Optimal. Leaf size=17 \[ \frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2701, 30}
\begin {gather*} \frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2701
Rubi steps
\begin {align*} \int \frac {\cos (a+b x)}{\sqrt {\csc (a+b x)}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} \frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 14, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {2}{3 b \csc \left (b x +a \right )^{\frac {3}{2}}}\) | \(14\) |
default | \(\frac {2}{3 b \csc \left (b x +a \right )^{\frac {3}{2}}}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 0.76 \begin {gather*} \frac {2 \, \sin \left (b x + a\right )^{\frac {3}{2}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 23, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )}}{3 \, b \sqrt {\sin \left (b x + a\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\cos {\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 13, normalized size = 0.76 \begin {gather*} \frac {2 \, \sin \left (b x + a\right )^{\frac {3}{2}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 15, normalized size = 0.88 \begin {gather*} \frac {2}{3\,b\,{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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