Optimal. Leaf size=129 \[ \frac {b \sin (d+e x)-c \cos (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {c-\sqrt {b^2+c^2} \sin (d+e x)}{3 c e \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))} \]
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Rubi [A] time = 0.09, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3116, 3114} \[ -\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \sqrt {b^2+c^2} \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {c-\sqrt {b^2+c^2} \sin (d+e x)}{3 c e \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))} \]
Antiderivative was successfully verified.
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Rule 3114
Rule 3116
Rubi steps
\begin {align*} \int \frac {1}{\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)}{3 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}+\frac {\int \frac {1}{\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx}{3 \sqrt {b^2+c^2}}\\ &=-\frac {c \cos (d+e x)-b \sin (d+e x)}{3 \sqrt {b^2+c^2} e \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^2}-\frac {c-\sqrt {b^2+c^2} \sin (d+e x)}{3 c \sqrt {b^2+c^2} e (c \cos (d+e x)-b \sin (d+e x))}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 98, normalized size = 0.76 \[ \frac {-2 c \sqrt {b^2+c^2}+b^2 \sin ^3(d+e x)+2 b c \cos ^3(d+e x)+2 c^2 \sin (d+e x)+c^2 \sin (d+e x) \cos ^2(d+e x)}{3 c e (c \cos (d+e x)-b \sin (d+e x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.74, size = 192, normalized size = 1.49 \[ -\frac {3 \, b^{3} \cos \left (e x + d\right ) - {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{3} + {\left (3 \, b^{2} c + 2 \, c^{3} - {\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right ) - 2 \, {\left (b^{2} + c^{2}\right )}^{\frac {3}{2}}}{3 \, {\left ({\left (3 \, b^{4} c + 2 \, b^{2} c^{3} - c^{5}\right )} e \cos \left (e x + d\right )^{3} - 3 \, {\left (b^{4} c + b^{2} c^{3}\right )} e \cos \left (e x + d\right ) - {\left ({\left (b^{5} - 2 \, b^{3} c^{2} - 3 \, b c^{4}\right )} e \cos \left (e x + d\right )^{2} - {\left (b^{5} + b^{3} c^{2}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 160, normalized size = 1.24 \[ -\frac {2 \, {\left (8 \, b^{4} + 10 \, b^{2} c^{2} + 2 \, c^{4} + 3 \, {\left (2 \, b^{2} c^{2} + c^{4} + 2 \, \sqrt {b^{2} + c^{2}} b c^{2}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 3 \, {\left (4 \, b^{3} c + 3 \, b c^{3} + {\left (4 \, b^{2} c + c^{3}\right )} \sqrt {b^{2} + c^{2}}\right )} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 2 \, {\left (4 \, b^{3} + 3 \, b c^{2}\right )} \sqrt {b^{2} + c^{2}}\right )} e^{\left (-1\right )}}{3 \, {\left (c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b + \sqrt {b^{2} + c^{2}}\right )}^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 233, normalized size = 1.81 \[ \frac {2 \left (\sqrt {b^{2}+c^{2}}+b \right ) \left (-\frac {\left (\sqrt {b^{2}+c^{2}}+b \right ) \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{c^{2}}-\frac {\left (2 b^{2}+c^{2}+2 \sqrt {b^{2}+c^{2}}\, b \right ) \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{c^{3}}-\frac {2 \left (2 \sqrt {b^{2}+c^{2}}\, b^{2}+\sqrt {b^{2}+c^{2}}\, c^{2}+2 b^{3}+2 c^{2} b \right )}{3 c^{4}}\right )}{e \,c^{2} \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )+\frac {2 \sqrt {b^{2}+c^{2}}\, \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{c}+\frac {2 b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{c}+\frac {2 \sqrt {b^{2}+c^{2}}\, b}{c^{2}}+\frac {2 b^{2}}{c^{2}}+1\right ) \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 274, normalized size = 2.12 \[ -\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {4\,b^2+2\,c^2}{c^4}+\frac {4\,b\,\sqrt {b^2+c^2}}{c^4}\right )+\frac {\frac {16\,b^4}{3}+\frac {20\,b^2\,c^2}{3}+\frac {4\,c^4}{3}}{c^6}+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {8\,b^3+6\,b\,c^2}{c^5}+\frac {\left (8\,b^2+2\,c^2\right )\,\sqrt {b^2+c^2}}{c^5}\right )+\frac {\left (\frac {16\,b^3}{3}+4\,b\,c^2\right )\,\sqrt {b^2+c^2}}{c^6}}{e\,\left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {6\,b^2+3\,c^2}{c^2}+\frac {6\,b\,\sqrt {b^2+c^2}}{c^2}\right )+\frac {4\,b^3+3\,b\,c^2}{c^3}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (\frac {3\,\sqrt {b^2+c^2}}{c}+\frac {3\,b}{c}\right )+\frac {\left (4\,b^2+c^2\right )\,\sqrt {b^2+c^2}}{c^3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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