Optimal. Leaf size=23 \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d} \]
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Rubi [A] time = 0.03, 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 = {2669, 3767, 8} \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2669
Rule 3767
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 \operatorname {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 \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 40, normalized size = 1.74 \[ \frac {a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 19, normalized size = 0.83 \[ -\frac {2 \, a}{d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 24, normalized size = 1.04 \[ \frac {\frac {a}{\cos \left (d x +c \right )}+a \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 23, normalized size = 1.00 \[ \frac {a \tan \left (d x + c\right ) + \frac {a}{\cos \left (d x + c\right )}}{d} \]
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 \[ -\frac {2\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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 \[ a \left (\int \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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