Optimal. Leaf size=137 \[ \frac{a^3 \sec ^9(c+d x)}{9 d}+\frac{3 a^3 \sec ^8(c+d x)}{8 d}-\frac{4 a^3 \sec ^6(c+d x)}{3 d}-\frac{6 a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{2 d}+\frac{8 a^3 \sec ^3(c+d x)}{3 d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0771652, 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} \[ \frac{a^3 \sec ^9(c+d x)}{9 d}+\frac{3 a^3 \sec ^8(c+d x)}{8 d}-\frac{4 a^3 \sec ^6(c+d x)}{3 d}-\frac{6 a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{2 d}+\frac{8 a^3 \sec ^3(c+d x)}{3 d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^3 \tan ^7(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^3 (a+a x)^6}{x^{10}} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^9}{x^{10}}+\frac{3 a^9}{x^9}-\frac{8 a^9}{x^7}-\frac{6 a^9}{x^6}+\frac{6 a^9}{x^5}+\frac{8 a^9}{x^4}-\frac{3 a^9}{x^2}-\frac{a^9}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac{a^3 \log (\cos (c+d x))}{d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{8 a^3 \sec ^3(c+d x)}{3 d}+\frac{3 a^3 \sec ^4(c+d x)}{2 d}-\frac{6 a^3 \sec ^5(c+d x)}{5 d}-\frac{4 a^3 \sec ^6(c+d x)}{3 d}+\frac{3 a^3 \sec ^8(c+d x)}{8 d}+\frac{a^3 \sec ^9(c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.392007, size = 110, normalized size = 0.8 \[ \frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (40 \sec ^9(c+d x)+135 \sec ^8(c+d x)-480 \sec ^6(c+d x)-432 \sec ^5(c+d x)+540 \sec ^4(c+d x)+960 \sec ^3(c+d x)-1080 \sec (c+d x)+360 \log (\cos (c+d x))\right )}{2880 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 288, normalized size = 2.1 \begin{align*}{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{4\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{9\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}-{\frac{4\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{45\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{4\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{45\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{9\,d\cos \left ( dx+c \right ) }}-{\frac{64\,{a}^{3}\cos \left ( dx+c \right ) }{45\,d}}-{\frac{4\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{9\,d}}-{\frac{8\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{15\,d}}-{\frac{32\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{45\,d}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{9\,d \left ( \cos \left ( dx+c \right ) \right ) ^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40997, size = 149, normalized size = 1.09 \begin{align*} \frac{360 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac{1080 \, a^{3} \cos \left (d x + c\right )^{8} - 960 \, a^{3} \cos \left (d x + c\right )^{6} - 540 \, a^{3} \cos \left (d x + c\right )^{5} + 432 \, a^{3} \cos \left (d x + c\right )^{4} + 480 \, a^{3} \cos \left (d x + c\right )^{3} - 135 \, a^{3} \cos \left (d x + c\right ) - 40 \, a^{3}}{\cos \left (d x + c\right )^{9}}}{360 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06437, size = 308, normalized size = 2.25 \begin{align*} \frac{360 \, a^{3} \cos \left (d x + c\right )^{9} \log \left (-\cos \left (d x + c\right )\right ) - 1080 \, a^{3} \cos \left (d x + c\right )^{8} + 960 \, a^{3} \cos \left (d x + c\right )^{6} + 540 \, a^{3} \cos \left (d x + c\right )^{5} - 432 \, a^{3} \cos \left (d x + c\right )^{4} - 480 \, a^{3} \cos \left (d x + c\right )^{3} + 135 \, a^{3} \cos \left (d x + c\right ) + 40 \, a^{3}}{360 \, 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 = 60.509, size = 350, normalized size = 2.55 \begin{align*} \begin{cases} - \frac{a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{9 d} + \frac{3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} + \frac{3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{7 d} + \frac{a^{3} \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac{2 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{21 d} - \frac{3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac{18 a^{3} \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{35 d} - \frac{a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac{8 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{105 d} + \frac{3 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} + \frac{24 a^{3} \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{35 d} + \frac{a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac{16 a^{3} \sec ^{3}{\left (c + d x \right )}}{315 d} - \frac{3 a^{3} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac{48 a^{3} \sec{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right )^{3} \tan ^{7}{\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 = 9.10637, size = 428, normalized size = 3.12 \begin{align*} -\frac{2520 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{14297 \, a^{3} + \frac{133713 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{560052 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1384068 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1594782 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1336734 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{781956 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{302004 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{69201 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{7129 \, a^{3}{\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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