Optimal. Leaf size=407 \[ \frac{4 b \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 \sqrt{b^2-a^2}}-\frac{4 b \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{4 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{4 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 \sqrt{b^2-a^2}}+\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d \sqrt{b^2-a^2}}-\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d \sqrt{b^2-a^2}}+\frac{2 x^{3/2}}{3 a} \]
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Rubi [A] time = 0.865294, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4205, 4191, 3323, 2264, 2190, 2531, 2282, 6589} \[ \frac{4 b \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 \sqrt{b^2-a^2}}-\frac{4 b \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{4 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{4 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 \sqrt{b^2-a^2}}+\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d \sqrt{b^2-a^2}}-\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d \sqrt{b^2-a^2}}+\frac{2 x^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 4205
Rule 4191
Rule 3323
Rule 2264
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{a+b \csc \left (c+d \sqrt{x}\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{a+b \csc (c+d x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^2}{a}-\frac{b x^2}{a (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{3/2}}{3 a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{b+a \sin (c+d x)} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 x^{3/2}}{3 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 x^{3/2}}{3 a}+\frac{(4 i b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b-2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}-\frac{(4 i b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b+2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}\\ &=\frac{2 x^{3/2}}{3 a}+\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{(4 i b) \operatorname{Subst}\left (\int x \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}+\frac{(4 i b) \operatorname{Subst}\left (\int x \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}\\ &=\frac{2 x^{3/2}}{3 a}+\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{4 b \sqrt{x} \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{4 b \sqrt{x} \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{(4 b) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{(4 b) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}\\ &=\frac{2 x^{3/2}}{3 a}+\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{4 b \sqrt{x} \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{4 b \sqrt{x} \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{(4 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{b-\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{(4 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a \sqrt{-a^2+b^2} d^3}\\ &=\frac{2 x^{3/2}}{3 a}+\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{4 b \sqrt{x} \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{4 b \sqrt{x} \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{4 i b \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{4 i b \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}\\ \end{align*}
Mathematica [A] time = 10.8972, size = 512, normalized size = 1.26 \[ \frac{2 \csc \left (c+d \sqrt{x}\right ) \left (a \sin \left (c+d \sqrt{x}\right )+b\right ) \left (x^{3/2}+\frac{3 b e^{3 i c} \left (a^2-b^2\right ) \left (-2 i d \sqrt{x} \text{PolyLog}\left (2,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{i \sqrt{e^{2 i c} \left (a^2-b^2\right )}+b e^{i c}}\right )+2 i d \sqrt{x} \text{PolyLog}\left (2,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{\sqrt{e^{2 i c} \left (a^2-b^2\right )}+i b e^{i c}}\right )+2 \text{PolyLog}\left (3,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{i \sqrt{e^{2 i c} \left (a^2-b^2\right )}+b e^{i c}}\right )-2 \text{PolyLog}\left (3,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{\sqrt{e^{2 i c} \left (a^2-b^2\right )}+i b e^{i c}}\right )+d^2 x \log \left (1+\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i b e^{i c}-\sqrt{e^{2 i c} \left (a^2-b^2\right )}}\right )-d^2 x \log \left (1+\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{\sqrt{e^{2 i c} \left (a^2-b^2\right )}+i b e^{i c}}\right )\right )}{d^3 \sqrt{e^{2 i c} \left (b^2-a^2\right )} \sqrt{e^{4 i c} \left (-\left (a^2-b^2\right )^2\right )}}\right )}{3 a \left (a+b \csc \left (c+d \sqrt{x}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{x} \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (\frac{\sqrt{x}}{b \csc \left (d \sqrt{x} + c\right ) + a}, 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 \frac{\sqrt{x}}{a + b \csc{\left (c + d \sqrt{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 \frac{\sqrt{x}}{b \csc \left (d \sqrt{x} + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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