Optimal. Leaf size=47 \[ \frac {(a-a \sin (c+d x))^4}{4 a^5 d}-\frac {2 (a-a \sin (c+d x))^3}{3 a^4 d} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {(a-a \sin (c+d x))^4}{4 a^5 d}-\frac {2 (a-a \sin (c+d x))^3}{3 a^4 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a (a-x)^2-(a-x)^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {2 (a-a \sin (c+d x))^3}{3 a^4 d}+\frac {(a-a \sin (c+d x))^4}{4 a^5 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 46, normalized size = 0.98 \[ \frac {\sin (c+d x) \left (3 \sin ^3(c+d x)-4 \sin ^2(c+d x)-6 \sin (c+d x)+12\right )}{12 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 37, normalized size = 0.79 \[ \frac {3 \, \cos \left (d x + c\right )^{4} + 4 \, {\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right )}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 47, normalized size = 1.00 \[ \frac {3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} + 12 \, \sin \left (d x + c\right )}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 45, normalized size = 0.96 \[ \frac {\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 47, normalized size = 1.00 \[ \frac {3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} + 12 \, \sin \left (d x + c\right )}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 54, normalized size = 1.15 \[ \frac {\frac {\sin \left (c+d\,x\right )}{a}-\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}+\frac {{\sin \left (c+d\,x\right )}^4}{4\,a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.77, size = 530, normalized size = 11.28 \[ \begin {cases} \frac {6 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} - \frac {6 \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} + \frac {10 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} + \frac {10 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} - \frac {6 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} + \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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