Optimal. Leaf size=151 \[ \frac{a \sec ^9(c+d x)}{9 d}+\frac{a \sec ^8(c+d x)}{8 d}-\frac{4 a \sec ^7(c+d x)}{7 d}-\frac{2 a \sec ^6(c+d x)}{3 d}+\frac{6 a \sec ^5(c+d x)}{5 d}+\frac{3 a \sec ^4(c+d x)}{2 d}-\frac{4 a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0722948, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3879, 88} \[ \frac{a \sec ^9(c+d x)}{9 d}+\frac{a \sec ^8(c+d x)}{8 d}-\frac{4 a \sec ^7(c+d x)}{7 d}-\frac{2 a \sec ^6(c+d x)}{3 d}+\frac{6 a \sec ^5(c+d x)}{5 d}+\frac{3 a \sec ^4(c+d x)}{2 d}-\frac{4 a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec ^2(c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \tan ^9(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^4 (a+a x)^5}{x^{10}} \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^9}{x^{10}}+\frac{a^9}{x^9}-\frac{4 a^9}{x^8}-\frac{4 a^9}{x^7}+\frac{6 a^9}{x^6}+\frac{6 a^9}{x^5}-\frac{4 a^9}{x^4}-\frac{4 a^9}{x^3}+\frac{a^9}{x^2}+\frac{a^9}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac{a \log (\cos (c+d x))}{d}+\frac{a \sec (c+d x)}{d}-\frac{2 a \sec ^2(c+d x)}{d}-\frac{4 a \sec ^3(c+d x)}{3 d}+\frac{3 a \sec ^4(c+d x)}{2 d}+\frac{6 a \sec ^5(c+d x)}{5 d}-\frac{2 a \sec ^6(c+d x)}{3 d}-\frac{4 a \sec ^7(c+d x)}{7 d}+\frac{a \sec ^8(c+d x)}{8 d}+\frac{a \sec ^9(c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.457898, size = 134, normalized size = 0.89 \[ \frac{a \sec ^9(c+d x)}{9 d}-\frac{4 a \sec ^7(c+d x)}{7 d}+\frac{6 a \sec ^5(c+d x)}{5 d}-\frac{4 a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}-\frac{a \left (-3 \tan ^8(c+d x)+4 \tan ^6(c+d x)-6 \tan ^4(c+d x)+12 \tan ^2(c+d x)+24 \log (\cos (c+d x))\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 273, normalized size = 1.8 \begin{align*}{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{8}}{8\,d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{9\,d \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{63\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{105\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{63\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{9\,d\cos \left ( dx+c \right ) }}+{\frac{128\,a\cos \left ( dx+c \right ) }{315\,d}}+{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{8}a}{9\,d}}+{\frac{8\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{63\,d}}+{\frac{16\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{105\,d}}+{\frac{64\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{315\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08038, size = 157, normalized size = 1.04 \begin{align*} -\frac{2520 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac{2520 \, a \cos \left (d x + c\right )^{8} - 5040 \, a \cos \left (d x + c\right )^{7} - 3360 \, a \cos \left (d x + c\right )^{6} + 3780 \, a \cos \left (d x + c\right )^{5} + 3024 \, a \cos \left (d x + c\right )^{4} - 1680 \, a \cos \left (d x + c\right )^{3} - 1440 \, a \cos \left (d x + c\right )^{2} + 315 \, a \cos \left (d x + c\right ) + 280 \, a}{\cos \left (d x + c\right )^{9}}}{2520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08978, size = 362, normalized size = 2.4 \begin{align*} -\frac{2520 \, a \cos \left (d x + c\right )^{9} \log \left (-\cos \left (d x + c\right )\right ) - 2520 \, a \cos \left (d x + c\right )^{8} + 5040 \, a \cos \left (d x + c\right )^{7} + 3360 \, a \cos \left (d x + c\right )^{6} - 3780 \, a \cos \left (d x + c\right )^{5} - 3024 \, a \cos \left (d x + c\right )^{4} + 1680 \, a \cos \left (d x + c\right )^{3} + 1440 \, a \cos \left (d x + c\right )^{2} - 315 \, a \cos \left (d x + c\right ) - 280 \, a}{2520 \, d \cos \left (d x + c\right )^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 53.3889, size = 184, normalized size = 1.22 \begin{align*} \begin{cases} \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \tan ^{8}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{9 d} + \frac{a \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac{8 a \tan ^{6}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{63 d} - \frac{a \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac{16 a \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{105 d} + \frac{a \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{64 a \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{315 d} - \frac{a \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{128 a \sec{\left (c + d x \right )}}{315 d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right ) \tan ^{9}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 17.8114, size = 396, normalized size = 2.62 \begin{align*} \frac{2520 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{9177 \, a + \frac{87633 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{375732 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{953988 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1594782 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1336734 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{781956 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{302004 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{69201 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{7129 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{9}}}{2520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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