Optimal. Leaf size=87 \[ -\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^2 x}{8} \]
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Rubi [A] time = 0.0793729, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3788, 2633, 4045, 2635, 8} \[ -\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^2 x}{8} \]
Antiderivative was successfully verified.
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Rule 3788
Rule 2633
Rule 4045
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \, dx &=\left (2 a^2\right ) \int \cos ^3(c+d x) \, dx+\int \cos ^4(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} \left (7 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{2 a^2 \sin (c+d x)}{d}+\frac{7 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{1}{8} \left (7 a^2\right ) \int 1 \, dx\\ &=\frac{7 a^2 x}{8}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{7 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.137689, size = 53, normalized size = 0.61 \[ \frac{a^2 (144 \sin (c+d x)+48 \sin (2 (c+d x))+16 \sin (3 (c+d x))+3 \sin (4 (c+d x))+84 d x)}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 90, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{2\,{a}^{2} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) }{3}}+{a}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1238, size = 112, normalized size = 1.29 \begin{align*} -\frac{64 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} - 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70655, size = 154, normalized size = 1.77 \begin{align*} \frac{21 \, a^{2} d x +{\left (6 \, a^{2} \cos \left (d x + c\right )^{3} + 16 \, a^{2} \cos \left (d x + c\right )^{2} + 21 \, a^{2} \cos \left (d x + c\right ) + 32 \, a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \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 [A] time = 1.37486, size = 130, normalized size = 1.49 \begin{align*} \frac{21 \,{\left (d x + c\right )} a^{2} + \frac{2 \,{\left (21 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 77 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 83 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 75 \, a^{2} \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}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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