Optimal. Leaf size=33 \[ -\frac {2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2619} \[ -\frac {2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2619
Rubi steps
\begin {align*} \int \frac {(d \csc (a+b x))^{5/2}}{\sqrt {c \sec (a+b x)}} \, dx &=-\frac {2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 33, normalized size = 1.00 \[ -\frac {2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 51, normalized size = 1.55 \[ -\frac {2 \, d^{2} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, b c \sin \left (b x + a\right )} \]
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 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}}}{\sqrt {c \sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 42, normalized size = 1.27 \[ -\frac {2 \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {5}{2}} \cos \left (b x +a \right ) \sin \left (b x +a \right )}{3 b \sqrt {\frac {c}{\cos \left (b x +a \right )}}} \]
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 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}}}{\sqrt {c \sec \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 49, normalized size = 1.48 \[ -\frac {d^2\,\sin \left (2\,a+2\,b\,x\right )\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{3\,b\,{\sin \left (a+b\,x\right )}^2\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}} \]
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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