 a^{4} d}-\frac {43 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B}{a^{4} d}+\frac {43 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B}{a^{4} d}-\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{2 a^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 a^{4} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {2 i \left (3675 A \,{\mathrm e}^{6 i \left (d x +c \right )}-2100 B \,{\mathrm e}^{6 i \left (d x +c \right )}+19845 A \,{\mathrm e}^{5 i \left (d x +c \right )}-11025 B \,{\mathrm e}^{5 i \left (d x +c \right )}+46550 A \,{\mathrm e}^{4 i \left (d x +c \right )}-25515 B \,{\mathrm e}^{4 i \left (d x +c \right )}+59570 A \,{\mathrm e}^{3 i \left (d x +c \right )}-32340 B \,{\mathrm e}^{3 i \left (d x +c \right )}+43827 A \,{\mathrm e}^{2 i \left (d x +c \right )}-23688 B \,{\mathrm e}^{2 i \left (d x +c \right )}+17549 \,{\mathrm e}^{i \left (d x +c \right )} A -9471 B \,{\mathrm e}^{i \left (d x +c \right )}+3032 A -1653 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(379\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 452, normalized size = 1.77 \begin {gather*} \frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \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 {36960 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 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 {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.31, size = 257, normalized size = 1.00 \begin {gather*} -\frac {105 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (44 \, A - 21 \, B\right )} d x - {\left (70 \, A \cos \left (d x + c\right )^{6} - 35 \, {\left (4 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{5} + 140 \, {\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3196 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (7184 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (24436 \, A - 11619 \, B\right )} \cos \left (d x + c\right ) + 7264 \, A - 3456 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {A \cos ^{3}{\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 \cos ^{3}{\left (c + d x \right )} \sec {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.55, size = 261, normalized size = 1.02 \begin {gather*} -\frac {\frac {420 \, {\left (d x + c\right )} {\left (44 \, A - 21 \, B\right )}}{a^{4}} - \frac {280 \, {\left (78 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 124 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, 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 )}^{3} 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} - 231 \, 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} + 2065 \, 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} - 21945 \, 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.05, size = 300, normalized size = 1.17 \begin {gather*} \frac {\left (26\,A-9\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {124\,A}{3}-16\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A-7\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {x\,\left (44\,A-21\,B\right )}{2\,a^4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,\left (A-B\right )}{12\,a^4}+\frac {7\,A-5\,B}{6\,a^4}+\frac {21\,A-9\,B}{24\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-B}{10\,a^4}+\frac {7\,A-5\,B}{40\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{2\,a^4}+\frac {5\,\left (7\,A-5\,B\right )}{4\,a^4}+\frac {21\,A-9\,B}{2\,a^4}+\frac {35\,A-5\,B}{8\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________