Optimal. Leaf size=24 \[ \frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3657, 4122, 191} \[ \frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 3657
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx &=\int \frac {1}{\sqrt {a \sec ^2(c+d x)}} \, dx\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 24, normalized size = 1.00 \[ \frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 38, normalized size = 1.58 \[ \frac {\sqrt {a \tan \left (d x + c\right )^{2} + a} \tan \left (d x + c\right )}{a d \tan \left (d x + c\right )^{2} + a d} \]
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 \[ \int \frac {1}{\sqrt {a \tan \left (d x + c\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 25, normalized size = 1.04 \[ \frac {\tan \left (d x +c \right )}{d \sqrt {a +a \left (\tan ^{2}\left (d x +c \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 13, normalized size = 0.54 \[ \frac {\sin \left (d x + c\right )}{\sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.07, size = 55, normalized size = 2.29 \[ \frac {\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )}{4\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+3}}}{a\,d} \]
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 \[ \int \frac {1}{\sqrt {a \tan ^{2}{\left (c + d x \right )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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