Optimal. Leaf size=77 \[ \frac {1}{2} a (A+2 C) x+\frac {a (2 A+3 C) \sin (c+d x)}{3 d}+\frac {a A \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4160, 4132,
2717, 4130, 8} \begin {gather*} \frac {a (2 A+3 C) \sin (c+d x)}{3 d}+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (A+2 C) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2717
Rule 4130
Rule 4132
Rule 4160
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) \left (-3 a A-a (2 A+3 C) \sec (c+d x)-3 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) \left (-3 a A-3 a C \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} (a (2 A+3 C)) \int \cos (c+d x) \, dx\\ &=\frac {a (2 A+3 C) \sin (c+d x)}{3 d}+\frac {a A \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {1}{2} (a (A+2 C)) \int 1 \, dx\\ &=\frac {1}{2} a (A+2 C) x+\frac {a (2 A+3 C) \sin (c+d x)}{3 d}+\frac {a A \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 59, normalized size = 0.77 \begin {gather*} \frac {a (6 A c+6 A d x+12 C d x+3 (3 A+4 C) \sin (c+d x)+3 A \sin (2 (c+d x))+A \sin (3 (c+d x)))}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 68, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {A a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \sin \left (d x +c \right )+a C \left (d x +c \right )}{d}\) | \(68\) |
default | \(\frac {\frac {A a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \sin \left (d x +c \right )+a C \left (d x +c \right )}{d}\) | \(68\) |
risch | \(\frac {a A x}{2}+a x C +\frac {3 a A \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) a C}{d}+\frac {A a \sin \left (3 d x +3 c \right )}{12 d}+\frac {A a \sin \left (2 d x +2 c \right )}{4 d}\) | \(68\) |
norman | \(\frac {\frac {a \left (A +2 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (3 A +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (A +2 C \right ) x}{2}-\frac {14 A a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 A a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 a \left (A -3 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a \left (A +2 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-a \left (A +2 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a \left (A +2 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a \left (A +2 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \left (A +2 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 67, normalized size = 0.87 \begin {gather*} -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 12 \, {\left (d x + c\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 = 2.74, size = 56, normalized size = 0.73 \begin {gather*} \frac {3 \, {\left (A + 2 \, C\right )} a d x + {\left (2 \, A a \cos \left (d x + c\right )^{2} + 3 \, A a \cos \left (d x + c\right ) + 2 \, {\left (2 \, A + 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a \left (\int A \cos ^{3}{\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 125, normalized size = 1.62 \begin {gather*} \frac {3 \, {\left (A a + 2 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A 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 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, 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 = 2.57, size = 67, normalized size = 0.87 \begin {gather*} \frac {A\,a\,x}{2}+C\,a\,x+\frac {3\,A\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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