Optimal. Leaf size=103 \[ \frac{24 \sin (c+d x)}{5 a^3 d}-\frac{3 \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{3 x}{a^3}-\frac{3 \sin (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.220573, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3817, 4020, 3787, 2637, 8} \[ \frac{24 \sin (c+d x)}{5 a^3 d}-\frac{3 \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{3 x}{a^3}-\frac{3 \sin (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3817
Rule 4020
Rule 3787
Rule 2637
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac{\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{\int \frac{\cos (c+d x) (-6 a+3 a \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac{\int \frac{\cos (c+d x) \left (-27 a^2+18 a^2 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac{3 \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\int \cos (c+d x) \left (-72 a^3+45 a^3 \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac{3 \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{3 \int 1 \, dx}{a^3}+\frac{24 \int \cos (c+d x) \, dx}{5 a^3}\\ &=-\frac{3 x}{a^3}+\frac{24 \sin (c+d x)}{5 a^3 d}-\frac{\sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{3 \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac{3 \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.561546, size = 169, normalized size = 1.64 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (20 (\sin (c+d x)-3 d x) \cos ^5\left (\frac{1}{2} (c+d x)\right )-12 \tan \left (\frac{c}{2}\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right )+\tan \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )+96 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^4\left (\frac{1}{2} (c+d x)\right )-12 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{5 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 107, normalized size = 1. \begin{align*}{\frac{1}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{17}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-6\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71797, size = 185, normalized size = 1.8 \begin{align*} \frac{\frac{40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac{a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{120 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90152, size = 325, normalized size = 3.16 \begin{align*} -\frac{15 \, d x \cos \left (d x + c\right )^{3} + 45 \, d x \cos \left (d x + c\right )^{2} + 45 \, d x \cos \left (d x + c\right ) + 15 \, d x -{\left (5 \, \cos \left (d x + c\right )^{3} + 39 \, \cos \left (d x + c\right )^{2} + 57 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right )}{5 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37433, size = 130, normalized size = 1.26 \begin{align*} -\frac{\frac{60 \,{\left (d x + c\right )}}{a^{3}} - \frac{40 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac{a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 10 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 85 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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