Optimal. Leaf size=45 \[ \frac {(a \sin (c+d x)+a)^4}{2 a^2 d}-\frac {(a \sin (c+d x)+a)^5}{5 a^3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 45, 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} \[ \frac {(a \sin (c+d x)+a)^4}{2 a^2 d}-\frac {(a \sin (c+d x)+a)^5}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int (a-x) (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a (a+x)^3-(a+x)^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {(a+a \sin (c+d x))^4}{2 a^2 d}-\frac {(a+a \sin (c+d x))^5}{5 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 46, normalized size = 1.02 \[ -\frac {a^2 \sin (c+d x) \left (2 \sin ^4(c+d x)+5 \sin ^3(c+d x)-10 \sin (c+d x)-10\right )}{10 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 58, normalized size = 1.29 \[ -\frac {5 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2}\right )} \sin \left (d x + c\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 56, normalized size = 1.24 \[ -\frac {2 \, a^{2} \sin \left (d x + c\right )^{5} + 5 \, a^{2} \sin \left (d x + c\right )^{4} - 10 \, a^{2} \sin \left (d x + c\right )^{2} - 10 \, a^{2} \sin \left (d x + c\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 79, normalized size = 1.76 \[ \frac {a^{2} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) a^{2}}{2}+\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 56, normalized size = 1.24 \[ -\frac {2 \, a^{2} \sin \left (d x + c\right )^{5} + 5 \, a^{2} \sin \left (d x + c\right )^{4} - 10 \, a^{2} \sin \left (d x + c\right )^{2} - 10 \, a^{2} \sin \left (d x + c\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 53, normalized size = 1.18 \[ \frac {-\frac {a^2\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {a^2\,{\sin \left (c+d\,x\right )}^4}{2}+a^2\,{\sin \left (c+d\,x\right )}^2+a^2\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.05, size = 107, normalized size = 2.38 \[ \begin {cases} \frac {2 a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a^{2} \cos ^{4}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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