Optimal. Leaf size=34 \[ -\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2671} \[ -\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}} \, dx &=-\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 34, normalized size = 1.00 \[ -\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 41, normalized size = 1.21 \[ -\frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{a d e \sin \left (d x + c\right ) + a d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 34, normalized size = 1.00 \[ -\frac {2 \cos \left (d x +c \right )}{d \sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+\sin \left (d x +c \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.81, size = 130, normalized size = 3.82 \[ -\frac {2 \, {\left (\sqrt {a} \sqrt {e} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{{\left (a e + \frac {a e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.66, size = 46, normalized size = 1.35 \[ -\frac {2\,\cos \left (c+d\,x\right )\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}}{a\,d\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sqrt {e \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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