Optimal. Leaf size=58 \[ -\frac{\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8} \]
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Rubi [A] time = 0.0803006, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2672, 2671} \[ -\frac{\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac{\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}+\frac{\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{9 a}\\ &=-\frac{\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}-\frac{\cos ^7(c+d x)}{63 a d (a+a \sin (c+d x))^7}\\ \end{align*}
Mathematica [A] time = 0.0830468, size = 36, normalized size = 0.62 \[ -\frac{(\sin (c+d x)+8) \cos ^7(c+d x)}{63 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.131, size = 145, normalized size = 2.5 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( 7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}+{\frac{496}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{928}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}-{\frac{86}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-{\frac{128}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1}-136\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-5}+76\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}+64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-8} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02298, size = 506, normalized size = 8.72 \begin{align*} -\frac{2 \,{\left (\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{189 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{693 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{315 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{483 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{63 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 8\right )}}{63 \,{\left (a^{8} + \frac{9 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{36 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{84 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{126 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{126 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{84 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{36 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{9 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54647, size = 614, normalized size = 10.59 \begin{align*} \frac{\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{4} + 19 \, \cos \left (d x + c\right )^{3} + 52 \, \cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + 24 \, \cos \left (d x + c\right )^{2} - 28 \, \cos \left (d x + c\right ) - 56\right )} \sin \left (d x + c\right ) - 28 \, \cos \left (d x + c\right ) - 56}{63 \,{\left (a^{8} d \cos \left (d x + c\right )^{5} + 5 \, a^{8} d \cos \left (d x + c\right )^{4} - 8 \, a^{8} d \cos \left (d x + c\right )^{3} - 20 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d +{\left (a^{8} d \cos \left (d x + c\right )^{4} - 4 \, a^{8} d \cos \left (d x + c\right )^{3} - 12 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\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 [B] time = 1.20341, size = 169, normalized size = 2.91 \begin{align*} -\frac{2 \,{\left (63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 483 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 189 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 225 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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