Optimal. Leaf size=75 \[ \frac{\frac{a}{b^2}+\frac{1}{a}}{d (a \cot (c+d x)+b)}-\frac{2 a \log (\tan (c+d x))}{b^3 d}-\frac{2 a \log (a \cot (c+d x)+b)}{b^3 d}+\frac{\tan (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.0962912, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 894} \[ \frac{\frac{a}{b^2}+\frac{1}{a}}{d (a \cot (c+d x)+b)}-\frac{2 a \log (\tan (c+d x))}{b^3 d}-\frac{2 a \log (a \cot (c+d x)+b)}{b^3 d}+\frac{\tan (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 894
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^2 (b+a x)^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^2 x^2}-\frac{2 a}{b^3 x}+\frac{a^2+b^2}{b^2 (b+a x)^2}+\frac{2 a^2}{b^3 (b+a x)}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{\frac{1}{a}+\frac{a}{b^2}}{d (b+a \cot (c+d x))}-\frac{2 a \log (b+a \cot (c+d x))}{b^3 d}-\frac{2 a \log (\tan (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b^2 d}\\ \end{align*}
Mathematica [A] time = 0.25982, size = 51, normalized size = 0.68 \[ \frac{-\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.204, size = 78, normalized size = 1. \begin{align*}{\frac{\tan \left ( dx+c \right ) }{{b}^{2}d}}-2\,{\frac{a\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{b}^{3}d}}-{\frac{{a}^{2}}{{b}^{3}d \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{1}{db \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19236, size = 81, normalized size = 1.08 \begin{align*} -\frac{\frac{a^{2} + b^{2}}{b^{4} \tan \left (d x + c\right ) + a b^{3}} + \frac{2 \, a \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{3}} - \frac{\tan \left (d x + c\right )}{b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.535613, size = 440, normalized size = 5.87 \begin{align*} -\frac{2 \, b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - b^{2} +{\left (a^{2} \cos \left (d x + c\right )^{2} + a b \cos \left (d x + c\right ) \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^{2} \cos \left (d x + c\right )^{2} + a b \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right )}{a b^{3} d \cos \left (d x + c\right )^{2} + b^{4} d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (c + d x \right )}}{\left (a \cos{\left (c + d x \right )} + b \sin{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13786, size = 96, normalized size = 1.28 \begin{align*} -\frac{\frac{2 \, a \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{3}} - \frac{\tan \left (d x + c\right )}{b^{2}} - \frac{2 \, a b \tan \left (d x + c\right ) + a^{2} - b^{2}}{{\left (b \tan \left (d x + c\right ) + a\right )} b^{3}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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