Optimal. Leaf size=27 \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d}-a x \]
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Rubi [A] time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2838, 2606, 8, 3473} \[ \frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d}-a x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 2838
Rule 3473
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx &=a \int \sec (c+d x) \tan (c+d x) \, dx+a \int \tan ^2(c+d x) \, dx\\ &=\frac {a \tan (c+d x)}{d}-a \int 1 \, dx+\frac {a \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-a x+\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 1.33 \[ -\frac {a \tan ^{-1}(\tan (c+d x))}{d}+\frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 60, normalized size = 2.22 \[ -\frac {a d x + {\left (a d x - a\right )} \cos \left (d x + c\right ) - {\left (a d x + a\right )} \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 29, normalized size = 1.07 \[ -\frac {{\left (d x + c\right )} a + \frac {2 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 32, normalized size = 1.19 \[ \frac {a \left (\tan \left (d x +c \right )-d x -c \right )+\frac {a}{\cos \left (d x +c \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 32, normalized size = 1.19 \[ -\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} a - \frac {a}{\cos \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.98, size = 24, normalized size = 0.89 \[ -a\,x-\frac {2\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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 \[ a \left (\int \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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