Optimal. Leaf size=61 \[ -\frac{(4 a+3 b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a+3 b)-\frac{b \sin ^3(c+d x) \cos (c+d x)}{4 d} \]
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Rubi [A] time = 0.0404287, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3014, 2635, 8} \[ -\frac{(4 a+3 b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a+3 b)-\frac{b \sin ^3(c+d x) \cos (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3014
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^2(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{b \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{4} (4 a+3 b) \int \sin ^2(c+d x) \, dx\\ &=-\frac{(4 a+3 b) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{b \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{8} (4 a+3 b) \int 1 \, dx\\ &=\frac{1}{8} (4 a+3 b) x-\frac{(4 a+3 b) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{b \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0938271, size = 45, normalized size = 0.74 \[ \frac{4 (4 a+3 b) (c+d x)-8 (a+b) \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 65, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( b \left ( -{\frac{\cos \left ( dx+c \right ) }{4} \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\sin \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +a \left ( -{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43521, size = 100, normalized size = 1.64 \begin{align*} \frac{{\left (d x + c\right )}{\left (4 \, a + 3 \, b\right )} - \frac{{\left (4 \, a + 5 \, b\right )} \tan \left (d x + c\right )^{3} +{\left (4 \, a + 3 \, b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64944, size = 119, normalized size = 1.95 \begin{align*} \frac{{\left (4 \, a + 3 \, b\right )} d x +{\left (2 \, b \cos \left (d x + c\right )^{3} -{\left (4 \, a + 5 \, b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.12638, size = 158, normalized size = 2.59 \begin{align*} \begin{cases} \frac{a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{a x \cos ^{2}{\left (c + d x \right )}}{2} - \frac{a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 b x \cos ^{4}{\left (c + d x \right )}}{8} - \frac{5 b \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} - \frac{3 b \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin ^{2}{\left (c \right )}\right ) \sin ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11987, size = 58, normalized size = 0.95 \begin{align*} \frac{1}{8} \,{\left (4 \, a + 3 \, b\right )} x + \frac{b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac{{\left (a + b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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