Optimal. Leaf size=29 \[ \frac {2 a \cos (d+e x)}{e}+2 a x+\frac {2 b \sin (d+e x)}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2637, 2638} \[ \frac {2 a \cos (d+e x)}{e}+2 a x+\frac {2 b \sin (d+e x)}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 2638
Rubi steps
\begin {align*} \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx &=2 a x-(2 a) \int \sin (d+e x) \, dx+(2 b) \int \cos (d+e x) \, dx\\ &=2 a x+\frac {2 a \cos (d+e x)}{e}+\frac {2 b \sin (d+e x)}{e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 53, normalized size = 1.83 \[ -\frac {2 a \sin (d) \sin (e x)}{e}+\frac {2 a \cos (d) \cos (e x)}{e}+2 a x+\frac {2 b \sin (d) \cos (e x)}{e}+\frac {2 b \cos (d) \sin (e x)}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.90, size = 26, normalized size = 0.90 \[ \frac {2 \, {\left (a e x + a \cos \left (e x + d\right ) + b \sin \left (e x + d\right )\right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 29, normalized size = 1.00 \[ 2 \, a \cos \left (x e + d\right ) e^{\left (-1\right )} + 2 \, b e^{\left (-1\right )} \sin \left (x e + d\right ) + 2 \, a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 30, normalized size = 1.03 \[ 2 a x +\frac {2 a \cos \left (e x +d \right )}{e}+\frac {2 b \sin \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 29, normalized size = 1.00 \[ 2 \, a x + \frac {2 \, a \cos \left (e x + d\right )}{e} + \frac {2 \, b \sin \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.44, size = 29, normalized size = 1.00 \[ 2\,a\,x+\frac {2\,a\,\cos \left (d+e\,x\right )}{e}+\frac {2\,b\,\sin \left (d+e\,x\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 39, normalized size = 1.34 \[ 2 a x - 2 a \left (\begin {cases} - \frac {\cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \sin {\relax (d )} & \text {otherwise} \end {cases}\right ) + 2 b \left (\begin {cases} \frac {\sin {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \cos {\relax (d )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________