Optimal. Leaf size=17 \[ -\frac {a \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.02, 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 = {2667, 31} \[ -\frac {a \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {a \operatorname {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 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 17, normalized size = 1.00 \[ -\frac {a \log \left (-\sin \left (d x + c\right ) + 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.94, size = 37, normalized size = 2.18 \[ \frac {a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 16, normalized size = 0.94 \[ -\frac {a \ln \left (\sin \left (d x +c \right )-1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 15, normalized size = 0.88 \[ -\frac {a \log \left (\sin \left (d x + c\right ) - 1\right )}{d} \]
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 \[ -\frac {a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{d} \]
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 \[ a \left (\int \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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