Optimal. Leaf size=55 \[ \frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^3(c+d x)}{3 a d}-\frac {\cos ^4(c+d x)}{4 a d} \]
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Rubi [A] time = 0.11, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2835, 2565, 30, 2564, 14} \[ \frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^3(c+d x)}{3 a d}-\frac {\cos ^4(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2564
Rule 2565
Rule 2835
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^3(c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^3(c+d x) \sin ^2(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,\cos (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a d}\\ &=-\frac {\cos ^4(c+d x)}{4 a d}-\frac {\operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (c+d x)\right )}{a d}\\ &=-\frac {\cos ^4(c+d x)}{4 a d}-\frac {\sin ^3(c+d x)}{3 a d}+\frac {\sin ^5(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 48, normalized size = 0.87 \[ \frac {\sin ^2(c+d x) \left (12 \sin ^3(c+d x)-15 \sin ^2(c+d x)-20 \sin (c+d x)+30\right )}{60 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 49, normalized size = 0.89 \[ -\frac {15 \, \cos \left (d x + c\right )^{4} - 4 \, {\left (3 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{60 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 49, normalized size = 0.89 \[ \frac {12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 30 \, \sin \left (d x + c\right )^{2}}{60 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 49, normalized size = 0.89 \[ \frac {\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 49, normalized size = 0.89 \[ \frac {12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 30 \, \sin \left (d x + c\right )^{2}}{60 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 57, normalized size = 1.04 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}-\frac {{\sin \left (c+d\,x\right )}^4}{4\,a}+\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.74, size = 741, normalized size = 13.47 \[ \begin {cases} \frac {30 \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {40 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {30 \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {16 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {30 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {40 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {30 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \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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