Optimal. Leaf size=20 \[ -\frac {2}{b d \sqrt {d \tan (a+b x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 20, 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 = {2607, 32} \[ -\frac {2}{b d \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int \frac {\sec ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(d x)^{3/2}} \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac {2}{b d \sqrt {d \tan (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 20, normalized size = 1.00 \[ -\frac {2}{b d \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 40, normalized size = 2.00 \[ -\frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{b d^{2} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 18, normalized size = 0.90 \[ -\frac {2}{\sqrt {d \tan \left (b x + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 19, normalized size = 0.95 \[ -\frac {2}{b d \sqrt {d \tan \left (b x +a \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 18, normalized size = 0.90 \[ -\frac {2}{\sqrt {d \tan \left (b x + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.57, size = 51, normalized size = 2.55 \[ -\frac {\sin \left (2\,a+2\,b\,x\right )\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}}{b\,d^2\,{\sin \left (a+b\,x\right )}^2} \]
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 \[ \int \frac {\sec ^{2}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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