Optimal. Leaf size=99 \[ \frac{a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{4 d}+\frac{2 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.066601, antiderivative size = 99, 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, 75} \[ \frac{a^3 \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec ^4(c+d x)}{4 d}+\frac{2 a^3 \sec ^3(c+d x)}{3 d}-\frac{a^3 \sec ^2(c+d x)}{d}-\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 75
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^3 \tan ^3(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x) (a+a x)^4}{x^6} \, dx,x,\cos (c+d x)\right )}{a^2 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^2 d}\\ &=\frac{a^3 \log (\cos (c+d x))}{d}-\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \sec ^2(c+d x)}{d}+\frac{2 a^3 \sec ^3(c+d x)}{3 d}+\frac{3 a^3 \sec ^4(c+d x)}{4 d}+\frac{a^3 \sec ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.299458, size = 92, normalized size = 0.93 \[ -\frac{a^3 \sec ^5(c+d x) (280 \cos (2 (c+d x))+90 \cos (4 (c+d x))+\cos (3 (c+d x)) (60-75 \log (\cos (c+d x)))-150 \cos (c+d x) \log (\cos (c+d x))-15 \cos (5 (c+d x)) \log (\cos (c+d x))+142)}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 164, normalized size = 1.7 \begin{align*}{\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{16\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{15\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{16\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{15\,d\cos \left ( dx+c \right ) }}-{\frac{16\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{15\,d}}-{\frac{32\,{a}^{3}\cos \left ( dx+c \right ) }{15\,d}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.32302, size = 113, normalized size = 1.14 \begin{align*} \frac{60 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac{180 \, a^{3} \cos \left (d x + c\right )^{4} + 60 \, a^{3} \cos \left (d x + c\right )^{3} - 40 \, a^{3} \cos \left (d x + c\right )^{2} - 45 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3}}{\cos \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01575, size = 232, normalized size = 2.34 \begin{align*} \frac{60 \, a^{3} \cos \left (d x + c\right )^{5} \log \left (-\cos \left (d x + c\right )\right ) - 180 \, a^{3} \cos \left (d x + c\right )^{4} - 60 \, a^{3} \cos \left (d x + c\right )^{3} + 40 \, a^{3} \cos \left (d x + c\right )^{2} + 45 \, a^{3} \cos \left (d x + c\right ) + 12 \, a^{3}}{60 \, d \cos \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.80314, size = 165, normalized size = 1.67 \begin{align*} \begin{cases} - \frac{a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{5 d} + \frac{3 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{4 d} + \frac{a^{3} \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{d} + \frac{a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac{2 a^{3} \sec ^{3}{\left (c + d x \right )}}{15 d} - \frac{3 a^{3} \sec ^{2}{\left (c + d x \right )}}{4 d} - \frac{2 a^{3} \sec{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right )^{3} \tan ^{3}{\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 = 2.07299, size = 293, normalized size = 2.96 \begin{align*} -\frac{60 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{393 \, a^{3} + \frac{2085 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{2610 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1970 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{805 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{137 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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