Optimal. Leaf size=118 \[ \frac{2 \sqrt{\sin (d+e x)} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{e \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}}} \]
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Rubi [A] time = 0.140733, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3164, 3119, 2653} \[ \frac{2 \sqrt{\sin (d+e x)} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{e \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}}} \]
Antiderivative was successfully verified.
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Rule 3164
Rule 3119
Rule 2653
Rubi steps
\begin{align*} \int \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)} \, dx &=\frac{\left (\sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)}\right ) \int \sqrt{b+c \cos (d+e x)+a \sin (d+e x)} \, dx}{\sqrt{b+c \cos (d+e x)+a \sin (d+e x)}}\\ &=\frac{\left (\sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)}\right ) \int \sqrt{\frac{b}{b+\sqrt{a^2+c^2}}+\frac{\sqrt{a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt{a^2+c^2}}} \, dx}{\sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}\\ &=\frac{2 \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right ) \sqrt{\sin (d+e x)}}{e \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}\\ \end{align*}
Mathematica [F] time = 12.6705, size = 0, normalized size = 0. \[ \int \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.435, size = 12362, normalized size = 104.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt{\sin \left (e x + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt{\sin \left (e x + d\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \csc{\left (d + e x \right )} + c \cot{\left (d + e x \right )}} \sqrt{\sin{\left (d + e x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt{\sin \left (e x + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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