Optimal. Leaf size=88 \[ \frac {1}{2} a^2 (3 B+4 C) x+\frac {a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac {B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4157, 4102,
4081, 3855} \begin {gather*} \frac {a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac {B \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d}+\frac {1}{2} a^2 x (3 B+4 C)+\frac {a^2 C \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 4081
Rule 4102
Rule 4157
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 (B+C \sec (c+d x)) \, dx\\ &=\frac {B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) (a+a \sec (c+d x)) (a (3 B+2 C)+2 a C \sec (c+d x)) \, dx\\ &=\frac {a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac {B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^2 (3 B+4 C)-2 a^2 C \sec (c+d x)\right ) \, dx\\ &=\frac {1}{2} a^2 (3 B+4 C) x+\frac {a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac {B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}+\left (a^2 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^2 (3 B+4 C) x+\frac {a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac {B \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 96, normalized size = 1.09 \begin {gather*} \frac {a^2 \left (6 B d x+8 C d x-4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 (2 B+C) \sin (c+d x)+B \sin (2 (c+d x))\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 96, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {a^{2} B \left (d x +c \right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 a^{2} B \sin \left (d x +c \right )+2 a^{2} C \left (d x +c \right )+a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \sin \left (d x +c \right )}{d}\) | \(96\) |
default | \(\frac {a^{2} B \left (d x +c \right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 a^{2} B \sin \left (d x +c \right )+2 a^{2} C \left (d x +c \right )+a^{2} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \sin \left (d x +c \right )}{d}\) | \(96\) |
risch | \(\frac {3 a^{2} B x}{2}+2 a^{2} x C -\frac {i a^{2} B \,{\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{2} C}{2 d}+\frac {i a^{2} B \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{2} C}{2 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{2} B \sin \left (2 d x +2 c \right )}{4 d}\) | \(153\) |
norman | \(\frac {\left (-\frac {3}{2} a^{2} B -2 a^{2} C \right ) x +\left (-\frac {9}{2} a^{2} B -6 a^{2} C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} a^{2} B +2 a^{2} C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {9}{2} a^{2} B +6 a^{2} C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a^{2} \left (3 B +2 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a^{2} \left (7 B +2 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2} \left (B +2 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \left (3 B +2 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2} \left (5 B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{2} \left (5 B +2 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {a^{2} C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{2} C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(318\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 101, normalized size = 1.15 \begin {gather*} \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 4 \, {\left (d x + c\right )} B a^{2} + 8 \, {\left (d x + c\right )} C a^{2} + 2 \, C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, B a^{2} \sin \left (d x + c\right ) + 4 \, C a^{2} \sin \left (d x + c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.78, size = 79, normalized size = 0.90 \begin {gather*} \frac {{\left (3 \, B + 4 \, C\right )} a^{2} d x + C a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - C a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (B a^{2} \cos \left (d x + c\right ) + 2 \, {\left (2 \, B + C\right )} a^{2}\right )} \sin \left (d x + c\right )}{2 \, 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^{2} \left (\int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 2 B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{4}{\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 = 0.56, size = 145, normalized size = 1.65 \begin {gather*} \frac {2 \, C a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, C a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (3 \, B a^{2} + 4 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, 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 )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.98, size = 141, normalized size = 1.60 \begin {gather*} \frac {2\,B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,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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