Optimal. Leaf size=22 \[ -\frac {1}{2 b d (a+b \tan (c+d x))^2} \]
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Rubi [A]
time = 0.03, 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 = {3587, 32}
\begin {gather*} -\frac {1}{2 b d (a+b \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3587
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {\text {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] Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(22)=44\).
time = 0.21, size = 58, normalized size = 2.64 \begin {gather*} \frac {-b \sec ^2(c+d x)+2 \tan (c+d x) (a+b \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 21, normalized size = 0.95
method | result | size |
derivativedivides | \(-\frac {1}{2 b d \left (a +b \tan \left (d x +c \right )\right )^{2}}\) | \(21\) |
default | \(-\frac {1}{2 b d \left (a +b \tan \left (d x +c \right )\right )^{2}}\) | \(21\) |
risch | \(\frac {2 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 i a}{\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} d \left (i a +b \right )^{2}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{2 \, {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs.
\(2 (20) = 40\).
time = 0.38, size = 142, normalized size = 6.45 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{2 \, {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b d} \end {gather*}
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 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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