Optimal. Leaf size=471 \[ -\frac {(4 A b-a B) x}{a^5}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
[Out]
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Rubi [A]
time = 8.21, antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4185, 4189,
4004, 3916, 2738, 214} \begin {gather*} -\frac {x (4 A b-a B)}{a^5}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\sin (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {\sin (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{6 a^4 d \left (a^2-b^2\right )^3}-\frac {\sin (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 2738
Rule 3916
Rule 4004
Rule 4185
Rule 4189
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (4 A b^2-a b B-a^2 (3 A-C)+3 a (A b-a B+b C) \sec (c+d x)-3 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B+a^4 (6 A-5 C)+2 a \left (A b^3+3 a^3 B+2 a b^2 B-a^2 b (6 A+5 C)\right ) \sec (c+d x)-2 \left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-68 a^2 A b^4+24 A b^6-26 a^5 b B+17 a^3 b^3 B-6 a b^5 B-a^6 (6 A-11 C)+a^4 b^2 (65 A+4 C)+a \left (4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B-a^2 b^3 (7 A-4 C)+a^4 b (18 A+11 C)\right ) \sec (c+d x)+3 \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {-6 \left (a^2-b^2\right )^3 (4 A b-a B)-3 a \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3}\\ &=-\frac {(4 A b-a B) x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac {(4 A b-a B) x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 b \left (a^2-b^2\right )^3}\\ &=-\frac {(4 A b-a B) x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 b \left (a^2-b^2\right )^3 d}\\ &=-\frac {(4 A b-a B) x}{a^5}+\frac {\left (20 a^6 A b^2-35 a^4 A b^4+28 a^2 A b^6-8 A b^8-8 a^7 b B+8 a^5 b^3 B-7 a^3 b^5 B+2 a b^7 B+2 a^8 C+3 a^6 b^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 8.30, size = 1367, normalized size = 2.90 \begin {gather*} -\frac {2 (4 A b-a B) x (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a^5 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {\left (-20 a^6 A b^2+35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-3 a^6 b^2 C\right ) (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {2 i \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \cos (c)}{a^5 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {2 \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \sin (c)}{a^5 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}\right )}{\left (-a^2+b^2\right )^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {2 (b+a \cos (c+d x)) \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (A b^6 \sin (c)-a b^5 B \sin (c)+a^2 b^4 C \sin (c)-a A b^5 \sin (d x)+a^2 b^4 B \sin (d x)-a^3 b^3 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right ) d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {(b+a \cos (c+d x))^2 \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-17 a^2 A b^5 \sin (c)+12 A b^7 \sin (c)+14 a^3 b^4 B \sin (c)-9 a b^6 B \sin (c)-11 a^4 b^3 C \sin (c)+6 a^2 b^5 C \sin (c)+15 a^3 A b^4 \sin (d x)-10 a A b^6 \sin (d x)-12 a^4 b^3 B \sin (d x)+7 a^2 b^5 B \sin (d x)+9 a^5 b^2 C \sin (d x)-4 a^3 b^4 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right )^2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {(b+a \cos (c+d x))^3 \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (75 a^4 A b^4 \sin (c)-96 a^2 A b^6 \sin (c)+36 A b^8 \sin (c)-48 a^5 b^3 B \sin (c)+51 a^3 b^5 B \sin (c)-18 a b^7 B \sin (c)+27 a^6 b^2 C \sin (c)-18 a^4 b^4 C \sin (c)+6 a^2 b^6 C \sin (c)-60 a^5 A b^3 \sin (d x)+71 a^3 A b^5 \sin (d x)-26 a A b^7 \sin (d x)+36 a^6 b^2 B \sin (d x)-32 a^4 b^4 B \sin (d x)+11 a^2 b^6 B \sin (d x)-18 a^7 b C \sin (d x)+5 a^5 b^3 C \sin (d x)-2 a^3 b^5 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right )^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {2 A (b+a \cos (c+d x))^4 \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \tan (c+d x)}{a^4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
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Maple [A]
time = 0.61, size = 646, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+\left (4 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B -4 B \,a^{4} b^{2}+6 a^{3} b^{3} B +B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C +3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}-18 a^{5} b B +11 a^{3} b^{3} B -3 a \,b^{5} B +9 a^{6} C +a^{4} b^{2} C \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B +4 B \,a^{4} b^{2}+6 a^{3} b^{3} B -B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C -3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}-8 a^{7} b B +8 a^{5} b^{3} B -7 a^{3} b^{5} B +2 a \,b^{7} B +2 a^{8} C +3 a^{6} b^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) | \(646\) |
default | \(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+\left (4 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B -4 B \,a^{4} b^{2}+6 a^{3} b^{3} B +B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C +3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}-18 a^{5} b B +11 a^{3} b^{3} B -3 a \,b^{5} B +9 a^{6} C +a^{4} b^{2} C \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B +4 B \,a^{4} b^{2}+6 a^{3} b^{3} B -B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C -3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}-8 a^{7} b B +8 a^{5} b^{3} B -7 a^{3} b^{5} B +2 a \,b^{7} B +2 a^{8} C +3 a^{6} b^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) | \(646\) |
risch | \(\text {Expression too large to display}\) | \(2840\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1354 vs.
\(2 (448) = 896\).
time = 3.14, size = 2766, normalized size = 5.87 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1225 vs.
\(2 (448) = 896\).
time = 0.59, size = 1225, normalized size = 2.60 \begin {gather*} \frac {\frac {3 \, {\left (2 \, C a^{8} - 8 \, B a^{7} b + 20 \, A a^{6} b^{2} + 3 \, C a^{6} b^{2} + 8 \, B a^{5} b^{3} - 35 \, A a^{4} b^{4} - 7 \, B a^{3} b^{5} + 28 \, A a^{2} b^{6} + 2 \, B a b^{7} - 8 \, A b^{8}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {18 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, B a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, B a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, B a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 117 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 42 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, B a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 116 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 236 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 56 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 152 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 60 \, B a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, B a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 117 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}} + \frac {3 \, {\left (B a - 4 \, A b\right )} {\left (d x + c\right )}}{a^{5}} + \frac {6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.12, size = 2500, normalized size = 5.31 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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