Optimal. Leaf size=86 \[ \frac {\left (4 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x \left (4 a^2+b^2\right )-\frac {5 a b \cos ^3(c+d x)}{12 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d} \]
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Rubi [A] time = 0.10, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2692, 2669, 2635, 8} \[ \frac {\left (4 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x \left (4 a^2+b^2\right )-\frac {5 a b \cos ^3(c+d x)}{12 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}+\frac {1}{4} \int \cos ^2(c+d x) \left (4 a^2+b^2+5 a b \sin (c+d x)\right ) \, dx\\ &=-\frac {5 a b \cos ^3(c+d x)}{12 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}+\frac {1}{4} \left (4 a^2+b^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {5 a b \cos ^3(c+d x)}{12 d}+\frac {\left (4 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}+\frac {1}{8} \left (4 a^2+b^2\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (4 a^2+b^2\right ) x-\frac {5 a b \cos ^3(c+d x)}{12 d}+\frac {\left (4 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 85, normalized size = 0.99 \[ \frac {3 \left (8 a^2 \sin (2 (c+d x))+16 a^2 c+16 a^2 d x-b^2 \sin (4 (c+d x))+4 b^2 c+4 b^2 d x\right )-48 a b \cos (c+d x)-16 a b \cos (3 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 70, normalized size = 0.81 \[ -\frac {16 \, a b \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{2} + b^{2}\right )} d x + 3 \, {\left (2 \, b^{2} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 76, normalized size = 0.88 \[ \frac {1}{8} \, {\left (4 \, a^{2} + b^{2}\right )} x - \frac {a b \cos \left (3 \, d x + 3 \, c\right )}{6 \, d} - \frac {a b \cos \left (d x + c\right )}{2 \, d} - \frac {b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 86, normalized size = 1.00 \[ \frac {b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {2 a b \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 64, normalized size = 0.74 \[ -\frac {64 \, a b \cos \left (d x + c\right )^{3} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 3 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.38, size = 71, normalized size = 0.83 \[ \frac {6\,a^2\,\sin \left (2\,c+2\,d\,x\right )-\frac {3\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{4}-12\,a\,b\,\cos \left (c+d\,x\right )-4\,a\,b\,\cos \left (3\,c+3\,d\,x\right )+12\,a^2\,d\,x+3\,b^2\,d\,x}{24\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.41, size = 180, normalized size = 2.09 \[ \begin {cases} \frac {a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {2 a b \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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