Optimal. Leaf size=118 \[ \frac{a \sec ^7(c+d x)}{7 d}+\frac{a \sec ^6(c+d x)}{6 d}-\frac{3 a \sec ^5(c+d x)}{5 d}-\frac{3 a \sec ^4(c+d x)}{4 d}+\frac{a \sec ^3(c+d x)}{d}+\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0619443, antiderivative size = 118, 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} \[ \frac{a \sec ^7(c+d x)}{7 d}+\frac{a \sec ^6(c+d x)}{6 d}-\frac{3 a \sec ^5(c+d x)}{5 d}-\frac{3 a \sec ^4(c+d x)}{4 d}+\frac{a \sec ^3(c+d x)}{d}+\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{a \sec (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^3 (a+a x)^4}{x^8} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^7}{x^8}+\frac{a^7}{x^7}-\frac{3 a^7}{x^6}-\frac{3 a^7}{x^5}+\frac{3 a^7}{x^4}+\frac{3 a^7}{x^3}-\frac{a^7}{x^2}-\frac{a^7}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac{a \log (\cos (c+d x))}{d}-\frac{a \sec (c+d x)}{d}+\frac{3 a \sec ^2(c+d x)}{2 d}+\frac{a \sec ^3(c+d x)}{d}-\frac{3 a \sec ^4(c+d x)}{4 d}-\frac{3 a \sec ^5(c+d x)}{5 d}+\frac{a \sec ^6(c+d x)}{6 d}+\frac{a \sec ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.442353, size = 106, normalized size = 0.9 \[ \frac{a \sec ^7(c+d x)}{7 d}-\frac{3 a \sec ^5(c+d x)}{5 d}+\frac{a \sec ^3(c+d x)}{d}-\frac{a \sec (c+d x)}{d}+\frac{a \left (2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 216, normalized size = 1.8 \begin{align*}{\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 ) ^{8}}{7\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{7\,d\cos \left ( dx+c \right ) }}-{\frac{16\,a\cos \left ( dx+c \right ) }{35\,d}}-{\frac{a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{7\,d}}-{\frac{6\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d}}-{\frac{8\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{35\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16596, size = 127, normalized size = 1.08 \begin{align*} \frac{420 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac{420 \, a \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{5} - 420 \, a \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{3} + 252 \, a \cos \left (d x + c\right )^{2} - 70 \, a \cos \left (d x + c\right ) - 60 \, a}{\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 = 0.999816, size = 284, normalized size = 2.41 \begin{align*} \frac{420 \, a \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 420 \, a \cos \left (d x + c\right )^{6} + 630 \, a \cos \left (d x + c\right )^{5} + 420 \, a \cos \left (d x + c\right )^{4} - 315 \, a \cos \left (d x + c\right )^{3} - 252 \, a \cos \left (d x + c\right )^{2} + 70 \, a \cos \left (d x + c\right ) + 60 \, a}{420 \, 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 [A] time = 15.0235, size = 148, normalized size = 1.25 \begin{align*} \begin{cases} - \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \tan ^{6}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{7 d} + \frac{a \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac{6 a \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{35 d} - \frac{a \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac{8 a \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{35 d} + \frac{a \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac{16 a \sec{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right ) \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 = 8.04382, size = 333, normalized size = 2.82 \begin{align*} -\frac{420 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{1473 \, a + \frac{11151 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{36813 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{69475 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{56035 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{28749 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{8463 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{1089 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\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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