Optimal. Leaf size=105 \[ \frac{10 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right ) \sqrt{c \csc (a+b x)}}{21 b c^4}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}} \]
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Rubi [A] time = 0.0491781, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3769, 3771, 2641} \[ -\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{10 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \csc (a+b x)}}{21 b c^4}-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(c \csc (a+b x))^{7/2}} \, dx &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}+\frac{5 \int \frac{1}{(c \csc (a+b x))^{3/2}} \, dx}{7 c^2}\\ &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{5 \int \sqrt{c \csc (a+b x)} \, dx}{21 c^4}\\ &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{\left (5 \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{21 c^4}\\ &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{10 \sqrt{c \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 c^4}\\ \end{align*}
Mathematica [A] time = 0.143812, size = 70, normalized size = 0.67 \[ -\frac{\sqrt{c \csc (a+b x)} \left (40 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+26 \sin (2 (a+b x))-3 \sin (4 (a+b x))\right )}{84 b c^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.275, size = 216, normalized size = 2.1 \begin{align*} -{\frac{\sqrt{2}}{21\,b \left ( -1+\cos \left ( bx+a \right ) \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{3}} \left ( 5\,i\sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\sin \left ( bx+a \right ) \sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}-3\,\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{4}+3\,\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{3}+8\,\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{2}-8\,\sqrt{2}\cos \left ( bx+a \right ) \right ) \left ({\frac{c}{\sin \left ( bx+a \right ) }} \right ) ^{-{\frac{7}{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{1}{\left (c \csc \left (b x + a\right )\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{\sqrt{c \csc \left (b x + a\right )}}{c^{4} \csc \left (b x + a\right )^{4}}, 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}{\left (c \csc \left (b x + a\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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