Optimal. Leaf size=97 \[ \frac{3 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{19 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)}-\frac{2 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)^2}-\frac{11 x}{2 a^3} \]
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Rubi [A] time = 0.309781, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3872, 2874, 2966, 2637, 2635, 8, 2650, 2648} \[ \frac{3 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{19 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)}-\frac{2 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)^2}-\frac{11 x}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2874
Rule 2966
Rule 2637
Rule 2635
Rule 8
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac{\cos ^3(c+d x) \sin ^2(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=-\frac{\int \frac{\cos ^3(c+d x) (-a+a \cos (c+d x))}{(-a-a \cos (c+d x))^2} \, dx}{a^2}\\ &=-\frac{\int \left (\frac{5}{a}-\frac{3 \cos (c+d x)}{a}+\frac{\cos ^2(c+d x)}{a}+\frac{2}{a (1+\cos (c+d x))^2}-\frac{7}{a (1+\cos (c+d x))}\right ) \, dx}{a^2}\\ &=-\frac{5 x}{a^3}-\frac{\int \cos ^2(c+d x) \, dx}{a^3}-\frac{2 \int \frac{1}{(1+\cos (c+d x))^2} \, dx}{a^3}+\frac{3 \int \cos (c+d x) \, dx}{a^3}+\frac{7 \int \frac{1}{1+\cos (c+d x)} \, dx}{a^3}\\ &=-\frac{5 x}{a^3}+\frac{3 \sin (c+d x)}{a^3 d}-\frac{\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{2 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))^2}+\frac{7 \sin (c+d x)}{a^3 d (1+\cos (c+d x))}-\frac{\int 1 \, dx}{2 a^3}-\frac{2 \int \frac{1}{1+\cos (c+d x)} \, dx}{3 a^3}\\ &=-\frac{11 x}{2 a^3}+\frac{3 \sin (c+d x)}{a^3 d}-\frac{\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{2 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))^2}+\frac{19 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.417204, size = 177, normalized size = 1.82 \[ -\frac{\sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (1326 \sin \left (c+\frac{d x}{2}\right )-2012 \sin \left (c+\frac{3 d x}{2}\right )-498 \sin \left (2 c+\frac{3 d x}{2}\right )-135 \sin \left (2 c+\frac{5 d x}{2}\right )-135 \sin \left (3 c+\frac{5 d x}{2}\right )+15 \sin \left (3 c+\frac{7 d x}{2}\right )+15 \sin \left (4 c+\frac{7 d x}{2}\right )+1980 d x \cos \left (c+\frac{d x}{2}\right )+660 d x \cos \left (c+\frac{3 d x}{2}\right )+660 d x \cos \left (2 c+\frac{3 d x}{2}\right )-3216 \sin \left (\frac{d x}{2}\right )+1980 d x \cos \left (\frac{d x}{2}\right )\right )}{960 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 122, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+6\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3}}}+7\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+5\,{\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 ) ^{2}}}-11\,{\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 = 1.79722, size = 221, normalized size = 2.28 \begin{align*} \frac{\frac{3 \,{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac{2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{18 \, \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^{3}} - \frac{33 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81664, size = 259, normalized size = 2.67 \begin{align*} -\frac{33 \, d x \cos \left (d x + c\right )^{2} + 66 \, d x \cos \left (d x + c\right ) + 33 \, d x +{\left (3 \, \cos \left (d x + c\right )^{3} - 12 \, \cos \left (d x + c\right )^{2} - 71 \, \cos \left (d x + c\right ) - 52\right )} \sin \left (d x + c\right )}{6 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, 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{\sin ^{2}{\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 = 1.30563, size = 130, normalized size = 1.34 \begin{align*} -\frac{\frac{33 \,{\left (d x + c\right )}}{a^{3}} - \frac{6 \,{\left (7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5 \, \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 )}^{2} a^{3}} + \frac{2 \,{\left (a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 18 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{9}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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