Optimal. Leaf size=20 \[ -\frac {1}{b d (a+b \tan (c+d x))} \]
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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 = {3506, 32} \[ -\frac {1}{b d (a+b \tan (c+d x))} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3506
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac {1}{b d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 32, normalized size = 1.60 \[ \frac {\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 57, normalized size = 2.85 \[ -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{{\left (a^{3} + a b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{2} b + b^{3}\right )} d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.90, size = 20, normalized size = 1.00 \[ -\frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 21, normalized size = 1.05 \[ -\frac {1}{b d \left (a +b \tan \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 20, normalized size = 1.00 \[ -\frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 20, normalized size = 1.00 \[ -\frac {1}{b\,d\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )} \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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