Optimal. Leaf size=77 \[ \frac {1}{2} a (B+C) x+\frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4157, 4081,
3872, 2715, 8, 2717} \begin {gather*} \frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a B \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{2} a x (B+C) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4081
Rule 4157
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^3(c+d x) (a+a \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) (-3 a (B+C)-a (2 B+3 C) \sec (c+d x)) \, dx\\ &=\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}+(a (B+C)) \int \cos ^2(c+d x) \, dx+\frac {1}{3} (a (2 B+3 C)) \int \cos (c+d x) \, dx\\ &=\frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {1}{2} (a (B+C)) \int 1 \, dx\\ &=\frac {1}{2} a (B+C) x+\frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 65, normalized size = 0.84 \begin {gather*} \frac {a (6 B c+6 c C+6 B d x+6 C d x+3 (3 B+4 C) \sin (c+d x)+3 (B+C) \sin (2 (c+d x))+B \sin (3 (c+d x)))}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.07, size = 85, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \sin \left (d x +c \right )}{d}\) | \(85\) |
default | \(\frac {\frac {B a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \sin \left (d x +c \right )}{d}\) | \(85\) |
risch | \(\frac {a B x}{2}+\frac {a x C}{2}+\frac {3 \sin \left (d x +c \right ) B a}{4 d}+\frac {\sin \left (d x +c \right ) a C}{d}+\frac {\sin \left (3 d x +3 c \right ) B a}{12 d}+\frac {B a \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a C}{4 d}\) | \(85\) |
norman | \(\frac {\left (\frac {1}{2} B a +\frac {1}{2} a C \right ) x +\left (-\frac {1}{2} B a -\frac {1}{2} a C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} B a -\frac {1}{2} a C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} B a +\frac {1}{2} a C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 B a -2 a C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B a +a C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B a +a C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a \left (B +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (B -3 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {3 a \left (B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (B +9 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a \left (5 B -3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a \left (5 B +9 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}\) | \(301\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 79, normalized size = 1.03 \begin {gather*} -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 12 \, C a \sin \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.89, size = 56, normalized size = 0.73 \begin {gather*} \frac {3 \, {\left (B + C\right )} a d x + {\left (2 \, B a \cos \left (d x + c\right )^{2} + 3 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + 2 \, {\left (2 \, B + 3 \, C\right )} a\right )} \sin \left (d x + c\right )}{6 \, 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 = 0.46, size = 124, normalized size = 1.61 \begin {gather*} \frac {3 \, {\left (B a + C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a \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}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.86, size = 84, normalized size = 1.09 \begin {gather*} \frac {B\,a\,x}{2}+\frac {C\,a\,x}{2}+\frac {3\,B\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a\,\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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