Optimal. Leaf size=24 \[ \frac {a \sin (c+d x)}{d}-\frac {b \cos (c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3486, 2637} \[ \frac {a \sin (c+d x)}{d}-\frac {b \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3486
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {b \cos (c+d x)}{d}+a \int \cos (c+d x) \, dx\\ &=-\frac {b \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 1.92 \[ \frac {a \sin (c) \cos (d x)}{d}+\frac {a \cos (c) \sin (d x)}{d}+\frac {b \sin (c) \sin (d x)}{d}-\frac {b \cos (c) \cos (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 23, normalized size = 0.96 \[ -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 129, normalized size = 5.38 \[ -\frac {b \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 2 \, a \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a \tan \left (\frac {1}{2} \, d x\right ) - 2 \, a \tan \left (\frac {1}{2} \, c\right ) + b}{d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d \tan \left (\frac {1}{2} \, d x\right )^{2} + d \tan \left (\frac {1}{2} \, c\right )^{2} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 23, normalized size = 0.96 \[ \frac {a \sin \left (d x +c \right )-b \cos \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 23, normalized size = 0.96 \[ -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 38, normalized size = 1.58 \[ -\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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