Optimal. Leaf size=24 \[ -\frac {b \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d} \]
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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 = {3567, 2717}
\begin {gather*} \frac {a \sin (c+d x)}{d}-\frac {b \cos (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3567
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 = 0.03, size = 46, normalized size = 1.92 \begin {gather*} -\frac {b \cos (c) \cos (d x)}{d}+\frac {a \cos (d x) \sin (c)}{d}+\frac {a \cos (c) \sin (d x)}{d}+\frac {b \sin (c) \sin (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 23, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {-b \cos \left (d x +c \right )+a \sin \left (d x +c \right )}{d}\) | \(23\) |
default | \(\frac {-b \cos \left (d x +c \right )+a \sin \left (d x +c \right )}{d}\) | \(23\) |
risch | \(-\frac {b \cos \left (d x +c \right )}{d}+\frac {a \sin \left (d x +c \right )}{d}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 23, normalized size = 0.96 \begin {gather*} -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 23, normalized size = 0.96 \begin {gather*} -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \end {gather*}
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 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (24) = 48\).
time = 0.50, size = 129, normalized size = 5.38 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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