Optimal. Leaf size=22 \[ -\frac{1}{7 b d (a+b \sin (c+d x))^7} \]
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Rubi [A] time = 0.0269824, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2668, 32} \[ -\frac{1}{7 b d (a+b \sin (c+d x))^7} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac{1}{7 b d (a+b \sin (c+d x))^7}\\ \end{align*}
Mathematica [A] time = 0.0780185, size = 22, normalized size = 1. \[ -\frac{1}{7 b d (a+b \sin (c+d x))^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{7\,bd \left ( a+b\sin \left ( dx+c \right ) \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964449, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{7 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.18221, size = 482, normalized size = 21.91 \begin{align*} \frac{1}{7 \,{\left (7 \, a b^{7} d \cos \left (d x + c\right )^{6} - 7 \,{\left (5 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{4} + 7 \,{\left (3 \, a^{5} b^{3} + 10 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{2} -{\left (a^{7} b + 21 \, a^{5} b^{3} + 35 \, a^{3} b^{5} + 7 \, a b^{7}\right )} d +{\left (b^{8} d \cos \left (d x + c\right )^{6} - 3 \,{\left (7 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )^{4} +{\left (35 \, a^{4} b^{4} + 42 \, a^{2} b^{6} + 3 \, b^{8}\right )} d \cos \left (d x + c\right )^{2} -{\left (7 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 21 \, a^{2} b^{6} + b^{8}\right )} 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 [A] time = 55.0178, size = 167, normalized size = 7.59 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{8}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{8} d} & \text{for}\: b = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{8}} & \text{for}\: d = 0 \\- \frac{1}{7 a^{7} b d + 49 a^{6} b^{2} d \sin{\left (c + d x \right )} + 147 a^{5} b^{3} d \sin ^{2}{\left (c + d x \right )} + 245 a^{4} b^{4} d \sin ^{3}{\left (c + d x \right )} + 245 a^{3} b^{5} d \sin ^{4}{\left (c + d x \right )} + 147 a^{2} b^{6} d \sin ^{5}{\left (c + d x \right )} + 49 a b^{7} d \sin ^{6}{\left (c + d x \right )} + 7 b^{8} d \sin ^{7}{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39745, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{7 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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