Optimal. Leaf size=23 \[ \frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2748, 3852, 8}
\begin {gather*} \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 3852
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {a \sec (c+d x)}{d}+a \int \sec ^2(c+d x) \, dx\\ &=\frac {a \sec (c+d x)}{d}-\frac {a \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 24, normalized size = 1.04
method | result | size |
risch | \(\frac {2 a}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\) | \(21\) |
derivativedivides | \(\frac {\frac {a}{\cos \left (d x +c \right )}+a \tan \left (d x +c \right )}{d}\) | \(24\) |
default | \(\frac {\frac {a}{\cos \left (d x +c \right )}+a \tan \left (d x +c \right )}{d}\) | \(24\) |
norman | \(\frac {-\frac {2 a}{d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 23, normalized size = 1.00 \begin {gather*} \frac {a \tan \left (d x + c\right ) + \frac {a}{\cos \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 = 0.34, size = 40, normalized size = 1.74 \begin {gather*} \frac {a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + 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 \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{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 = 3.70, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2 \, a}{d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.69, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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