Optimal. Leaf size=120 \[ \frac {\sec ^6(c+d x)}{6 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {\sec ^4(c+d x)}{4 a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {\sec ^2(c+d x)}{2 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.00, 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} \[ \frac {\sec ^6(c+d x)}{6 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {\sec ^4(c+d x)}{4 a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {\sec ^2(c+d x)}{2 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^4 (a+a x)^2}{x^7} \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^6}{x^7}-\frac {2 a^6}{x^6}-\frac {a^6}{x^5}+\frac {4 a^6}{x^4}-\frac {a^6}{x^3}-\frac {2 a^6}{x^2}+\frac {a^6}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {\log (\cos (c+d x))}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {\sec ^2(c+d x)}{2 a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {\sec ^4(c+d x)}{4 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {\sec ^6(c+d x)}{6 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 125, normalized size = 1.04 \[ -\frac {\sec ^6(c+d x) (312 \cos (c+d x)+5 (28 \cos (3 (c+d x))+6 \cos (4 (c+d x))+12 \cos (5 (c+d x))+18 \cos (4 (c+d x)) \log (\cos (c+d x))+3 \cos (6 (c+d x)) \log (\cos (c+d x))+30 \log (\cos (c+d x))+9 \cos (2 (c+d x)) (5 \log (\cos (c+d x))+4)+14))}{480 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 85, normalized size = 0.71 \[ -\frac {60 \, \cos \left (d x + c\right )^{6} \log \left (-\cos \left (d x + c\right )\right ) + 120 \, \cos \left (d x + c\right )^{5} + 30 \, \cos \left (d x + c\right )^{4} - 80 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{2} + 24 \, \cos \left (d x + c\right ) - 10}{60 \, a^{2} d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 26.20, size = 223, normalized size = 1.86 \[ \frac {\frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{2}} + \frac {\frac {234 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1005 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2220 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2925 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1002 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {147 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 19}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 110, normalized size = 0.92 \[ \frac {\sec ^{6}\left (d x +c \right )}{6 a^{2} d}-\frac {2 \left (\sec ^{5}\left (d x +c \right )\right )}{5 a^{2} d}-\frac {\sec ^{4}\left (d x +c \right )}{4 a^{2} d}+\frac {4 \left (\sec ^{3}\left (d x +c \right )\right )}{3 a^{2} d}-\frac {\sec ^{2}\left (d x +c \right )}{2 a^{2} d}-\frac {2 \sec \left (d x +c \right )}{a^{2} d}+\frac {\ln \left (\sec \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 80, normalized size = 0.67 \[ -\frac {\frac {60 \, \log \left (\cos \left (d x + c\right )\right )}{a^{2}} + \frac {120 \, \cos \left (d x + c\right )^{5} + 30 \, \cos \left (d x + c\right )^{4} - 80 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{2} + 24 \, \cos \left (d x + c\right ) - 10}{a^{2} \cos \left (d x + c\right )^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.18, size = 193, normalized size = 1.61 \[ \frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^2\,d}+\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {54\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {32}{15}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tan ^{9}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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