Optimal. Leaf size=44 \[ \frac {a (c+d x)^2}{2 d}+\frac {a d \cos (e+f x)}{f^2}+\frac {a (c+d x) \sin (e+f x)}{f} \]
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Rubi [A]
time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3398, 3377,
2718} \begin {gather*} \frac {a (c+d x) \sin (e+f x)}{f}+\frac {a (c+d x)^2}{2 d}+\frac {a d \cos (e+f x)}{f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3398
Rubi steps
\begin {align*} \int (c+d x) (a+a \cos (e+f x)) \, dx &=\int (a (c+d x)+a (c+d x) \cos (e+f x)) \, dx\\ &=\frac {a (c+d x)^2}{2 d}+a \int (c+d x) \cos (e+f x) \, dx\\ &=\frac {a (c+d x)^2}{2 d}+\frac {a (c+d x) \sin (e+f x)}{f}-\frac {(a d) \int \sin (e+f x) \, dx}{f}\\ &=\frac {a (c+d x)^2}{2 d}+\frac {a d \cos (e+f x)}{f^2}+\frac {a (c+d x) \sin (e+f x)}{f}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 52, normalized size = 1.18 \begin {gather*} \frac {a (-2 (e+f x) (d e-2 c f-d f x)+4 d \cos (e+f x)+4 f (c+d x) \sin (e+f x))}{4 f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs.
\(2(42)=84\).
time = 0.05, size = 89, normalized size = 2.02
method | result | size |
risch | \(\frac {d a \,x^{2}}{2}+a c x +\frac {a d \cos \left (f x +e \right )}{f^{2}}+\frac {a \left (d x +c \right ) \sin \left (f x +e \right )}{f}\) | \(41\) |
derivativedivides | \(\frac {a c \sin \left (f x +e \right )-\frac {a d e \sin \left (f x +e \right )}{f}+\frac {a d \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f}+a c \left (f x +e \right )-\frac {a d e \left (f x +e \right )}{f}+\frac {a d \left (f x +e \right )^{2}}{2 f}}{f}\) | \(89\) |
default | \(\frac {a c \sin \left (f x +e \right )-\frac {a d e \sin \left (f x +e \right )}{f}+\frac {a d \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f}+a c \left (f x +e \right )-\frac {a d e \left (f x +e \right )}{f}+\frac {a d \left (f x +e \right )^{2}}{2 f}}{f}\) | \(89\) |
norman | \(\frac {a c x +a c x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {d a \,x^{2}}{2}-\frac {2 d a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f^{2}}+\frac {2 a c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {d a \,x^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {2 d a x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (44) = 88\).
time = 0.30, size = 101, normalized size = 2.30 \begin {gather*} \frac {2 \, {\left (f x + e\right )} a c + \frac {{\left (f x + e\right )}^{2} a d}{f} - \frac {2 \, {\left (f x + e\right )} a d e}{f} + 2 \, a c \sin \left (f x + e\right ) - \frac {2 \, a d e \sin \left (f x + e\right )}{f} + \frac {2 \, {\left ({\left (f x + e\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right )\right )} a d}{f}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 53, normalized size = 1.20 \begin {gather*} \frac {a d f^{2} x^{2} + 2 \, a c f^{2} x + 2 \, a d \cos \left (f x + e\right ) + 2 \, {\left (a d f x + a c f\right )} \sin \left (f x + e\right )}{2 \, f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 68, normalized size = 1.55 \begin {gather*} \begin {cases} a c x + \frac {a c \sin {\left (e + f x \right )}}{f} + \frac {a d x^{2}}{2} + \frac {a d x \sin {\left (e + f x \right )}}{f} + \frac {a d \cos {\left (e + f x \right )}}{f^{2}} & \text {for}\: f \neq 0 \\\left (a \cos {\left (e \right )} + a\right ) \left (c x + \frac {d x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, a d x^{2} + a c x + \frac {a d \cos \left (f x + e\right )}{f^{2}} + \frac {{\left (a d f x + a c f\right )} \sin \left (f x + e\right )}{f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 52, normalized size = 1.18 \begin {gather*} \frac {\frac {a\,f\,\left (2\,c\,\sin \left (e+f\,x\right )+2\,d\,x\,\sin \left (e+f\,x\right )\right )}{2}+a\,d\,\cos \left (e+f\,x\right )}{f^2}+\frac {a\,\left (d\,x^2+2\,c\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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