Optimal. Leaf size=112 \[ -\frac{2}{3} \sqrt{\sin (x)} \sqrt{\csc (x)} \text{EllipticF}\left (\frac{\pi }{4}-\frac{x}{2},2\right ) (3 a B+3 A b+b C)+2 \sqrt{\sin (x)} \sqrt{\csc (x)} E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) (b B-a (A-C))-2 \cos (x) \sqrt{\csc (x)} (a C+b B)-\frac{2}{3} b C \cos (x) \csc ^{\frac{3}{2}}(x) \]
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Rubi [A] time = 0.180087, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {4076, 4047, 3771, 2641, 4046, 2639} \[ -\frac{2}{3} \sqrt{\sin (x)} \sqrt{\csc (x)} F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) (3 a B+3 A b+b C)+2 \sqrt{\sin (x)} \sqrt{\csc (x)} E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) (b B-a (A-C))-2 \cos (x) \sqrt{\csc (x)} (a C+b B)-\frac{2}{3} b C \cos (x) \csc ^{\frac{3}{2}}(x) \]
Antiderivative was successfully verified.
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Rule 4076
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \csc (x)) \left (A+B \csc (x)+C \csc ^2(x)\right )}{\sqrt{\csc (x)}} \, dx &=-\frac{2}{3} b C \cos (x) \csc ^{\frac{3}{2}}(x)+\frac{2}{3} \int \frac{\frac{3 a A}{2}+\frac{1}{2} (3 A b+3 a B+b C) \csc (x)+\frac{3}{2} (b B+a C) \csc ^2(x)}{\sqrt{\csc (x)}} \, dx\\ &=-\frac{2}{3} b C \cos (x) \csc ^{\frac{3}{2}}(x)+\frac{2}{3} \int \frac{\frac{3 a A}{2}+\frac{3}{2} (b B+a C) \csc ^2(x)}{\sqrt{\csc (x)}} \, dx+\frac{1}{3} (3 A b+3 a B+b C) \int \sqrt{\csc (x)} \, dx\\ &=-2 (b B+a C) \cos (x) \sqrt{\csc (x)}-\frac{2}{3} b C \cos (x) \csc ^{\frac{3}{2}}(x)+(-b B+a (A-C)) \int \frac{1}{\sqrt{\csc (x)}} \, dx+\frac{1}{3} \left ((3 A b+3 a B+b C) \sqrt{\csc (x)} \sqrt{\sin (x)}\right ) \int \frac{1}{\sqrt{\sin (x)}} \, dx\\ &=-2 (b B+a C) \cos (x) \sqrt{\csc (x)}-\frac{2}{3} b C \cos (x) \csc ^{\frac{3}{2}}(x)-\frac{2}{3} (3 A b+3 a B+b C) \sqrt{\csc (x)} F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{\sin (x)}+\left ((-b B+a (A-C)) \sqrt{\csc (x)} \sqrt{\sin (x)}\right ) \int \sqrt{\sin (x)} \, dx\\ &=-2 (b B+a C) \cos (x) \sqrt{\csc (x)}-\frac{2}{3} b C \cos (x) \csc ^{\frac{3}{2}}(x)+2 (b B-a (A-C)) \sqrt{\csc (x)} E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{\sin (x)}-\frac{2}{3} (3 A b+3 a B+b C) \sqrt{\csc (x)} F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{\sin (x)}\\ \end{align*}
Mathematica [A] time = 0.74282, size = 133, normalized size = 1.19 \[ -\frac{4 (a+b \csc (x)) \left (A+B \csc (x)+C \csc ^2(x)\right ) \left (\sqrt{\sin (x)} \text{EllipticF}\left (\frac{1}{4} (\pi -2 x),2\right ) (3 a B+3 A b+b C)-3 \sqrt{\sin (x)} E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right ) (a (C-A)+b B)+3 a C \cos (x)+3 b B \cos (x)+b C \cot (x)\right )}{3 \csc ^{\frac{5}{2}}(x) (a \sin (x)+b) (A (-\cos (2 x))+A+2 B \sin (x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.643, size = 204, normalized size = 1.8 \begin{align*} -{\frac{1}{3\,\cos \left ( x \right ) } \left ( \left ( 6\,bB+6\,Ca \right ) \sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{2}+\sqrt{\sin \left ( x \right ) +1}\sqrt{-2\,\sin \left ( x \right ) +2}\sqrt{-\sin \left ( x \right ) } \left ( 6\,A{\it EllipticE} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) a-3\,A{\it EllipticF} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) a-3\,A{\it EllipticF} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) b-6\,B{\it EllipticE} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) b-3\,B{\it EllipticF} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) a+3\,B{\it EllipticF} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) b-6\,C{\it EllipticE} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) a+3\,C{\it EllipticF} \left ( \sqrt{\sin \left ( x \right ) +1},1/2\,\sqrt{2} \right ) a-C{\it EllipticF} \left ( \sqrt{\sin \left ( x \right ) +1},{\frac{\sqrt{2}}{2}} \right ) b \right ) \sin \left ( x \right ) +2\,Cb \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{-{\frac{3}{2}}}} \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 \frac{{\left (C \csc \left (x\right )^{2} + B \csc \left (x\right ) + A\right )}{\left (b \csc \left (x\right ) + a\right )}}{\sqrt{\csc \left (x\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 (\frac{C b \csc \left (x\right )^{3} +{\left (C a + B b\right )} \csc \left (x\right )^{2} + A a +{\left (B a + A b\right )} \csc \left (x\right )}{\sqrt{\csc \left (x\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (C \csc \left (x\right )^{2} + B \csc \left (x\right ) + A\right )}{\left (b \csc \left (x\right ) + a\right )}}{\sqrt{\csc \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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