Optimal. Leaf size=30 \[ -\frac{\cot ^3(c+d x)}{3 b d (a \cot (c+d x)+b)^3} \]
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Rubi [A] time = 0.0528595, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 37} \[ -\frac{\cot ^3(c+d x)}{3 b d (a \cot (c+d x)+b)^3} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 37
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{(b+a x)^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\cot ^3(c+d x)}{3 b d (b+a \cot (c+d x))^3}\\ \end{align*}
Mathematica [B] time = 0.662484, size = 124, normalized size = 4.13 \[ \frac{-6 a b \left (a^2+b^2\right ) \cos (c+d x)+\left (2 a b^3-6 a^3 b\right ) \cos (3 (c+d x))+2 \left (a^2-b^2\right ) \sin (c+d x) \left (\left (3 a^2-b^2\right ) \cos (2 (c+d x))+3 a^2+b^2\right )}{12 a d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 21, normalized size = 0.7 \begin{align*} -{\frac{1}{3\,db \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17786, size = 72, normalized size = 2.4 \begin{align*} -\frac{1}{3 \,{\left (b^{4} \tan \left (d x + c\right )^{3} + 3 \, a b^{3} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b^{2} \tan \left (d x + c\right ) + a^{3} b\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.526365, size = 552, normalized size = 18.4 \begin{align*} -\frac{{\left (9 \, a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right ) -{\left (a^{3} b^{2} - 3 \, a b^{4} +{\left (3 \, a^{5} - 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \,{\left ({\left (a^{9} - 6 \, a^{5} b^{4} - 8 \, a^{3} b^{6} - 3 \, a b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) +{\left ({\left (3 \, a^{8} b + 8 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - b^{9}\right )} d \cos \left (d x + c\right )^{2} +{\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11551, size = 27, normalized size = 0.9 \begin{align*} -\frac{1}{3 \,{\left (b \tan \left (d x + c\right ) + a\right )}^{3} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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