Optimal. Leaf size=61 \[ -\frac{a^2+b^2}{b^3 d (a+b \tan (c+d x))}-\frac{2 a \log (a+b \tan (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.0660522, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3506, 697} \[ -\frac{a^2+b^2}{b^3 d (a+b \tan (c+d x))}-\frac{2 a \log (a+b \tan (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 697
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+\frac{x^2}{b^2}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^2}+\frac{a^2+b^2}{b^2 (a+x)^2}-\frac{2 a}{b^2 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac{2 a \log (a+b \tan (c+d x))}{b^3 d}+\frac{\tan (c+d x)}{b^2 d}-\frac{a^2+b^2}{b^3 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0607277, size = 51, normalized size = 0.84 \[ \frac{-\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 78, normalized size = 1.3 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{{b}^{2}d}}-{\frac{{a}^{2}}{d{b}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{1}{bd \left ( a+b\tan \left ( dx+c \right ) \right ) }}-2\,{\frac{a\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06808, size = 81, normalized size = 1.33 \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 = 2.00202, size = 440, normalized size = 7.21 \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 ^{4}{\left (c + d x \right )}}{\left (a + b \tan{\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 = 1.35014, size = 96, normalized size = 1.57 \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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