Optimal. Leaf size=20 \[ -\frac {2 d}{3 b (d \sec (a+b x))^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2622, 30} \[ -\frac {2 d}{3 b (d \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2622
Rubi steps
\begin {align*} \int \frac {\sin (a+b x)}{\sqrt {d \sec (a+b x)}} \, dx &=\frac {d \operatorname {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=-\frac {2 d}{3 b (d \sec (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 20, normalized size = 1.00 \[ -\frac {2 d}{3 b (d \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 28, normalized size = 1.40 \[ -\frac {2 \, \sqrt {\frac {d}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.76, size = 35, normalized size = 1.75 \[ -\frac {2 \, \sqrt {d \cos \left (b x + a\right )} {\left | b \right |} \cos \left (b x + a\right ) \mathrm {sgn}\relax (b) \mathrm {sgn}\left (\cos \left (b x + a\right )\right )}{3 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 17, normalized size = 0.85 \[ -\frac {2 d}{3 b \left (d \sec \left (b x +a \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 23, normalized size = 1.15 \[ -\frac {2 \, \cos \left (b x + a\right )}{3 \, b \sqrt {\frac {d}{\cos \left (b x + a\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 28, normalized size = 1.40 \[ -\frac {2\,{\cos \left (a+b\,x\right )}^2\,\sqrt {\frac {d}{\cos \left (a+b\,x\right )}}}{3\,b\,d} \]
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 \[ \int \frac {\sin {\left (a + b x \right )}}{\sqrt {d \sec {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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