Optimal. Leaf size=90 \[ \frac{10 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{21 b}-\frac{2 \cos (a+b x)}{7 b \csc ^{\frac{5}{2}}(a+b x)}-\frac{10 \cos (a+b x)}{21 b \sqrt{\csc (a+b x)}} \]
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Rubi [A] time = 0.0383128, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3769, 3771, 2641} \[ -\frac{2 \cos (a+b x)}{7 b \csc ^{\frac{5}{2}}(a+b x)}-\frac{10 \cos (a+b x)}{21 b \sqrt{\csc (a+b x)}}+\frac{10 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{21 b} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\csc ^{\frac{7}{2}}(a+b x)} \, dx &=-\frac{2 \cos (a+b x)}{7 b \csc ^{\frac{5}{2}}(a+b x)}+\frac{5}{7} \int \frac{1}{\csc ^{\frac{3}{2}}(a+b x)} \, dx\\ &=-\frac{2 \cos (a+b x)}{7 b \csc ^{\frac{5}{2}}(a+b x)}-\frac{10 \cos (a+b x)}{21 b \sqrt{\csc (a+b x)}}+\frac{5}{21} \int \sqrt{\csc (a+b x)} \, dx\\ &=-\frac{2 \cos (a+b x)}{7 b \csc ^{\frac{5}{2}}(a+b x)}-\frac{10 \cos (a+b x)}{21 b \sqrt{\csc (a+b x)}}+\frac{1}{21} \left (5 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=-\frac{2 \cos (a+b x)}{7 b \csc ^{\frac{5}{2}}(a+b x)}-\frac{10 \cos (a+b x)}{21 b \sqrt{\csc (a+b x)}}+\frac{10 \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{21 b}\\ \end{align*}
Mathematica [A] time = 0.165642, size = 65, normalized size = 0.72 \[ -\frac{\sqrt{\csc (a+b x)} \left (40 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+26 \sin (2 (a+b x))-3 \sin (4 (a+b x))\right )}{84 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.198, size = 104, normalized size = 1.2 \begin{align*}{\frac{1}{\cos \left ( bx+a \right ) b} \left ({\frac{2\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{7}}+{\frac{5}{21}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }-{\frac{16\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) }{21}} \right ){\frac{1}{\sqrt{\sin \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 \frac{1}{\csc \left (b x + a\right )^{\frac{7}{2}}}\,{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{1}{\csc \left (b x + a\right )^{\frac{7}{2}}}, 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{1}{\csc \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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