Optimal. Leaf size=59 \[ \frac{\left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan ^2(c+d x)}{2 b d} \]
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Rubi [A] time = 0.0645737, antiderivative size = 59, 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} \[ \frac{\left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan ^2(c+d x)}{2 b d} \]
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)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+\frac{x^2}{b^2}}{a+x} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b^2}+\frac{a^2+b^2}{b^2 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan ^2(c+d x)}{2 b d}\\ \end{align*}
Mathematica [A] time = 0.126072, size = 52, normalized size = 0.88 \[ \frac{\left (a^2+b^2\right ) \log (a+b \tan (c+d x))-a b \tan (c+d x)+\frac{1}{2} b^2 \tan ^2(c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 72, normalized size = 1.2 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,bd}}-{\frac{a\tan \left ( dx+c \right ) }{{b}^{2}d}}+{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{3}}}+{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10986, size = 72, normalized size = 1.22 \begin{align*} \frac{\frac{b \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{b^{2}} + \frac{2 \,{\left (a^{2} + b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91328, size = 294, normalized size = 4.98 \begin{align*} \frac{{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} \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} + b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\cos \left (d x + c\right )^{2}\right ) - 2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b^{2}}{2 \, b^{3} d \cos \left (d x + c\right )^{2}} \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 )}}{a + b \tan{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50554, size = 73, normalized size = 1.24 \begin{align*} \frac{\frac{b \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{b^{2}} + \frac{2 \,{\left (a^{2} + b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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