Optimal. Leaf size=90 \[ -\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {6 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b} \]
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Rubi [A] time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3768, 3771, 2639} \[ -\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {6 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \csc ^{\frac {7}{2}}(a+b x) \, dx &=-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \int \csc ^{\frac {3}{2}}(a+b x) \, dx\\ &=-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx\\ &=-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {1}{5} \left (3 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx\\ &=-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 63, normalized size = 0.70 \[ \frac {\csc ^{\frac {5}{2}}(a+b x) \left (-7 \cos (a+b x)+3 \cos (3 (a+b x))+12 \sin ^{\frac {5}{2}}(a+b x) E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{10 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\csc \left (b x + a\right )^{\frac {7}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.89, size = 160, normalized size = 1.78 \[ \frac {6 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \left (\sin ^{2}\left (b x +a \right )\right ) \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \left (\sin ^{2}\left (b x +a \right )\right ) \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+6 \left (\sin ^{4}\left (b x +a \right )\right )-4 \left (\sin ^{2}\left (b x +a \right )\right )-2}{5 \sin \left (b x +a \right )^{\frac {5}{2}} \cos \left (b x +a \right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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