Optimal. Leaf size=175 \[ \frac {1}{8} \left (8 a b B+4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {\left (4 a A b+2 a^2 B+3 b^2 B+6 a b C\right ) \sin (c+d x)}{3 d}+\frac {\left (2 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d} \]
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Rubi [A]
time = 0.32, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4179, 4159,
4132, 2717, 4130, 8} \begin {gather*} \frac {\sin (c+d x) \left (2 a^2 B+4 a A b+6 a b C+3 b^2 B\right )}{3 d}+\frac {\sin (c+d x) \cos (c+d x) \left (a^2 (3 A+4 C)+8 a b B+2 A b^2\right )}{8 d}+\frac {1}{8} x \left (a^2 (3 A+4 C)+8 a b B+4 b^2 (A+2 C)\right )+\frac {a (2 a B+A b) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2717
Rule 4130
Rule 4132
Rule 4159
Rule 4179
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (2 (A b+2 a B)+(3 a A+4 b B+4 a C) \sec (c+d x)+b (A+4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a (A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac {1}{12} \int \cos ^2(c+d x) \left (-3 \left (2 A b^2+8 a b B+a^2 (3 A+4 C)\right )-4 \left (4 a A b+2 a^2 B+3 b^2 B+6 a b C\right ) \sec (c+d x)-3 b^2 (A+4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a (A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac {1}{12} \int \cos ^2(c+d x) \left (-3 \left (2 A b^2+8 a b B+a^2 (3 A+4 C)\right )-3 b^2 (A+4 C) \sec ^2(c+d x)\right ) \, dx-\frac {1}{3} \left (-4 a A b-2 a^2 B-3 b^2 B-6 a b C\right ) \int \cos (c+d x) \, dx\\ &=\frac {\left (4 a A b+2 a^2 B+3 b^2 B+6 a b C\right ) \sin (c+d x)}{3 d}+\frac {\left (2 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac {1}{8} \left (-8 a b B-4 b^2 (A+2 C)-a^2 (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (8 a b B+4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {\left (4 a A b+2 a^2 B+3 b^2 B+6 a b C\right ) \sin (c+d x)}{3 d}+\frac {\left (2 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a (A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 134, normalized size = 0.77 \begin {gather*} \frac {12 \left (8 a b B+4 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) (c+d x)+24 \left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \sin (c+d x)+24 \left (A b^2+2 a b B+a^2 (A+C)\right ) \sin (2 (c+d x))+8 a (2 A b+a B) \sin (3 (c+d x))+3 a^2 A \sin (4 (c+d x))}{96 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 200, normalized size = 1.14
method | result | size |
derivativedivides | \(\frac {A \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a A b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a b B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} B \sin \left (d x +c \right )+2 a b C \sin \left (d x +c \right )+b^{2} C \left (d x +c \right )}{d}\) | \(200\) |
default | \(\frac {A \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a A b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a b B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} B \sin \left (d x +c \right )+2 a b C \sin \left (d x +c \right )+b^{2} C \left (d x +c \right )}{d}\) | \(200\) |
risch | \(\frac {3 a^{2} A x}{8}+\frac {x A \,b^{2}}{2}+B a b x +\frac {x \,a^{2} C}{2}+x \,b^{2} C +\frac {3 a A b \sin \left (d x +c \right )}{2 d}+\frac {3 a^{2} B \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) b^{2} B}{d}+\frac {2 \sin \left (d x +c \right ) a b C}{d}+\frac {A \,a^{2} \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a A b}{6 d}+\frac {a^{2} B \sin \left (3 d x +3 c \right )}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a b B}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{4 d}\) | \(215\) |
norman | \(\frac {\left (-\frac {3}{8} A \,a^{2}-\frac {1}{2} A \,b^{2}-a b B -\frac {1}{2} a^{2} C -b^{2} C \right ) x +\left (-\frac {9}{8} A \,a^{2}-\frac {3}{2} A \,b^{2}-3 a b B -\frac {3}{2} a^{2} C -3 b^{2} C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {9}{8} A \,a^{2}-\frac {3}{2} A \,b^{2}-3 a b B -\frac {3}{2} a^{2} C -3 b^{2} C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} A \,a^{2}-\frac {1}{2} A \,b^{2}-a b B -\frac {1}{2} a^{2} C -b^{2} C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} A \,a^{2}+\frac {1}{2} A \,b^{2}+a b B +\frac {1}{2} a^{2} C +b^{2} C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} A \,a^{2}+\frac {1}{2} A \,b^{2}+a b B +\frac {1}{2} a^{2} C +b^{2} C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {9}{8} A \,a^{2}+\frac {3}{2} A \,b^{2}+3 a b B +\frac {3}{2} a^{2} C +3 b^{2} C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {9}{8} A \,a^{2}+\frac {3}{2} A \,b^{2}+3 a b B +\frac {3}{2} a^{2} C +3 b^{2} C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (7 A \,a^{2}-4 A \,b^{2}-8 a b B -4 a^{2} C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (27 A \,a^{2}-32 a A b +12 A \,b^{2}-16 a^{2} B +24 a b B +12 a^{2} C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {\left (27 A \,a^{2}+32 a A b +12 A \,b^{2}+16 a^{2} B +24 a b B +12 a^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {\left (5 A \,a^{2}-16 a A b +4 A \,b^{2}-8 a^{2} B +8 a b B -8 b^{2} B +4 a^{2} C -16 a b C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (5 A \,a^{2}+16 a A b +4 A \,b^{2}+8 a^{2} B +8 a b B +8 b^{2} B +4 a^{2} C +16 a b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (81 A \,a^{2}-16 a A b -12 A \,b^{2}-8 a^{2} B -24 a b B -72 b^{2} B -12 a^{2} C -144 a b C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (81 A \,a^{2}+16 a A b -12 A \,b^{2}+8 a^{2} B -24 a b B +72 b^{2} B -12 a^{2} C +144 a b C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(749\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 187, normalized size = 1.07 \begin {gather*} \frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b + 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{2} + 96 \, {\left (d x + c\right )} C b^{2} + 192 \, C a b \sin \left (d x + c\right ) + 96 \, B b^{2} \sin \left (d x + c\right )}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.74, size = 134, normalized size = 0.77 \begin {gather*} \frac {3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{2} + 8 \, B a b + 4 \, {\left (A + 2 \, C\right )} b^{2}\right )} d x + {\left (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 16 \, B a^{2} + 16 \, {\left (2 \, A + 3 \, C\right )} a b + 24 \, B b^{2} + 8 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{2} + 8 \, B a b + 4 \, A b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 577 vs.
\(2 (165) = 330\).
time = 0.52, size = 577, normalized size = 3.30 \begin {gather*} \frac {3 \, {\left (3 \, A a^{2} + 4 \, C a^{2} + 8 \, B a b + 4 \, A b^{2} + 8 \, C b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 80 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 144 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 144 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B b^{2} \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 )}^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.13, size = 214, normalized size = 1.22 \begin {gather*} \frac {3\,A\,a^2\,x}{8}+\frac {A\,b^2\,x}{2}+\frac {C\,a^2\,x}{2}+C\,b^2\,x+\frac {3\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,b^2\,\sin \left (c+d\,x\right )}{d}+B\,a\,b\,x+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {A\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,A\,a\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {2\,C\,a\,b\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a\,b\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {B\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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