Optimal. Leaf size=15 \[ a x+\frac {a \sin (c+d x)}{d} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3872, 2717, 8}
\begin {gather*} \frac {a \sin (c+d x)}{d}+a x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2717
Rule 3872
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sec (c+d x)) \, dx &=a \int 1 \, dx+a \int \cos (c+d x) \, dx\\ &=a x+\frac {a \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.73 \begin {gather*} a x+\frac {a \cos (d x) \sin (c)}{d}+\frac {a \cos (c) \sin (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 21, normalized size = 1.40
method | result | size |
risch | \(a x +\frac {a \sin \left (d x +c \right )}{d}\) | \(16\) |
derivativedivides | \(\frac {a \sin \left (d x +c \right )+\left (d x +c \right ) a}{d}\) | \(21\) |
default | \(\frac {a \sin \left (d x +c \right )+\left (d x +c \right ) a}{d}\) | \(21\) |
norman | \(\frac {a x +a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 20, normalized size = 1.33 \begin {gather*} \frac {{\left (d x + c\right )} a + a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.61, size = 17, normalized size = 1.13 \begin {gather*} \frac {a d x + a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.13, size = 17, normalized size = 1.13 \begin {gather*} a x + a \left (\begin {cases} x \cos {\left (c \right )} & \text {for}\: d = 0 \\\frac {\sin {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (15) = 30\).
time = 0.43, size = 39, normalized size = 2.60 \begin {gather*} \frac {{\left (d x + c\right )} a + \frac {2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 15, normalized size = 1.00 \begin {gather*} a\,x+\frac {a\,\sin \left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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