Optimal. Leaf size=37 \[ x \sqrt {b^2+c^2}+\frac {b \sin (d+e x)}{e}-\frac {c \cos (d+e x)}{e} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2637, 2638} \[ x \sqrt {b^2+c^2}+\frac {b \sin (d+e x)}{e}-\frac {c \cos (d+e x)}{e} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin {align*} \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right ) \, dx &=\sqrt {b^2+c^2} x+b \int \cos (d+e x) \, dx+c \int \sin (d+e x) \, dx\\ &=\sqrt {b^2+c^2} x-\frac {c \cos (d+e x)}{e}+\frac {b \sin (d+e x)}{e}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.97 \[ \frac {e x \sqrt {b^2+c^2}+b \sin (d+e x)-c \cos (d+e x)}{e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 34, normalized size = 0.92 \[ \frac {\sqrt {b^{2} + c^{2}} e x - c \cos \left (e x + d\right ) + b \sin \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 35, normalized size = 0.95 \[ -c \cos \left (x e + d\right ) e^{\left (-1\right )} + b e^{\left (-1\right )} \sin \left (x e + d\right ) + \sqrt {b^{2} + c^{2}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 0.97 \[ -\frac {c \cos \left (e x +d \right )}{e}+\frac {b \sin \left (e x +d \right )}{e}+x \sqrt {b^{2}+c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 35, normalized size = 0.95 \[ \sqrt {b^{2} + c^{2}} x - \frac {c \cos \left (e x + d\right )}{e} + \frac {b \sin \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 48, normalized size = 1.30 \[ x\,\sqrt {b^2+c^2}-\frac {2\,c-2\,b\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 42, normalized size = 1.14 \[ b \left (\begin {cases} \frac {\sin {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \cos {\relax (d )} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {\cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \sin {\relax (d )} & \text {otherwise} \end {cases}\right ) + x \sqrt {b^{2} + c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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