Optimal. Leaf size=18 \[ -\frac {2 d}{b \sqrt {d \tan (a+b x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2591, 30} \[ -\frac {2 d}{b \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2591
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx &=\frac {d \operatorname {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac {2 d}{b \sqrt {d \tan (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 18, normalized size = 1.00 \[ -\frac {2 d}{b \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 37, normalized size = 2.06 \[ -\frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 16, normalized size = 0.89 \[ -\frac {2 \, d}{\sqrt {d \tan \left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.49, size = 38, normalized size = 2.11 \[ -\frac {2 \sqrt {\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}}\, \cos \left (b x +a \right )}{b \sin \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 23, normalized size = 1.28 \[ -\frac {2 \, \sqrt {d \tan \left (b x + a\right )}}{b \tan \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.05, size = 48, normalized size = 2.67 \[ -\frac {\sin \left (2\,a+2\,b\,x\right )\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}}{b\,{\sin \left (a+b\,x\right )}^2} \]
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 \[ \int \sqrt {d \tan {\left (a + b x \right )}} \csc ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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