Optimal. Leaf size=31 \[ -\frac {2 c d \sqrt {d \csc (a+b x)}}{b \sqrt {c \sec (a+b x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 31, 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} \[ -\frac {2 c d \sqrt {d \csc (a+b x)}}{b \sqrt {c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2619
Rubi steps
\begin {align*} \int (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)} \, dx &=-\frac {2 c d \sqrt {d \csc (a+b x)}}{b \sqrt {c \sec (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 31, normalized size = 1.00 \[ -\frac {2 c d \sqrt {d \csc (a+b x)}}{b \sqrt {c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 36, normalized size = 1.16 \[ -\frac {2 \, d \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )}{b} \]
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 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}} \sqrt {c \sec \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.22, size = 42, normalized size = 1.35 \[ -\frac {2 \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {3}{2}} \sqrt {\frac {c}{\cos \left (b x +a \right )}}\, \cos \left (b x +a \right ) \sin \left (b x +a \right )}{b} \]
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 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}} \sqrt {c \sec \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 36, normalized size = 1.16 \[ -\frac {2\,d\,\cos \left (a+b\,x\right )\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{b} \]
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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