Optimal. Leaf size=121 \[ \frac{\left (a^2-b^2\right )^2}{a b^4 d (a+b \sec (c+d x))}+\frac{\left (3 a^2+b^2\right ) \left (a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\sec ^2(c+d x)}{2 b^2 d} \]
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Rubi [A] time = 0.104316, antiderivative size = 121, 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} \[ \frac{\left (a^2-b^2\right )^2}{a b^4 d (a+b \sec (c+d x))}+\frac{\left (3 a^2+b^2\right ) \left (a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\sec ^2(c+d x)}{2 b^2 d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{x (a+x)^2} \, dx,x,b \sec (c+d x)\right )}{b^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 a+\frac{b^4}{a^2 x}+x-\frac{\left (a^2-b^2\right )^2}{a (a+x)^2}+\frac{\left (a^2-b^2\right ) \left (3 a^2+b^2\right )}{a^2 (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^4 d}\\ &=-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\left (a^2-b^2\right ) \left (3 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\sec ^2(c+d x)}{2 b^2 d}+\frac{\left (a^2-b^2\right )^2}{a b^4 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.536959, size = 187, normalized size = 1.55 \[ \frac{b \left (a^2 b^2 \sec ^2(c+d x)-2 \left (a^2 \left (3 a^2-2 b^2\right ) \log (\cos (c+d x))+\left (2 a^2 b^2-3 a^4+b^4\right ) \log (a \cos (c+d x)+b)-2 a^2 b^2+3 a^4+b^4\right )-3 a^3 b \sec (c+d x)\right )-2 a \cos (c+d x) \left (a^2 \left (3 a^2-2 b^2\right ) \log (\cos (c+d x))+\left (2 a^2 b^2-3 a^4+b^4\right ) \log (a \cos (c+d x)+b)\right )}{2 a^2 b^4 d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 192, normalized size = 1.6 \begin{align*} -{\frac{{a}^{2}}{d{b}^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+2\,{\frac{1}{db \left ( b+a\cos \left ( dx+c \right ) \right ) }}-{\frac{b}{d{a}^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+3\,{\frac{{a}^{2}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{4}}}-2\,{\frac{\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}-{\frac{\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{2}}}-3\,{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{4}}}+2\,{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}+{\frac{1}{2\,d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{a}{d{b}^{3}\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989591, size = 201, normalized size = 1.66 \begin{align*} -\frac{\frac{3 \, a^{3} b \cos \left (d x + c\right ) - a^{2} b^{2} + 2 \,{\left (3 \, a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}}{a^{3} b^{3} \cos \left (d x + c\right )^{3} + a^{2} b^{4} \cos \left (d x + c\right )^{2}} + \frac{2 \,{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{4}} - \frac{2 \,{\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} b^{4}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26654, size = 485, normalized size = 4.01 \begin{align*} -\frac{3 \, a^{3} b^{2} \cos \left (d x + c\right ) - a^{2} b^{3} + 2 \,{\left (3 \, a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} - 2 \,{\left ({\left (3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + 2 \,{\left ({\left (3 \, a^{5} - 2 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, a^{4} b - 2 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right )}{2 \,{\left (a^{3} b^{4} d \cos \left (d x + c\right )^{3} + a^{2} b^{5} d \cos \left (d x + c\right )^{2}\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 ^{5}{\left (c + d x \right )}}{\left (a + b \sec{\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 [B] time = 3.70442, size = 767, normalized size = 6.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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