Optimal. Leaf size=17 \[ -\frac {a \log (1-\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2746, 31}
\begin {gather*} -\frac {a \log (1-\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2746
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {a \text {Subst}\left (\int \frac {1}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \log (1-\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.53 \begin {gather*} \frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 16, normalized size = 0.94
method | result | size |
derivativedivides | \(-\frac {a \ln \left (\sin \left (d x +c \right )-1\right )}{d}\) | \(16\) |
default | \(-\frac {a \ln \left (\sin \left (d x +c \right )-1\right )}{d}\) | \(16\) |
risch | \(i a x +\frac {2 i a c}{d}-\frac {2 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(34\) |
norman | \(\frac {a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 0.88 \begin {gather*} -\frac {a \log \left (\sin \left (d x + c\right ) - 1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 17, normalized size = 1.00 \begin {gather*} -\frac {a \log \left (-\sin \left (d x + c\right ) + 1\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 {\left (c + d x \right )}\, dx + \int \sec {\left (c + d x \right )}\, dx\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. 37 vs.
\(2 (17) = 34\).
time = 4.06, size = 37, normalized size = 2.18 \begin {gather*} \frac {a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} -\frac {a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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