Optimal. Leaf size=99 \[ \frac {\sec ^5(c+d x)}{5 a^3 d}-\frac {3 \sec ^4(c+d x)}{4 a^3 d}+\frac {2 \sec ^3(c+d x)}{3 a^3 d}+\frac {\sec ^2(c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\log (\cos (c+d x))}{a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 99, 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, 75} \[ \frac {\sec ^5(c+d x)}{5 a^3 d}-\frac {3 \sec ^4(c+d x)}{4 a^3 d}+\frac {2 \sec ^3(c+d x)}{3 a^3 d}+\frac {\sec ^2(c+d x)}{a^3 d}-\frac {3 \sec (c+d x)}{a^3 d}-\frac {\log (\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 75
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^4 (a+a x)}{x^6} \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^5}{x^6}-\frac {3 a^5}{x^5}+\frac {2 a^5}{x^4}+\frac {2 a^5}{x^3}-\frac {3 a^5}{x^2}+\frac {a^5}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^8 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)}{a^3 d}+\frac {2 \sec ^3(c+d x)}{3 a^3 d}-\frac {3 \sec ^4(c+d x)}{4 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 93, normalized size = 0.94 \[ -\frac {\sec ^5(c+d x) (280 \cos (2 (c+d x))+90 \cos (4 (c+d x))+150 \cos (c+d x) \log (\cos (c+d x))+15 \cos (5 (c+d x)) \log (\cos (c+d x))+15 \cos (3 (c+d x)) (5 \log (\cos (c+d x))-4)+142)}{240 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 75, normalized size = 0.76 \[ -\frac {60 \, \cos \left (d x + c\right )^{5} \log \left (-\cos \left (d x + c\right )\right ) + 180 \, \cos \left (d x + c\right )^{4} - 60 \, \cos \left (d x + c\right )^{3} - 40 \, \cos \left (d x + c\right )^{2} + 45 \, \cos \left (d x + c\right ) - 12}{60 \, a^{3} d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 20.75, size = 202, normalized size = 2.04 \[ \frac {\frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} - \frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3}} - \frac {\frac {475 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {590 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {50 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {805 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 119}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 93, normalized size = 0.94 \[ \frac {\sec ^{5}\left (d x +c \right )}{5 a^{3} d}-\frac {3 \left (\sec ^{4}\left (d x +c \right )\right )}{4 a^{3} d}+\frac {2 \left (\sec ^{3}\left (d x +c \right )\right )}{3 a^{3} d}+\frac {\sec ^{2}\left (d x +c \right )}{a^{3} d}-\frac {3 \sec \left (d x +c \right )}{a^{3} d}+\frac {\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 70, normalized size = 0.71 \[ -\frac {\frac {60 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} + \frac {180 \, \cos \left (d x + c\right )^{4} - 60 \, \cos \left (d x + c\right )^{3} - 40 \, \cos \left (d x + c\right )^{2} + 45 \, \cos \left (d x + c\right ) - 12}{a^{3} \cos \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.92, size = 167, normalized size = 1.69 \[ \frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^3\,d}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {98\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {58\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {64}{15}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )} \]
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 \[ \frac {\int \frac {\tan ^{9}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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