Optimal. Leaf size=34 \[ \frac {2 \sqrt {a \sin (c+d x)+a}}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.07, 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} \[ \frac {2 \sqrt {a \sin (c+d x)+a}}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {2 \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 34, normalized size = 1.00 \[ \frac {2 \sqrt {a (\sin (c+d x)+1)}}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 38, normalized size = 1.12 \[ \frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{d e^{2} \cos \left (d x + c\right )} \]
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 \[ \int \frac {\sqrt {a \sin \left (d x + c\right ) + a}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 34, normalized size = 1.00 \[ \frac {2 \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}{d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 131, normalized size = 3.85 \[ \frac {2 \, {\left (\sqrt {a} \sqrt {e} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{{\left (e^{2} + \frac {e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.31, size = 30, normalized size = 0.88 \[ \frac {2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{d\,e\,\sqrt {e\,\cos \left (c+d\,x\right )}} \]
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 \[ \int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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