Optimal. Leaf size=77 \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b c^2 \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3769, 3771, 2639} \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b c^2 \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx &=-\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx}{5 c^2}\\ &=-\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac {3 \int \sqrt {\sin (a+b x)} \, dx}{5 c^2 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}}\\ &=-\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac {6 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{5 b c^2 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 60, normalized size = 0.78 \[ \frac {-2 \sin (2 (a+b x))-\frac {12 E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )}{\sqrt {\sin (a+b x)}}}{10 b c^2 \sqrt {c \csc (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \csc \left (b x + a\right )}}{c^{3} \csc \left (b x + a\right )^{3}}, x\right ) \]
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 \[ \int \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.85, size = 534, normalized size = 6.94 \[ \frac {\left (-6 \cos \left (b x +a \right ) \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+3 \cos \left (b x +a \right ) \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+\left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}-6 \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+3 \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-4 \cos \left (b x +a \right ) \sqrt {2}+3 \sqrt {2}\right ) \sqrt {2}}{5 b \left (\frac {c}{\sin \left (b x +a \right )}\right )^{\frac {5}{2}} \sin \left (b x +a \right )^{3}} \]
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 \[ \int \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (c \csc {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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