Optimal. Leaf size=24 \[ -\frac {1}{d (a \sin (c+d x)+b \sec (c+d x))} \]
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Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {4385} \[ -\frac {1}{d (a \sin (c+d x)+b \sec (c+d x))} \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin {align*} \int \frac {a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{(b \sec (c+d x)+a \sin (c+d x))^2} \, dx &=-\frac {1}{d (b \sec (c+d x)+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 27, normalized size = 1.12 \[ -\frac {2 \cos (c+d x)}{d (a \sin (2 (c+d x))+2 b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 29, normalized size = 1.21 \[ -\frac {\cos \left (d x + c\right )}{a d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.73, size = 108, normalized size = 4.50 \[ \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b\right )}}{{\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 25, normalized size = 1.04 \[ -\frac {1}{d \left (b \sec \left (d x +c \right )+a \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 24, normalized size = 1.00 \[ -\frac {1}{{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.24, size = 47, normalized size = 1.96 \[ -\frac {b\,\left (\cos \left (c+d\,x\right )+1\right )+\frac {a\,\sin \left (2\,c+2\,d\,x\right )}{2}}{b\,d\,\left (b+\frac {a\,\sin \left (2\,c+2\,d\,x\right )}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.18, size = 49, normalized size = 2.04 \[ \begin {cases} - \frac {1}{a d \sin {\left (c + d x \right )} + b d \sec {\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right )}{\left (a \sin {\relax (c )} + b \sec {\relax (c )}\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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