Optimal. Leaf size=32 \[ -\frac {2 \sqrt {d \sec (e+f x)}}{b f \sqrt {b \tan (e+f x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 32, 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 = {2605} \[ -\frac {2 \sqrt {d \sec (e+f x)}}{b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2605
Rubi steps
\begin {align*} \int \frac {\sqrt {d \sec (e+f x)}}{(b \tan (e+f x))^{3/2}} \, dx &=-\frac {2 \sqrt {d \sec (e+f x)}}{b f \sqrt {b \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 32, normalized size = 1.00 \[ -\frac {2 \sqrt {d \sec (e+f x)}}{b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 52, normalized size = 1.62 \[ -\frac {2 \, \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{b^{2} f \sin \left (f x + e\right )} \]
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 \[ \int \frac {\sqrt {d \sec \left (f x + e\right )}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 50, normalized size = 1.56 \[ -\frac {2 \sin \left (f x +e \right ) \sqrt {\frac {d}{\cos \left (f x +e \right )}}}{f \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \cos \left (f x +e \right )} \]
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 \[ \int \frac {\sqrt {d \sec \left (f x + e\right )}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 46, normalized size = 1.44 \[ -\frac {2\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}}{b\,f\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.01, size = 53, normalized size = 1.66 \[ \begin {cases} - \frac {2 \sqrt {d} \sqrt {\sec {\left (e + f x \right )}}}{b^{\frac {3}{2}} f \sqrt {\tan {\left (e + f x \right )}}} & \text {for}\: f \neq 0 \\\frac {x \sqrt {d \sec {\relax (e )}}}{\left (b \tan {\relax (e )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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