Optimal. Leaf size=32 \[ \frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3075} \[ \frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3075
Rubi steps
\begin {align*} \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx &=\frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 1.00 \[ \frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 57, normalized size = 1.78 \[ -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{{\left (a^{3} + a b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{2} b + b^{3}\right )} d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 20, normalized size = 0.62 \[ -\frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 21, normalized size = 0.66 \[ -\frac {1}{d b \left (a +b \tan \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 21, normalized size = 0.66 \[ -\frac {1}{{\left (b^{2} \tan \left (d x + c\right ) + a b\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 47, normalized size = 1.47 \[ \frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )} \]
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 \[ \int \frac {1}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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