Optimal. Leaf size=99 \[ \frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a} \]
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Rubi [A] time = 0.177497, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3872, 2839, 2564, 30, 2568, 2635, 8} \[ \frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2839
Rule 2564
Rule 30
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) \sin ^6(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac{\int \cos (c+d x) \sin ^4(c+d x) \, dx}{a}-\frac{\int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a}\\ &=\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}-\frac{\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{2 a}+\frac{\operatorname{Subst}\left (\int x^4 \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}+\frac{\sin ^5(c+d x)}{5 a d}-\frac{\int \cos ^2(c+d x) \, dx}{8 a}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}+\frac{\sin ^5(c+d x)}{5 a d}-\frac{\int 1 \, dx}{16 a}\\ &=-\frac{x}{16 a}-\frac{\cos (c+d x) \sin (c+d x)}{16 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}+\frac{\sin ^5(c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.695006, size = 112, normalized size = 1.13 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (120 \sin (c+d x)+15 \sin (2 (c+d x))-60 \sin (3 (c+d x))+15 \sin (4 (c+d x))+12 \sin (5 (c+d x))-5 \sin (6 (c+d x))+75 c-75 \tan \left (\frac{c}{2}\right )-60 d x\right )}{480 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.08, size = 222, normalized size = 2.2 \begin{align*} -{\frac{1}{8\,da} \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{17}{24\,da} \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{223}{20\,da} \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{33}{20\,da} \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}{24\,da} \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{1}{8\,da}\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{1}{8\,da}\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 = 1.51048, size = 375, normalized size = 3.79 \begin{align*} \frac{\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{85 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{198 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{1338 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{85 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{15 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a + \frac{6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac{15 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70145, size = 190, normalized size = 1.92 \begin{align*} -\frac{15 \, d x +{\left (40 \, \cos \left (d x + c\right )^{5} - 48 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{3} + 96 \, \cos \left (d x + c\right )^{2} + 15 \, \cos \left (d x + c\right ) - 48\right )} \sin \left (d x + c\right )}{240 \, a 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 = 1.3087, size = 153, normalized size = 1.55 \begin{align*} -\frac{\frac{15 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 85 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1338 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 198 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 85 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, \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}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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