Optimal. Leaf size=115 \[ \frac{a^2 \sec ^4(c+d x)}{4 d}-\frac{a^2 \sec ^2(c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^5(c+d x)}{5 d}-\frac{4 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.091351, antiderivative size = 131, normalized size of antiderivative = 1.14, 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 = {3885, 948} \[ \frac{\left (a^2-2 b^2\right ) \sec ^4(c+d x)}{4 d}-\frac{\left (2 a^2-b^2\right ) \sec ^2(c+d x)}{2 d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^5(c+d x)}{5 d}-\frac{4 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 948
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^2 \tan ^5(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^2 \left (b^2-x^2\right )^2}{x} \, dx,x,b \sec (c+d x)\right )}{b^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a b^4+\frac{a^2 b^4}{x}-b^2 \left (2 a^2-b^2\right ) x-4 a b^2 x^2+\left (a^2-2 b^2\right ) x^3+2 a x^4+x^5\right ) \, dx,x,b \sec (c+d x)\right )}{b^4 d}\\ &=-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{\left (2 a^2-b^2\right ) \sec ^2(c+d x)}{2 d}-\frac{4 a b \sec ^3(c+d x)}{3 d}+\frac{\left (a^2-2 b^2\right ) \sec ^4(c+d x)}{4 d}+\frac{2 a b \sec ^5(c+d x)}{5 d}+\frac{b^2 \sec ^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.257131, size = 105, normalized size = 0.91 \[ \frac{15 \left (a^2-2 b^2\right ) \sec ^4(c+d x)+30 \left (b^2-2 a^2\right ) \sec ^2(c+d x)-60 a^2 \log (\cos (c+d x))+24 a b \sec ^5(c+d x)-80 a b \sec ^3(c+d x)+120 a b \sec (c+d x)+10 b^2 \sec ^6(c+d x)}{60 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 197, normalized size = 1.7 \begin{align*}{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{15\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{5\,d\cos \left ( dx+c \right ) }}+{\frac{16\,a\cos \left ( dx+c \right ) b}{15\,d}}+{\frac{2\,a\cos \left ( dx+c \right ) b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{8\,a\cos \left ( dx+c \right ) b \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98191, size = 146, normalized size = 1.27 \begin{align*} -\frac{60 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac{120 \, a b \cos \left (d x + c\right )^{5} - 80 \, a b \cos \left (d x + c\right )^{3} - 30 \,{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 24 \, a b \cos \left (d x + c\right ) + 15 \,{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 10 \, b^{2}}{\cos \left (d x + c\right )^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.804059, size = 293, normalized size = 2.55 \begin{align*} -\frac{60 \, a^{2} \cos \left (d x + c\right )^{6} \log \left (-\cos \left (d x + c\right )\right ) - 120 \, a b \cos \left (d x + c\right )^{5} + 80 \, a b \cos \left (d x + c\right )^{3} + 30 \,{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} - 24 \, a b \cos \left (d x + c\right ) - 15 \,{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 10 \, b^{2}}{60 \, d \cos \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.82042, size = 189, normalized size = 1.64 \begin{align*} \begin{cases} \frac{a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{2 a b \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{5 d} - \frac{8 a b \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{15 d} + \frac{16 a b \sec{\left (c + d x \right )}}{15 d} + \frac{b^{2} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{6 d} - \frac{b^{2} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{6 d} + \frac{b^{2} \sec ^{2}{\left (c + d x \right )}}{6 d} & \text{for}\: d \neq 0 \\x \left (a + b \sec{\left (c \right )}\right )^{2} \tan ^{5}{\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 = 3.56158, size = 460, normalized size = 4. \begin{align*} \frac{60 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{147 \, a^{2} + 128 \, a b + \frac{1002 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{768 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{2925 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1920 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{4140 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1280 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{640 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{2925 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1002 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{147 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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