Optimal. Leaf size=45 \[ \frac {(a \sin (c+d x)+a)^6}{6 a^2 d}-\frac {(a \sin (c+d x)+a)^5}{5 a d} \]
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Rubi [A] time = 0.04, 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} \[ \frac {(a \sin (c+d x)+a)^6}{6 a^2 d}-\frac {(a \sin (c+d x)+a)^5}{5 a d} \]
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))^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+x)^4}{a} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int x (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a (a+x)^4+(a+x)^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {(a+a \sin (c+d x))^5}{5 a d}+\frac {(a+a \sin (c+d x))^6}{6 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 30, normalized size = 0.67 \[ \frac {a^4 (\sin (c+d x)+1)^5 (5 \sin (c+d x)-1)}{30 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 85, normalized size = 1.89 \[ -\frac {5 \, a^{4} \cos \left (d x + c\right )^{6} - 60 \, a^{4} \cos \left (d x + c\right )^{4} + 120 \, a^{4} \cos \left (d x + c\right )^{2} - 8 \, {\left (3 \, a^{4} \cos \left (d x + c\right )^{4} - 11 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 71, normalized size = 1.58 \[ \frac {5 \, a^{4} \sin \left (d x + c\right )^{6} + 24 \, a^{4} \sin \left (d x + c\right )^{5} + 45 \, a^{4} \sin \left (d x + c\right )^{4} + 40 \, a^{4} \sin \left (d x + c\right )^{3} + 15 \, a^{4} \sin \left (d x + c\right )^{2}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 71, normalized size = 1.58 \[ \frac {\frac {a^{4} \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {4 a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 a^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {4 a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 71, normalized size = 1.58 \[ \frac {5 \, a^{4} \sin \left (d x + c\right )^{6} + 24 \, a^{4} \sin \left (d x + c\right )^{5} + 45 \, a^{4} \sin \left (d x + c\right )^{4} + 40 \, a^{4} \sin \left (d x + c\right )^{3} + 15 \, a^{4} \sin \left (d x + c\right )^{2}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 70, normalized size = 1.56 \[ \frac {\frac {a^4\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {4\,a^4\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {3\,a^4\,{\sin \left (c+d\,x\right )}^4}{2}+\frac {4\,a^4\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a^4\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.87, size = 97, normalized size = 2.16 \[ \begin {cases} \frac {a^{4} \sin ^{6}{\left (c + d x \right )}}{6 d} + \frac {4 a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a^{4} \sin ^{4}{\left (c + d x \right )}}{2 d} + \frac {4 a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{4} \sin {\relax (c )} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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