Optimal. Leaf size=65 \[ \frac{\sec ^4(c+d x)}{4 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}+\frac{\log (\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.0578504, antiderivative size = 65, 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 = {3879, 75} \[ \frac{\sec ^4(c+d x)}{4 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}+\frac{\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 75
Rubi steps
\begin{align*} \int \frac{\tan ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^3 (a+a x)}{x^5} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^4}{x^5}-\frac{2 a^4}{x^4}+\frac{2 a^4}{x^2}-\frac{a^4}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac{\log (\cos (c+d x))}{a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}+\frac{\sec ^4(c+d x)}{4 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.20201, size = 83, normalized size = 1.28 \[ \frac{\sec ^4(c+d x) (20 \cos (c+d x)+3 (4 \cos (3 (c+d x))+4 \cos (2 (c+d x)) \log (\cos (c+d x))+\cos (4 (c+d x)) \log (\cos (c+d x))+3 \log (\cos (c+d x))+2))}{24 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 63, normalized size = 1. \begin{align*}{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{4\,d{a}^{2}}}-{\frac{2\, \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{3\,d{a}^{2}}}+2\,{\frac{\sec \left ( dx+c \right ) }{d{a}^{2}}}-{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13033, size = 68, normalized size = 1.05 \begin{align*} \frac{\frac{12 \, \log \left (\cos \left (d x + c\right )\right )}{a^{2}} + \frac{24 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right ) + 3}{a^{2} \cos \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18856, size = 147, normalized size = 2.26 \begin{align*} \frac{12 \, \cos \left (d x + c\right )^{4} \log \left (-\cos \left (d x + c\right )\right ) + 24 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right ) + 3}{12 \, a^{2} d \cos \left (d x + c\right )^{4}} \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*} \frac{\int \frac{\tan ^{7}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 7.42592, size = 243, normalized size = 3.74 \begin{align*} -\frac{\frac{12 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac{12 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{2}} - \frac{\frac{4 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{54 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{124 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{25 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 7}{a^{2}{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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