Optimal. Leaf size=16 \[ a x-\frac {b \cos (e+f x)}{f} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2718}
\begin {gather*} a x-\frac {b \cos (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rubi steps
\begin {align*} \int (a+b \sin (e+f x)) \, dx &=a x+b \int \sin (e+f x) \, dx\\ &=a x-\frac {b \cos (e+f x)}{f}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.69 \begin {gather*} a x-\frac {b \cos (e) \cos (f x)}{f}+\frac {b \sin (e) \sin (f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 17, normalized size = 1.06
method | result | size |
default | \(a x -\frac {b \cos \left (f x +e \right )}{f}\) | \(17\) |
risch | \(a x -\frac {b \cos \left (f x +e \right )}{f}\) | \(17\) |
derivativedivides | \(\frac {a \left (f x +e \right )-b \cos \left (f x +e \right )}{f}\) | \(22\) |
norman | \(\frac {a x +a x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {2 b \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 1.06 \begin {gather*} a x - \frac {b \cos \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 19, normalized size = 1.19 \begin {gather*} \frac {a f x - b \cos \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 1.19 \begin {gather*} a x + b \left (\begin {cases} - \frac {\cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \sin {\left (e \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 17, normalized size = 1.06 \begin {gather*} a x - \frac {b \cos \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.51, size = 25, normalized size = 1.56 \begin {gather*} a\,x-\frac {2\,b}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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