Optimal. Leaf size=106 \[ \frac{\left (4 a^2+b^2\right ) \sin ^3(c+d x)}{30 d}+\frac{\sin ^3(c+d x) (a \cos (c+d x)+b)^2}{5 d}+\frac{b \sin ^3(c+d x) (a \cos (c+d x)+b)}{10 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4} \]
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Rubi [A] time = 0.348789, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4397, 2862, 2669, 2635, 8} \[ \frac{\left (4 a^2+b^2\right ) \sin ^3(c+d x)}{30 d}+\frac{\sin ^3(c+d x) (a \cos (c+d x)+b)^2}{5 d}+\frac{b \sin ^3(c+d x) (a \cos (c+d x)+b)}{10 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2862
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx &=\int \cos (c+d x) (b+a \cos (c+d x))^2 \sin ^2(c+d x) \, dx\\ &=\frac{(b+a \cos (c+d x))^2 \sin ^3(c+d x)}{5 d}+\frac{1}{5} \int (b+a \cos (c+d x)) (2 a+2 b \cos (c+d x)) \sin ^2(c+d x) \, dx\\ &=\frac{b (b+a \cos (c+d x)) \sin ^3(c+d x)}{10 d}+\frac{(b+a \cos (c+d x))^2 \sin ^3(c+d x)}{5 d}+\frac{1}{20} \int \left (10 a b+2 \left (4 a^2+b^2\right ) \cos (c+d x)\right ) \sin ^2(c+d x) \, dx\\ &=\frac{\left (4 a^2+b^2\right ) \sin ^3(c+d x)}{30 d}+\frac{b (b+a \cos (c+d x)) \sin ^3(c+d x)}{10 d}+\frac{(b+a \cos (c+d x))^2 \sin ^3(c+d x)}{5 d}+\frac{1}{2} (a b) \int \sin ^2(c+d x) \, dx\\ &=-\frac{a b \cos (c+d x) \sin (c+d x)}{4 d}+\frac{\left (4 a^2+b^2\right ) \sin ^3(c+d x)}{30 d}+\frac{b (b+a \cos (c+d x)) \sin ^3(c+d x)}{10 d}+\frac{(b+a \cos (c+d x))^2 \sin ^3(c+d x)}{5 d}+\frac{1}{4} (a b) \int 1 \, dx\\ &=\frac{a b x}{4}-\frac{a b \cos (c+d x) \sin (c+d x)}{4 d}+\frac{\left (4 a^2+b^2\right ) \sin ^3(c+d x)}{30 d}+\frac{b (b+a \cos (c+d x)) \sin ^3(c+d x)}{10 d}+\frac{(b+a \cos (c+d x))^2 \sin ^3(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.396168, size = 77, normalized size = 0.73 \[ \frac{30 \left (a^2+2 b^2\right ) \sin (c+d x)-5 \left (a^2+4 b^2\right ) \sin (3 (c+d x))-3 a (a \sin (5 (c+d x))-20 b (c+d x)+5 b \sin (4 (c+d x)))}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 100, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{ \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{15}} \right ) +2\,ab \left ( -1/4\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1/8\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/8\,dx+c/8 \right ) +{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11882, size = 92, normalized size = 0.87 \begin{align*} \frac{80 \, b^{2} \sin \left (d x + c\right )^{3} - 16 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{3}\right )} a^{2} + 15 \,{\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a b}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.4998, size = 211, normalized size = 1.99 \begin{align*} \frac{15 \, a b d x -{\left (12 \, a^{2} \cos \left (d x + c\right )^{4} + 30 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right ) - 4 \,{\left (a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 8 \, a^{2} - 20 \, b^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right )^{2} \cos ^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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