Optimal. Leaf size=22 \[ -\frac {1}{7 a d (a+a \sin (c+d x))^7} \]
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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2746, 32}
\begin {gather*} -\frac {1}{7 a d (a \sin (c+d x)+a)^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(a+x)^8} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {1}{7 a d (a+a \sin (c+d x))^7}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 33, normalized size = 1.50 \begin {gather*} -\frac {1}{7 a^8 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{14}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 21, normalized size = 0.95
method | result | size |
derivativedivides | \(-\frac {1}{7 a d \left (a +a \sin \left (d x +c \right )\right )^{7}}\) | \(21\) |
default | \(-\frac {1}{7 a d \left (a +a \sin \left (d x +c \right )\right )^{7}}\) | \(21\) |
risch | \(\frac {128 i {\mathrm e}^{7 i \left (d x +c \right )}}{7 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{14}}\) | \(33\) |
norman | \(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {2 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {14 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {14 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {66 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {66 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {490 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {490 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {206 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {206 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {902 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {902 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {11370 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}+\frac {11370 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}+\frac {9382 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}+\frac {9382 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}\) | \(336\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{7 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (20) = 40\).
time = 0.34, size = 108, normalized size = 4.91 \begin {gather*} \frac {1}{7 \, {\left (7 \, a^{8} d \cos \left (d x + c\right )^{6} - 56 \, a^{8} d \cos \left (d x + c\right )^{4} + 112 \, a^{8} d \cos \left (d x + c\right )^{2} - 64 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{6} - 24 \, a^{8} d \cos \left (d x + c\right )^{4} + 80 \, a^{8} d \cos \left (d x + c\right )^{2} - 64 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (19) = 38\).
time = 14.20, size = 128, normalized size = 5.82 \begin {gather*} \begin {cases} - \frac {1}{7 a^{8} d \sin ^{7}{\left (c + d x \right )} + 49 a^{8} d \sin ^{6}{\left (c + d x \right )} + 147 a^{8} d \sin ^{5}{\left (c + d x \right )} + 245 a^{8} d \sin ^{4}{\left (c + d x \right )} + 245 a^{8} d \sin ^{3}{\left (c + d x \right )} + 147 a^{8} d \sin ^{2}{\left (c + d x \right )} + 49 a^{8} d \sin {\left (c + d x \right )} + 7 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.68, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{7 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.67, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{7\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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