Optimal. Leaf size=59 \[ -\frac{\left (a^2-b^2\right ) \log (a+b \sec (c+d x))}{a b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{b d} \]
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Rubi [A] time = 0.0710039, 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 = {3885, 894} \[ -\frac{\left (a^2-b^2\right ) \log (a+b \sec (c+d x))}{a b^2 d}+\frac{\log (\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x)}{a+b \sec (c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{b^2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-1+\frac{b^2}{a x}+\frac{a^2-b^2}{a (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^2 d}\\ &=\frac{\log (\cos (c+d x))}{a d}-\frac{\left (a^2-b^2\right ) \log (a+b \sec (c+d x))}{a b^2 d}+\frac{\sec (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.120668, size = 52, normalized size = 0.88 \[ \frac{\left (b^2-a^2\right ) \log (a \cos (c+d x)+b)+a^2 \log (\cos (c+d x))+a b \sec (c+d x)}{a b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 70, normalized size = 1.2 \begin{align*} -{\frac{a\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}+{\frac{\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{ad}}+{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}+{\frac{1}{db\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971969, size = 77, normalized size = 1.31 \begin{align*} \frac{\frac{a \log \left (\cos \left (d x + c\right )\right )}{b^{2}} - \frac{{\left (a^{2} - b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a b^{2}} + \frac{1}{b \cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.929986, size = 161, normalized size = 2.73 \begin{align*} \frac{a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right ) \log \left (a \cos \left (d x + c\right ) + b\right ) + a b}{a b^{2} d \cos \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{\tan ^{3}{\left (c + d x \right )}}{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.97962, size = 267, normalized size = 4.53 \begin{align*} -\frac{\frac{{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \log \left ({\left | a + b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{2} b^{2} - a b^{3}} + \frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac{a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{b^{2}} + \frac{a - 2 \, b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{b^{2}{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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