Optimal. Leaf size=116 \[ \frac {3 b \sqrt {b^2+c^2} \sin (d+e x)}{2 e}-\frac {3 c \sqrt {b^2+c^2} \cos (d+e x)}{2 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{2 e}+\frac {3}{2} x \left (b^2+c^2\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3113, 2637, 2638} \[ \frac {3 b \sqrt {b^2+c^2} \sin (d+e x)}{2 e}-\frac {3 c \sqrt {b^2+c^2} \cos (d+e x)}{2 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{2 e}+\frac {3}{2} x \left (b^2+c^2\right ) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3113
Rubi steps
\begin {align*} \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2 \, dx &=-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{2 e}+\frac {1}{2} \left (3 \sqrt {b^2+c^2}\right ) \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right ) \, dx\\ &=\frac {3}{2} \left (b^2+c^2\right ) x-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{2 e}+\frac {1}{2} \left (3 b \sqrt {b^2+c^2}\right ) \int \cos (d+e x) \, dx+\frac {1}{2} \left (3 c \sqrt {b^2+c^2}\right ) \int \sin (d+e x) \, dx\\ &=\frac {3}{2} \left (b^2+c^2\right ) x-\frac {3 c \sqrt {b^2+c^2} \cos (d+e x)}{2 e}+\frac {3 b \sqrt {b^2+c^2} \sin (d+e x)}{2 e}-\frac {(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 111, normalized size = 0.96 \[ \frac {8 b \sqrt {b^2+c^2} \sin (d+e x)-8 c \sqrt {b^2+c^2} \cos (d+e x)+b^2 \sin (2 (d+e x))+6 b^2 d+6 b^2 e x-2 b c \cos (2 (d+e x))-c^2 \sin (2 (d+e x))+6 c^2 d+6 c^2 e x}{4 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 81, normalized size = 0.70 \[ -\frac {2 \, b c \cos \left (e x + d\right )^{2} - 3 \, {\left (b^{2} + c^{2}\right )} e x - {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) + 4 \, \sqrt {b^{2} + c^{2}} {\left (c \cos \left (e x + d\right ) - b \sin \left (e x + d\right )\right )}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 92, normalized size = 0.79 \[ -\frac {1}{2} \, b c \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} - 2 \, \sqrt {b^{2} + c^{2}} c \cos \left (x e + d\right ) e^{\left (-1\right )} + 2 \, \sqrt {b^{2} + c^{2}} b e^{\left (-1\right )} \sin \left (x e + d\right ) + \frac {1}{4} \, {\left (b^{2} - c^{2}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + \frac {3}{2} \, {\left (b^{2} + c^{2}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 124, normalized size = 1.07 \[ \frac {b^{2} \left (\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-\left (\cos ^{2}\left (e x +d \right )\right ) b c +c^{2} \left (-\frac {\sin \left (e x +d \right ) \cos \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+2 \sqrt {b^{2}+c^{2}}\, b \sin \left (e x +d \right )-2 \sqrt {b^{2}+c^{2}}\, c \cos \left (e x +d \right )+b^{2} \left (e x +d \right )+c^{2} \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 113, normalized size = 0.97 \[ b^{2} x + c^{2} x - \frac {b c \cos \left (e x + d\right )^{2}}{e} + \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{4 \, e} + \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{4 \, e} - 2 \, \sqrt {b^{2} + c^{2}} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {b \sin \left (e x + d\right )}{e}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.08, size = 100, normalized size = 0.86 \[ \frac {b^2\,\sin \left (2\,d+2\,e\,x\right )-c^2\,\sin \left (2\,d+2\,e\,x\right )+16\,c\,{\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\sqrt {b^2+c^2}+8\,b\,\sin \left (d+e\,x\right )\,\sqrt {b^2+c^2}+4\,b\,c\,{\sin \left (d+e\,x\right )}^2+6\,b^2\,e\,x+6\,c^2\,e\,x}{4\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 192, normalized size = 1.66 \[ \begin {cases} \frac {b^{2} x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {b^{2} x \cos ^{2}{\left (d + e x \right )}}{2} + b^{2} x + \frac {b^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} - \frac {b c \cos ^{2}{\left (d + e x \right )}}{e} + \frac {2 b \sqrt {b^{2} + c^{2}} \sin {\left (d + e x \right )}}{e} + \frac {c^{2} x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {c^{2} x \cos ^{2}{\left (d + e x \right )}}{2} + c^{2} x - \frac {c^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} - \frac {2 c \sqrt {b^{2} + c^{2}} \cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \left (b \cos {\relax (d )} + c \sin {\relax (d )} + \sqrt {b^{2} + c^{2}}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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