Optimal. Leaf size=23 \[ -\frac {(a-a \sin (c+d x))^4}{4 a^7 d} \]
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Rubi [A] time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {2667, 32} \[ -\frac {(a-a \sin (c+d x))^4}{4 a^7 d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {(a-a \sin (c+d x))^4}{4 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 44, normalized size = 1.91 \[ -\frac {\sin (c+d x) \left (\sin ^3(c+d x)-4 \sin ^2(c+d x)+6 \sin (c+d x)-4\right )}{4 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 45, normalized size = 1.96 \[ -\frac {\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{4 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.74, size = 45, normalized size = 1.96 \[ -\frac {\sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{4 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 19, normalized size = 0.83 \[ -\frac {\left (\sin \left (d x +c \right )-1\right )^{4}}{4 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 45, normalized size = 1.96 \[ -\frac {\sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{4 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.55, size = 53, normalized size = 2.30 \[ \frac {\frac {\sin \left (c+d\,x\right )}{a^3}-\frac {3\,{\sin \left (c+d\,x\right )}^2}{2\,a^3}+\frac {{\sin \left (c+d\,x\right )}^3}{a^3}-\frac {{\sin \left (c+d\,x\right )}^4}{4\,a^3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 161.61, size = 654, normalized size = 28.43 \[ \begin {cases} \frac {2 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {6 \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} + \frac {14 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {16 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} + \frac {14 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {6 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} + \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{7}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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