Optimal. Leaf size=103 \[ \frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 x}{4} \]
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Rubi [A] time = 0.109805, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3788, 2635, 8, 4044, 3013, 373} \[ \frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 x}{4} \]
Antiderivative was successfully verified.
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Rule 3788
Rule 2635
Rule 8
Rule 4044
Rule 3013
Rule 373
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \, dx &=\left (2 a^2\right ) \int \cos ^4(c+d x) \, dx+\int \cos ^5(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)}{2 d}+\frac{1}{2} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx+\int \cos ^3(c+d x) \left (a^2+a^2 \cos ^2(c+d x)\right ) \, dx\\ &=\frac{3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}+\frac{1}{4} \left (3 a^2\right ) \int 1 \, dx-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (2 a^2-a^2 x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{3 a^2 x}{4}+\frac{3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac{\operatorname{Subst}\left (\int \left (2 a^2-3 a^2 x^2+a^2 x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{3 a^2 x}{4}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{a^2 \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.136428, size = 61, normalized size = 0.59 \[ \frac{a^2 (110 \sin (c+d x)+40 \sin (2 (c+d x))+15 \sin (3 (c+d x))+5 \sin (4 (c+d x))+\sin (5 (c+d x))+60 d x)}{80 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 96, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+2\,{a}^{2} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{{a}^{2} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07577, size = 128, normalized size = 1.24 \begin{align*} \frac{16 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} - 80 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} + 15 \,{\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}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70839, size = 186, normalized size = 1.81 \begin{align*} \frac{15 \, a^{2} d x +{\left (4 \, a^{2} \cos \left (d x + c\right )^{4} + 10 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 15 \, a^{2} \cos \left (d x + c\right ) + 24 \, a^{2}\right )} \sin \left (d x + c\right )}{20 \, 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 = 1.37274, size = 151, normalized size = 1.47 \begin{align*} \frac{15 \,{\left (d x + c\right )} a^{2} + \frac{2 \,{\left (15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 70 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 144 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 90 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 65 \, 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 )}^{5}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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