Optimal. Leaf size=45 \[ \frac {3 a^2 x}{2}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3873, 2717,
4130, 8} \begin {gather*} \frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3 a^2 x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2717
Rule 3873
Rule 4130
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx &=\left (2 a^2\right ) \int \cos (c+d x) \, dx+\int \cos ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac {3 a^2 x}{2}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 34, normalized size = 0.76 \begin {gather*} \frac {a^2 (6 (c+d x)+8 \sin (c+d x)+\sin (2 (c+d x)))}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 52, normalized size = 1.16
method | result | size |
risch | \(\frac {3 a^{2} x}{2}+\frac {2 a^{2} \sin \left (d x +c \right )}{d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{4 d}\) | \(39\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} \sin \left (d x +c \right )+a^{2} \left (d x +c \right )}{d}\) | \(52\) |
default | \(\frac {a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} \sin \left (d x +c \right )+a^{2} \left (d x +c \right )}{d}\) | \(52\) |
norman | \(\frac {-\frac {3 a^{2} x}{2}-\frac {5 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 48, normalized size = 1.07 \begin {gather*} \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 4 \, {\left (d x + c\right )} a^{2} + 8 \, 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 = 3.19, size = 36, normalized size = 0.80 \begin {gather*} \frac {3 \, a^{2} d x + {\left (a^{2} \cos \left (d x + c\right ) + 4 \, 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 2 \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{2}{\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 = 0.44, size = 64, normalized size = 1.42 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a^{2} + \frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, 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 = 1.07, size = 57, normalized size = 1.27 \begin {gather*} \frac {3\,a^2\,x}{2}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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