Optimal. Leaf size=90 \[ -\frac {2 \cos (a+b x)}{7 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \cos (a+b x)}{21 b \sqrt {\csc (a+b x)}}+\frac {10 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{21 b} \]
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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 = {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 2641
Rule 3769
Rule 3771
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*}
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Mathematica [A] time = 0.15, size = 65, normalized size = 0.72 \[ -\frac {\sqrt {\csc (a+b x)} \left (26 \sin (2 (a+b x))-3 \sin (4 (a+b x))+40 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{84 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\csc \left (b x + a\right )^{\frac {7}{2}}}, x\right ) \]
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 \[ \int \frac {1}{\csc \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.20, size = 104, normalized size = 1.16 \[ \frac {\frac {2 \left (\cos ^{4}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{7}+\frac {5 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{21}-\frac {16 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{21}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]
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 \[ \int \frac {1}{\csc \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\csc ^{\frac {7}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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