Optimal. Leaf size=108 \[ \frac{3 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{3 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}+\frac{(a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]
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Rubi [A] time = 0.0969453, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac{3 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{3 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}+\frac{(a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3 (a+x)^m}{a^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x^3 (a+x)^m \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^3 (a+x)^m+3 a^2 (a+x)^{1+m}-3 a (a+x)^{2+m}+(a+x)^{3+m}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=-\frac{(a+a \sin (c+d x))^{1+m}}{a d (1+m)}+\frac{3 (a+a \sin (c+d x))^{2+m}}{a^2 d (2+m)}-\frac{3 (a+a \sin (c+d x))^{3+m}}{a^3 d (3+m)}+\frac{(a+a \sin (c+d x))^{4+m}}{a^4 d (4+m)}\\ \end{align*}
Mathematica [A] time = 0.571176, size = 94, normalized size = 0.87 \[ \frac{\left (\left (m^3+6 m^2+11 m+6\right ) \sin ^3(c+d x)-3 \left (m^2+3 m+2\right ) \sin ^2(c+d x)+6 (m+1) \sin (c+d x)-6\right ) (a (\sin (c+d x)+1))^{m+1}}{a d (m+1) (m+2) (m+3) (m+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.652, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03799, size = 161, normalized size = 1.49 \begin{align*} \frac{{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} a^{m} \sin \left (d x + c\right )^{4} +{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (d x + c\right )^{3} - 3 \,{\left (m^{2} + m\right )} a^{m} \sin \left (d x + c\right )^{2} + 6 \, a^{m} m \sin \left (d x + c\right ) - 6 \, a^{m}\right )}{\left (\sin \left (d x + c\right ) + 1\right )}^{m}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80017, size = 336, normalized size = 3.11 \begin{align*} \frac{{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} \cos \left (d x + c\right )^{4} + m^{3} -{\left (2 \, m^{3} + 9 \, m^{2} + 19 \, m + 12\right )} \cos \left (d x + c\right )^{2} + 3 \, m^{2} +{\left (m^{3} -{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} \cos \left (d x + c\right )^{2} + 3 \, m^{2} + 8 \, m\right )} \sin \left (d x + c\right ) + 8 \, m\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{d m^{4} + 10 \, d m^{3} + 35 \, d m^{2} + 50 \, d m + 24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 70.7406, size = 1547, normalized size = 14.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24764, size = 686, normalized size = 6.35 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} - 3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{3} + 3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{3} -{\left (a \sin \left (d x + c\right ) + a\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{3} + 6 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} - 21 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{2} + 24 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{2} - 9 \,{\left (a \sin \left (d x + c\right ) + a\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{2} + 11 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 42 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 57 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m - 26 \,{\left (a \sin \left (d x + c\right ) + a\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m + 6 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{4}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 24 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a + 36 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} - 24 \,{\left (a \sin \left (d x + c\right ) + a\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3}}{{\left (a^{3} m^{4} + 10 \, a^{3} m^{3} + 35 \, a^{3} m^{2} + 50 \, a^{3} m + 24 \, a^{3}\right )} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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