Optimal. Leaf size=77 \[ \frac{2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{d \tan (a+b x)}}{3 b}-\frac{2 d \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}} \]
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Rubi [A] time = 0.10574, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2599, 2601, 2573, 2641} \[ \frac{2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{d \tan (a+b x)}}{3 b}-\frac{2 d \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2599
Rule 2601
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sqrt{d \tan (a+b x)} \, dx &=-\frac{2 d \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}}+\frac{2}{3} \int \csc (a+b x) \sqrt{d \tan (a+b x)} \, dx\\ &=-\frac{2 d \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}}+\frac{\left (2 \sqrt{\cos (a+b x)} \sqrt{d \tan (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)} \sqrt{\sin (a+b x)}} \, dx}{3 \sqrt{\sin (a+b x)}}\\ &=-\frac{2 d \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}}+\frac{1}{3} \left (2 \csc (a+b x) \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{2 d \csc (a+b x)}{3 b \sqrt{d \tan (a+b x)}}+\frac{2 \csc (a+b x) F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}{3 b}\\ \end{align*}
Mathematica [C] time = 0.566555, size = 115, normalized size = 1.49 \[ \frac{2 \cos (2 (a+b x)) \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \left (\sqrt{\sec ^2(a+b x)}+2 \sqrt [4]{-1} \tan ^{\frac{3}{2}}(a+b x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tan (a+b x)}\right )\right |-1\right )\right )}{3 b d \left (\tan ^2(a+b x)-1\right ) \sqrt{\sec ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.167, size = 301, normalized size = 3.9 \begin{align*}{\frac{\sqrt{2} \left ( \cos \left ( bx+a \right ) -1 \right ) ^{2} \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}{3\,b \left ( \sin \left ( bx+a \right ) \right ) ^{6}} \left ( 2\,\cos \left ( bx+a \right ) \sqrt{{\frac{\cos \left ( bx+a \right ) -1}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{\cos \left ( bx+a \right ) -1+\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{\cos \left ( bx+a \right ) -1-\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sin \left ( bx+a \right ){\it EllipticF} \left ( \sqrt{-{\frac{\cos \left ( bx+a \right ) -1-\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \right ) +2\,\sin \left ( bx+a \right ){\it EllipticF} \left ( \sqrt{-{\frac{\cos \left ( bx+a \right ) -1-\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{\cos \left ( bx+a \right ) -1}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{\cos \left ( bx+a \right ) -1+\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{\cos \left ( bx+a \right ) -1-\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}-\cos \left ( bx+a \right ) \sqrt{2} \right ) \sqrt{{\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }}}} \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{d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{3}\,{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{d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{3}, 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 \sqrt{d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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