Optimal. Leaf size=16 \[ a x-\frac {a \tan (c+d x)}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3767, 8} \[ a x-\frac {a \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \left (a-a \sec ^2(c+d x)\right ) \, dx &=a x-a \int \sec ^2(c+d x) \, dx\\ &=a x+\frac {a \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a x-\frac {a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.62 \[ -a \left (\frac {\tan (c+d x)}{d}-\frac {\tan ^{-1}(\tan (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 32, normalized size = 2.00 \[ \frac {a d x \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 16, normalized size = 1.00 \[ a x - \frac {a \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.78, size = 17, normalized size = 1.06 \[ a x -\frac {a \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 16, normalized size = 1.00 \[ a x - \frac {a \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.44, size = 16, normalized size = 1.00 \[ a\,x-\frac {a\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
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 \[ - a \left (\int \left (-1\right )\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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