Optimal. Leaf size=178 \[ \frac{5 b \left (b^2+c^2\right ) \sin (d+e x)}{2 e}-\frac{5 c \left (b^2+c^2\right ) \cos (d+e x)}{2 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{3 e}-\frac{5 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{6 e}+\frac{5}{2} x \left (b^2+c^2\right )^{3/2} \]
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Rubi [A] time = 0.101832, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3113, 2637, 2638} \[ \frac{5 b \left (b^2+c^2\right ) \sin (d+e x)}{2 e}-\frac{5 c \left (b^2+c^2\right ) \cos (d+e x)}{2 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}{3 e}-\frac{5 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )}{6 e}+\frac{5}{2} x \left (b^2+c^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 3113
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 )^3 \, 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}{3 e}+\frac{1}{3} \left (5 \sqrt{b^2+c^2}\right ) \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2 \, dx\\ &=-\frac{5 \sqrt{b^2+c^2} (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 )}{6 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}{3 e}+\frac{1}{2} \left (5 \left (b^2+c^2\right )\right ) \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right ) \, dx\\ &=\frac{5}{2} \left (b^2+c^2\right )^{3/2} x-\frac{5 \sqrt{b^2+c^2} (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 )}{6 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}{3 e}+\frac{1}{2} \left (5 b \left (b^2+c^2\right )\right ) \int \cos (d+e x) \, dx+\frac{1}{2} \left (5 c \left (b^2+c^2\right )\right ) \int \sin (d+e x) \, dx\\ &=\frac{5}{2} \left (b^2+c^2\right )^{3/2} x-\frac{5 c \left (b^2+c^2\right ) \cos (d+e x)}{2 e}+\frac{5 b \left (b^2+c^2\right ) \sin (d+e x)}{2 e}-\frac{5 \sqrt{b^2+c^2} (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 )}{6 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}{3 e}\\ \end{align*}
Mathematica [C] time = 0.661051, size = 163, normalized size = 0.92 \[ \frac{30 (b-i c) (b+i c) \sqrt{b^2+c^2} (d+e x)+45 b \left (b^2+c^2\right ) \sin (d+e x)+9 \left (b^2-c^2\right ) \sqrt{b^2+c^2} \sin (2 (d+e x))+b \left (b^2-3 c^2\right ) \sin (3 (d+e x))-45 c \left (b^2+c^2\right ) \cos (d+e x)-18 b c \sqrt{b^2+c^2} \cos (2 (d+e x))+c \left (c^2-3 b^2\right ) \cos (3 (d+e x))}{12 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 250, normalized size = 1.4 \begin{align*}{\frac{1}{e} \left ({\frac{{b}^{3} \left ( 2+ \left ( \cos \left ( ex+d \right ) \right ) ^{2} \right ) \sin \left ( ex+d \right ) }{3}}- \left ( \cos \left ( ex+d \right ) \right ) ^{3}{b}^{2}c+3\,\sqrt{{b}^{2}+{c}^{2}}{b}^{2} \left ( 1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) +b{c}^{2} \left ( \sin \left ( ex+d \right ) \right ) ^{3}-3\,\sqrt{{b}^{2}+{c}^{2}}bc \left ( \cos \left ( ex+d \right ) \right ) ^{2}+3\,\sin \left ( ex+d \right ){b}^{3}+3\,b{c}^{2}\sin \left ( ex+d \right ) -{\frac{{c}^{3} \left ( 2+ \left ( \sin \left ( ex+d \right ) \right ) ^{2} \right ) \cos \left ( ex+d \right ) }{3}}+3\,\sqrt{{b}^{2}+{c}^{2}}{c}^{2} \left ( -1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) -3\,\cos \left ( ex+d \right ){b}^{2}c-3\,\cos \left ( ex+d \right ){c}^{3}+\sqrt{{b}^{2}+{c}^{2}}{b}^{2} \left ( ex+d \right ) +\sqrt{{b}^{2}+{c}^{2}}{c}^{2} \left ( ex+d \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00097, size = 279, normalized size = 1.57 \begin{align*} -\frac{b^{2} c \cos \left (e x + d\right )^{3}}{e} + \frac{b c^{2} \sin \left (e x + d\right )^{3}}{e} - \frac{{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{3 \, e} + \frac{{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{3 \, e} +{\left (b^{2} + c^{2}\right )}^{\frac{3}{2}} x - 3 \,{\left (b^{2} + c^{2}\right )}{\left (\frac{c \cos \left (e x + d\right )}{e} - \frac{b \sin \left (e x + d\right )}{e}\right )} - \frac{3}{4} \,{\left (\frac{4 \, 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}}{e} - \frac{{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} \sqrt{b^{2} + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25459, size = 342, normalized size = 1.92 \begin{align*} -\frac{2 \,{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} + 6 \,{\left (3 \, b^{2} c + 4 \, c^{3}\right )} \cos \left (e x + d\right ) - 2 \,{\left (11 \, b^{3} + 12 \, b c^{2} +{\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right ) + 3 \,{\left (6 \, b c \cos \left (e x + d\right )^{2} - 5 \,{\left (b^{2} + c^{2}\right )} e x - 3 \,{\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right )\right )} \sqrt{b^{2} + c^{2}}}{6 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.881, size = 415, normalized size = 2.33 \begin{align*} \begin{cases} \frac{2 b^{3} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac{b^{3} \sin{\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} + \frac{3 b^{3} \sin{\left (d + e x \right )}}{e} - \frac{b^{2} c \cos ^{3}{\left (d + e x \right )}}{e} - \frac{3 b^{2} c \cos{\left (d + e x \right )}}{e} + \frac{3 b^{2} x \sqrt{b^{2} + c^{2}} \sin ^{2}{\left (d + e x \right )}}{2} + \frac{3 b^{2} x \sqrt{b^{2} + c^{2}} \cos ^{2}{\left (d + e x \right )}}{2} + b^{2} x \sqrt{b^{2} + c^{2}} + \frac{3 b^{2} \sqrt{b^{2} + c^{2}} \sin{\left (d + e x \right )} \cos{\left (d + e x \right )}}{2 e} + \frac{b c^{2} \sin ^{3}{\left (d + e x \right )}}{e} + \frac{3 b c^{2} \sin{\left (d + e x \right )}}{e} + \frac{3 b c \sqrt{b^{2} + c^{2}} \sin ^{2}{\left (d + e x \right )}}{e} - \frac{c^{3} \sin ^{2}{\left (d + e x \right )} \cos{\left (d + e x \right )}}{e} - \frac{2 c^{3} \cos ^{3}{\left (d + e x \right )}}{3 e} - \frac{3 c^{3} \cos{\left (d + e x \right )}}{e} + \frac{3 c^{2} x \sqrt{b^{2} + c^{2}} \sin ^{2}{\left (d + e x \right )}}{2} + \frac{3 c^{2} x \sqrt{b^{2} + c^{2}} \cos ^{2}{\left (d + e x \right )}}{2} + c^{2} x \sqrt{b^{2} + c^{2}} - \frac{3 c^{2} \sqrt{b^{2} + c^{2}} \sin{\left (d + e x \right )} \cos{\left (d + e x \right )}}{2 e} & \text{for}\: e \neq 0 \\x \left (b \cos{\left (d \right )} + c \sin{\left (d \right )} + \sqrt{b^{2} + c^{2}}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15752, size = 269, normalized size = 1.51 \begin{align*} -\frac{3}{2} \, \sqrt{b^{2} + c^{2}} b c \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} - \frac{1}{12} \,{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (3 \, x e + 3 \, d\right ) e^{\left (-1\right )} - \frac{15}{4} \,{\left (b^{2} c + c^{3}\right )} \cos \left (x e + d\right ) e^{\left (-1\right )} + \frac{1}{12} \,{\left (b^{3} - 3 \, b c^{2}\right )} e^{\left (-1\right )} \sin \left (3 \, x e + 3 \, d\right ) + \frac{3}{4} \,{\left (\sqrt{b^{2} + c^{2}} b^{2} - \sqrt{b^{2} + c^{2}} c^{2}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) + \frac{15}{4} \,{\left (b^{3} + b c^{2}\right )} e^{\left (-1\right )} \sin \left (x e + d\right ) +{\left (b^{2} + c^{2}\right )}^{\frac{3}{2}} x + \frac{3}{2} \,{\left (\sqrt{b^{2} + c^{2}} b^{2} + \sqrt{b^{2} + c^{2}} c^{2}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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