Optimal. Leaf size=82 \[ -\frac{b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.136527, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3086, 3483, 3531, 3530} \[ -\frac{b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3086
Rule 3483
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx &=\int \frac{1}{(a+b \tan (c+d x))^2} \, dx\\ &=-\frac{b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{\int \frac{a-b \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{(2 a b) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac{2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.389831, size = 192, normalized size = 2.34 \[ \frac{b \sin (c+d x) \left (a^2 b \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )+(a+i b) \left (a^2 (c+d x)+a b (i c+i d x+1)-i b^2\right )\right )+a^2 \cos (c+d x) \left (a b \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )+(a+i b)^2 (c+d x)\right )-2 i a^2 b \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.155, size = 130, normalized size = 1.6 \begin{align*} -{\frac{b}{ \left ({a}^{2}+{b}^{2} \right ) d \left ( a+b\tan \left ( dx+c \right ) \right ) }}+2\,{\frac{ab\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{ab\ln \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.79684, size = 177, normalized size = 2.16 \begin{align*} \frac{\frac{2 \, a b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{b}{a^{3} + a b^{2} +{\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.518388, size = 389, normalized size = 4.74 \begin{align*} -\frac{{\left (b^{3} -{\left (a^{3} - a b^{2}\right )} d x\right )} \cos \left (d x + c\right ) -{\left (a^{2} b \cos \left (d x + c\right ) + a b^{2} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) -{\left (a b^{2} +{\left (a^{2} b - b^{3}\right )} d x\right )} \sin \left (d x + c\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cos \left (d x + c\right ) +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13989, size = 215, normalized size = 2.62 \begin{align*} \frac{\frac{2 \, a b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac{a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{2 \, a b^{2} \tan \left (d x + c\right ) + 3 \, a^{2} b + b^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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