Optimal. Leaf size=104 \[ -\frac{\sin ^5(c+d x)}{10 a^2 d}-\frac{\sin ^3(c+d x) (a-a \cos (c+d x))^3}{6 a^5 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{8 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2} \]
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Rubi [A] time = 0.311139, antiderivative size = 104, 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 = {3872, 2875, 2870, 2669, 2635, 8} \[ -\frac{\sin ^5(c+d x)}{10 a^2 d}-\frac{\sin ^3(c+d x) (a-a \cos (c+d x))^3}{6 a^5 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{8 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2875
Rule 2870
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin ^6(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac{\int \cos ^2(c+d x) (-a+a \cos (c+d x))^2 \sin ^2(c+d x) \, dx}{a^4}\\ &=-\frac{(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac{\int (-a+a \cos (c+d x)) \sin ^4(c+d x) \, dx}{2 a^3}\\ &=-\frac{(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac{\sin ^5(c+d x)}{10 a^2 d}+\frac{\int \sin ^4(c+d x) \, dx}{2 a^2}\\ &=-\frac{\cos (c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac{(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac{\sin ^5(c+d x)}{10 a^2 d}+\frac{3 \int \sin ^2(c+d x) \, dx}{8 a^2}\\ &=-\frac{3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac{(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac{\sin ^5(c+d x)}{10 a^2 d}+\frac{3 \int 1 \, dx}{16 a^2}\\ &=\frac{3 x}{16 a^2}-\frac{3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{8 a^2 d}-\frac{(a-a \cos (c+d x))^3 \sin ^3(c+d x)}{6 a^5 d}-\frac{\sin ^5(c+d x)}{10 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.869438, size = 111, normalized size = 1.07 \[ \frac{\cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (-480 \sin (c+d x)+30 \sin (2 (c+d x))+80 \sin (3 (c+d x))-90 \sin (4 (c+d x))+48 \sin (5 (c+d x))-10 \sin (6 (c+d x))+25 \tan \left (\frac{c}{2}\right )+360 d x\right )}{480 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.086, size = 222, normalized size = 2.1 \begin{align*}{\frac{3}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{205}{24\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{29}{20\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{99}{20\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}-{\frac{17}{8\,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 ) ^{-6}}-{\frac{3}{8\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-6}}+{\frac{3}{8\,d{a}^{2}}\arctan \left ( \tan \left ({\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 [B] time = 1.54564, size = 394, normalized size = 3.79 \begin{align*} -\frac{\frac{\frac{45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{255 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{594 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{174 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{1025 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{45 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{2} + \frac{6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac{45 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7213, size = 192, normalized size = 1.85 \begin{align*} \frac{45 \, d x -{\left (40 \, \cos \left (d x + c\right )^{5} - 96 \, \cos \left (d x + c\right )^{4} + 50 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} - 45 \, \cos \left (d x + c\right ) + 64\right )} \sin \left (d x + c\right )}{240 \, a^{2} 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 = 1.29965, size = 153, normalized size = 1.47 \begin{align*} \frac{\frac{45 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (45 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1025 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 174 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 594 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 255 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 45 \, \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 )}^{6} a^{2}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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