Optimal. Leaf size=124 \[ -\frac{4 \sin ^3(c+d x)}{a^2 d}+\frac{12 \sin (c+d x)}{a^2 d}-\frac{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.190435, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3817, 4020, 3787, 2633, 2635, 8} \[ -\frac{4 \sin ^3(c+d x)}{a^2 d}+\frac{12 \sin (c+d x)}{a^2 d}-\frac{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3817
Rule 4020
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{\cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \frac{\cos ^3(c+d x) (-6 a+4 a \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac{10 \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \cos ^3(c+d x) \left (-36 a^2+30 a^2 \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{10 \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{10 \int \cos ^2(c+d x) \, dx}{a^2}+\frac{12 \int \cos ^3(c+d x) \, dx}{a^2}\\ &=-\frac{5 \cos (c+d x) \sin (c+d x)}{a^2 d}-\frac{10 \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{5 \int 1 \, dx}{a^2}-\frac{12 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=-\frac{5 x}{a^2}+\frac{12 \sin (c+d x)}{a^2 d}-\frac{5 \cos (c+d x) \sin (c+d x)}{a^2 d}-\frac{10 \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac{\cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{4 \sin ^3(c+d x)}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.427158, size = 199, normalized size = 1.6 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (-156 \sin \left (c+\frac{d x}{2}\right )+342 \sin \left (c+\frac{3 d x}{2}\right )+118 \sin \left (2 c+\frac{3 d x}{2}\right )+30 \sin \left (2 c+\frac{5 d x}{2}\right )+30 \sin \left (3 c+\frac{5 d x}{2}\right )-3 \sin \left (3 c+\frac{7 d x}{2}\right )-3 \sin \left (4 c+\frac{7 d x}{2}\right )+\sin \left (4 c+\frac{9 d x}{2}\right )+\sin \left (5 c+\frac{9 d x}{2}\right )-360 d x \cos \left (c+\frac{d x}{2}\right )-120 d x \cos \left (c+\frac{3 d x}{2}\right )-120 d x \cos \left (2 c+\frac{3 d x}{2}\right )+516 \sin \left (\frac{d x}{2}\right )-360 d x \cos \left (\frac{d x}{2}\right )\right )}{192 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 156, normalized size = 1.3 \begin{align*} -{\frac{1}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{9}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+10\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+{\frac{40}{3\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}+6\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-10\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74651, size = 279, normalized size = 2.25 \begin{align*} \frac{\frac{4 \,{\left (\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\frac{27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{60 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68341, size = 278, normalized size = 2.24 \begin{align*} -\frac{15 \, d x \cos \left (d x + c\right )^{2} + 30 \, d x \cos \left (d x + c\right ) + 15 \, d x -{\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 33 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \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 = 1.33315, size = 146, normalized size = 1.18 \begin{align*} -\frac{\frac{30 \,{\left (d x + c\right )}}{a^{2}} - \frac{4 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 20 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, \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 )}^{3} a^{2}} + \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 27 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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