Optimal. Leaf size=33 \[ \frac {b \sin ^2(c+d x)}{2 d}-\frac {a \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4377, 12, 2564, 30, 2565} \[ \frac {b \sin ^2(c+d x)}{2 d}-\frac {a \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2564
Rule 2565
Rule 4377
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \cos ^2(c+d x) \sin (c+d x) \, dx+\int b \cos (c+d x) \sin (c+d x) \, dx\\ &=b \int \cos (c+d x) \sin (c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos ^3(c+d x)}{3 d}+\frac {b \operatorname {Subst}(\int x \, dx,x,\sin (c+d x))}{d}\\ &=-\frac {a \cos ^3(c+d x)}{3 d}+\frac {b \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 38, normalized size = 1.15 \[ -\frac {3 a \cos (c+d x)+a \cos (3 (c+d x))+3 b \cos (2 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 28, normalized size = 0.85 \[ -\frac {2 \, a \cos \left (d x + c\right )^{3} + 3 \, b \cos \left (d x + c\right )^{2}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.26, size = 99, normalized size = 3.00 \[ -\frac {a \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {a \cos \left (d x + c\right )}{4 \, d} - \frac {b \tan \left (d x\right )^{2} \tan \relax (c)^{2} - b \tan \left (d x\right )^{2} - 4 \, b \tan \left (d x\right ) \tan \relax (c) - b \tan \relax (c)^{2} + b}{4 \, {\left (d \tan \left (d x\right )^{2} \tan \relax (c)^{2} + d \tan \left (d x\right )^{2} + d \tan \relax (c)^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 29, normalized size = 0.88 \[ -\frac {\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{3}+\frac {b \left (\cos ^{2}\left (d x +c \right )\right )}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 28, normalized size = 0.85 \[ -\frac {2 \, a \cos \left (d x + c\right )^{3} - 3 \, b \sin \left (d x + c\right )^{2}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 49, normalized size = 1.48 \[ -\frac {\left (\cos \left (c+d\,x\right )+1\right )\,\left (2\,a-3\,b-2\,a\,\cos \left (c+d\,x\right )+3\,b\,\cos \left (c+d\,x\right )+2\,a\,{\cos \left (c+d\,x\right )}^2\right )}{6\,d} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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