Optimal. Leaf size=45 \[ \frac {(a \sin (c+d x)+a)^5}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^4}{4 a d} \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {(a \sin (c+d x)+a)^5}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^4}{4 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+x)^3}{a} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int x (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a (a+x)^3+(a+x)^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {(a+a \sin (c+d x))^4}{4 a d}+\frac {(a+a \sin (c+d x))^5}{5 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 30, normalized size = 0.67 \[ \frac {a^3 (\sin (c+d x)+1)^4 (4 \sin (c+d x)-1)}{20 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 71, normalized size = 1.58 \[ \frac {15 \, a^{3} \cos \left (d x + c\right )^{4} - 40 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 7 \, a^{3} \cos \left (d x + c\right )^{2} + 6 \, a^{3}\right )} \sin \left (d x + c\right )}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 58, normalized size = 1.29 \[ \frac {4 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} + 10 \, a^{3} \sin \left (d x + c\right )^{2}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 57, normalized size = 1.27 \[ \frac {\frac {a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+a^{3} \left (\sin ^{3}\left (d x +c \right )\right )+\frac {a^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 58, normalized size = 1.29 \[ \frac {4 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} + 10 \, a^{3} \sin \left (d x + c\right )^{2}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 56, normalized size = 1.24 \[ \frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^4}{4}+a^3\,{\sin \left (c+d\,x\right )}^3+\frac {a^3\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.55, size = 76, normalized size = 1.69 \[ \begin {cases} \frac {a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {a^{3} \sin ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \sin {\relax (c )} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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