Optimal. Leaf size=137 \[ \frac{\sec ^7(c+d x)}{7 a^3 d}-\frac{\sec ^6(c+d x)}{2 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{5 \sec ^4(c+d x)}{4 a^3 d}-\frac{5 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d} \]
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Rubi [A] time = 0.0768255, antiderivative size = 137, 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, 88} \[ \frac{\sec ^7(c+d x)}{7 a^3 d}-\frac{\sec ^6(c+d x)}{2 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{5 \sec ^4(c+d x)}{4 a^3 d}-\frac{5 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec ^2(c+d x)}{2 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{\log (\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^5 (a+a x)^2}{x^8} \, dx,x,\cos (c+d x)\right )}{a^{10} d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^7}{x^8}-\frac{3 a^7}{x^7}+\frac{a^7}{x^6}+\frac{5 a^7}{x^5}-\frac{5 a^7}{x^4}-\frac{a^7}{x^3}+\frac{3 a^7}{x^2}-\frac{a^7}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^{10} d}\\ &=\frac{\log (\cos (c+d x))}{a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}-\frac{\sec ^2(c+d x)}{2 a^3 d}-\frac{5 \sec ^3(c+d x)}{3 a^3 d}+\frac{5 \sec ^4(c+d x)}{4 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}-\frac{\sec ^6(c+d x)}{2 a^3 d}+\frac{\sec ^7(c+d x)}{7 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.316213, size = 140, normalized size = 1.02 \[ \frac{\sec ^7(c+d x) (4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))+2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))+630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (35 \log (\cos (c+d x))+8)+3732)}{6720 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 127, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{7}}{7\,d{a}^{3}}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{2\,d{a}^{3}}}+{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{5\,d{a}^{3}}}+{\frac{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{4\,d{a}^{3}}}-{\frac{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{3\,d{a}^{3}}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d{a}^{3}}}+3\,{\frac{\sec \left ( dx+c \right ) }{d{a}^{3}}}-{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08609, size = 122, normalized size = 0.89 \begin{align*} \frac{\frac{420 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} + \frac{1260 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{5} - 700 \, \cos \left (d x + c\right )^{4} + 525 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) + 60}{a^{3} \cos \left (d x + c\right )^{7}}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24943, size = 269, normalized size = 1.96 \begin{align*} \frac{420 \, \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) + 1260 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{5} - 700 \, \cos \left (d x + c\right )^{4} + 525 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) + 60}{420 \, a^{3} d \cos \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 36.512, size = 332, normalized size = 2.42 \begin{align*} -\frac{\frac{420 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} - \frac{420 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3}} - \frac{\frac{1393 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{819 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{6755 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{20195 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{28749 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{8463 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{1089 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 319}{a^{3}{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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