Optimal. Leaf size=32 \[ \frac {a \sin (c+d x)}{d}-\frac {(a-b) \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3676} \[ \frac {a \sin (c+d x)}{d}-\frac {(a-b) \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3676
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (a-(a-b) x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a \sin (c+d x)}{d}-\frac {(a-b) \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.38 \[ -\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}+\frac {b \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 30, normalized size = 0.94 \[ \frac {{\left ({\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a + b\right )} \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.48, size = 36, normalized size = 1.12 \[ -\frac {a \sin \left (d x + c\right )^{3} - b \sin \left (d x + c\right )^{3} - 3 \, a \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 36, normalized size = 1.12 \[ \frac {\frac {b \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 0.91 \[ -\frac {{\left (a - b\right )} \sin \left (d x + c\right )^{3} - 3 \, a \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.10, size = 47, normalized size = 1.47 \[ \frac {9\,a\,\sin \left (c+d\,x\right )+3\,b\,\sin \left (c+d\,x\right )+a\,\sin \left (3\,c+3\,d\,x\right )-b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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