Optimal. Leaf size=87 \[ \frac{\left (a^2-b^2\right ) \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.069152, antiderivative size = 87, 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 = {3885, 894} \[ \frac{\left (a^2-b^2\right ) \sec ^2(c+d x)}{2 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^2 \tan ^3(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a+x)^2 \left (b^2-x^2\right )}{x} \, dx,x,b \sec (c+d x)\right )}{b^2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (2 a b^2+\frac{a^2 b^2}{x}-\left (a^2-b^2\right ) x-2 a x^2-x^3\right ) \, dx,x,b \sec (c+d x)\right )}{b^2 d}\\ &=\frac{a^2 \log (\cos (c+d x))}{d}-\frac{2 a b \sec (c+d x)}{d}+\frac{\left (a^2-b^2\right ) \sec ^2(c+d x)}{2 d}+\frac{2 a b \sec ^3(c+d x)}{3 d}+\frac{b^2 \sec ^4(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.435898, size = 74, normalized size = 0.85 \[ \frac{6 \left (a^2-b^2\right ) \sec ^2(c+d x)+12 a^2 \log (\cos (c+d x))+8 a b \sec ^3(c+d x)-24 a b \sec (c+d x)+3 b^2 \sec ^4(c+d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 136, normalized size = 1.6 \begin{align*}{\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 ) ^{4}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\,d\cos \left ( dx+c \right ) }}-{\frac{2\,ab\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3\,d}}-{\frac{4\,ab\cos \left ( dx+c \right ) }{3\,d}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999052, size = 101, normalized size = 1.16 \begin{align*} \frac{12 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac{24 \, a b \cos \left (d x + c\right )^{3} - 8 \, a b \cos \left (d x + c\right ) - 6 \,{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, b^{2}}{\cos \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.969567, size = 205, normalized size = 2.36 \begin{align*} \frac{12 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (-\cos \left (d x + c\right )\right ) - 24 \, a b \cos \left (d x + c\right )^{3} + 8 \, a b \cos \left (d x + c\right ) + 6 \,{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, b^{2}}{12 \, d \cos \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.02762, size = 126, normalized size = 1.45 \begin{align*} \begin{cases} - \frac{a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{2 a b \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{3 d} - \frac{4 a b \sec{\left (c + d x \right )}}{3 d} + \frac{b^{2} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{4 d} - \frac{b^{2} \sec ^{2}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (a + b \sec{\left (c \right )}\right )^{2} \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 = 1.93279, size = 360, normalized size = 4.14 \begin{align*} -\frac{12 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 12 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{25 \, a^{2} + 32 \, a b + \frac{124 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{128 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{198 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{96 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{48 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{124 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{25 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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