Optimal. Leaf size=149 \[ \frac {1}{8} a^2 (7 A+8 B+12 C) x+\frac {a^2 (7 A+8 B+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+8 B+12 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(A+2 B) \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d} \]
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Rubi [A]
time = 0.23, antiderivative size = 149, 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 = {4171, 4098,
3873, 2717, 4130, 8} \begin {gather*} \frac {a^2 (7 A+8 B+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+8 B+12 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (7 A+8 B+12 C)+\frac {(A+2 B) \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{6 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^2}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2717
Rule 3873
Rule 4098
Rule 4130
Rule 4171
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \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+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 (2 a (A+2 B)+a (A+4 C) \sec (c+d x)) \, dx}{4 a}\\ &=\frac {(A+2 B) \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{12} (7 A+8 B+12 C) \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx\\ &=\frac {(A+2 B) \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{12} (7 A+8 B+12 C) \int \cos ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{6} \left (a^2 (7 A+8 B+12 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac {a^2 (7 A+8 B+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+8 B+12 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(A+2 B) \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{8} \left (a^2 (7 A+8 B+12 C)\right ) \int 1 \, dx\\ &=\frac {1}{8} a^2 (7 A+8 B+12 C) x+\frac {a^2 (7 A+8 B+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+8 B+12 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(A+2 B) \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 95, normalized size = 0.64 \begin {gather*} \frac {a^2 (84 A d x+96 B d x+144 C d x+24 (6 A+7 B+8 C) \sin (c+d x)+24 (2 A+2 B+C) \sin (2 (c+d x))+16 A \sin (3 (c+d x))+8 B \sin (3 (c+d x))+3 A \sin (4 (c+d x)))}{96 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 203, normalized size = 1.36
method | result | size |
risch | \(\frac {7 a^{2} A x}{8}+a^{2} B x +\frac {3 a^{2} x C}{2}+\frac {3 a^{2} A \sin \left (d x +c \right )}{2 d}+\frac {7 a^{2} B \sin \left (d x +c \right )}{4 d}+\frac {2 \sin \left (d x +c \right ) a^{2} C}{d}+\frac {a^{2} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{2} A \sin \left (3 d x +3 c \right )}{6 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} B}{12 d}+\frac {a^{2} A \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} B}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{4 d}\) | \(175\) |
derivativedivides | \(\frac {a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} B \sin \left (d x +c \right )+2 a^{2} C \sin \left (d x +c \right )+a^{2} C \left (d x +c \right )}{d}\) | \(203\) |
default | \(\frac {a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} B \sin \left (d x +c \right )+2 a^{2} C \sin \left (d x +c \right )+a^{2} C \left (d x +c \right )}{d}\) | \(203\) |
norman | \(\frac {\frac {a^{2} \left (3 A -8 B -4 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2} \left (7 A +8 B +12 C \right ) x}{8}+\frac {a^{2} \left (7 A +8 B +12 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{2} \left (7 A +8 B +12 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a^{2} \left (7 A +8 B +12 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 a^{2} \left (7 A +8 B +12 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 a^{2} \left (7 A +8 B +12 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {3 a^{2} \left (7 A +8 B +12 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {3 a^{2} \left (7 A +8 B +12 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} \left (7 A +8 B +12 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} \left (7 A +8 B +12 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {a^{2} \left (25 A +24 B +20 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a^{2} \left (53 A -104 B -156 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{2} \left (71 A +40 B +12 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a^{2} \left (85 A +56 B +132 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(438\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 190, normalized size = 1.28 \begin {gather*} -\frac {64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 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} - 24 \, {\left (2 \, d x + 2 \, c + \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} - 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, 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} - 96 \, {\left (d x + c\right )} C a^{2} - 96 \, B a^{2} \sin \left (d x + c\right ) - 192 \, C a^{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 = 2.42, size = 99, normalized size = 0.66 \begin {gather*} \frac {3 \, {\left (7 \, A + 8 \, B + 12 \, C\right )} a^{2} d x + {\left (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (7 \, A + 8 \, B + 4 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (4 \, A + 5 \, B + 6 \, C\right )} a^{2}\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 [A]
time = 0.48, size = 248, normalized size = 1.66 \begin {gather*} \frac {3 \, {\left (7 \, A a^{2} + 8 \, B a^{2} + 12 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (21 \, 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} + 36 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 77 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 88 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 132 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 83 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 136 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 156 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a^{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 = 3.10, size = 174, normalized size = 1.17 \begin {gather*} \frac {7\,A\,a^2\,x}{8}+B\,a^2\,x+\frac {3\,C\,a^2\,x}{2}+\frac {3\,A\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {7\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {A\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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