Optimal. Leaf size=22 \[ -\frac {1}{2 b d (a+b \tan (c+d x))^2} \]
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Rubi [A] time = 0.04, antiderivative size = 22, 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} \[ -\frac {1}{2 b d (a+b \tan (c+d x))^2} \]
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))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^3} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac {1}{2 b d (a+b \tan (c+d x))^2}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 58, normalized size = 2.64 \[ \frac {2 \tan (c+d x) (a+b \tan (c+d x))-b \sec ^2(c+d x)}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 142, normalized size = 6.45 \[ -\frac {4 \, a^{2} b \cos \left (d x + c\right )^{2} - a^{2} b + b^{3} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 20, normalized size = 0.91 \[ -\frac {1}{2 \, {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 21, normalized size = 0.95 \[ -\frac {1}{2 b d \left (a +b \tan \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 20, normalized size = 0.91 \[ -\frac {1}{2 \, {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 39, normalized size = 1.77 \[ -\frac {1}{d\,\left (2\,a^2\,b+4\,a\,b^2\,\mathrm {tan}\left (c+d\,x\right )+2\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )} \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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