Optimal. Leaf size=133 \[ \frac{b \left (2 a^2+b^2\right ) \tan (c+d x)}{d}+\frac{3 a \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \left (a^2+4 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{3 a^2 b \tan (c+d x) \sec ^2(c+d x)}{4 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))}{4 d} \]
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Rubi [A] time = 0.203476, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2792, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{b \left (2 a^2+b^2\right ) \tan (c+d x)}{d}+\frac{3 a \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \left (a^2+4 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{3 a^2 b \tan (c+d x) \sec ^2(c+d x)}{4 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \sec ^5(c+d x) \, dx &=\frac{a^2 (a+b \cos (c+d x)) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \int \left (9 a^2 b+3 a \left (a^2+4 b^2\right ) \cos (c+d x)+2 b \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{3 a^2 b \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{12} \int \left (9 a \left (a^2+4 b^2\right )+12 b \left (2 a^2+b^2\right ) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{3 a^2 b \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\left (b \left (2 a^2+b^2\right )\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{4} \left (3 a \left (a^2+4 b^2\right )\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{3 a \left (a^2+4 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{3 a^2 b \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{8} \left (3 a \left (a^2+4 b^2\right )\right ) \int \sec (c+d x) \, dx-\frac{\left (b \left (2 a^2+b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{3 a \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left (2 a^2+b^2\right ) \tan (c+d x)}{d}+\frac{3 a \left (a^2+4 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{3 a^2 b \sec ^2(c+d x) \tan (c+d x)}{4 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.414429, size = 90, normalized size = 0.68 \[ \frac{3 a \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (8 b \left (a^2 \tan ^2(c+d x)+3 a^2+b^2\right )+3 a \left (a^2+4 b^2\right ) \sec (c+d x)+2 a^3 \sec ^3(c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 160, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+2\,{\frac{{a}^{2}b\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}b \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{3\,a{b}^{2}\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{2\,d}}+{\frac{3\,a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{b}^{3}\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966777, size = 213, normalized size = 1.6 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} b - a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a b^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 16 \, b^{3} \tan \left (d x + c\right )}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03084, size = 348, normalized size = 2.62 \begin{align*} \frac{3 \,{\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \, a^{2} b \cos \left (d x + c\right ) + 8 \,{\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} + 2 \, a^{3} + 3 \,{\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.69653, size = 446, normalized size = 3.35 \begin{align*} \frac{3 \,{\left (a^{3} + 4 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (a^{3} + 4 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 8 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 24 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{4}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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