Optimal. Leaf size=18 \[ \frac {\log (a+b \tan (c+d x))}{b d} \]
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Rubi [A]
time = 0.03, antiderivative size = 18, 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, 31}
\begin {gather*} \frac {\log (a+b \tan (c+d x))}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3587
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\log (a+b \tan (c+d x))}{b d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log (a+b \tan (c+d x))}{b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 19, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \tan \left (d x +c \right )\right )}{b d}\) | \(19\) |
default | \(\frac {\ln \left (a +b \tan \left (d x +c \right )\right )}{b d}\) | \(19\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b d}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log \left (b \tan \left (d x + c\right ) + a\right )}{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. 59 vs.
\(2 (18) = 36\).
time = 0.37, size = 59, normalized size = 3.28 \begin {gather*} \frac {\log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - \log \left (\cos \left (d x + c\right )^{2}\right )}{2 \, b d} \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 )}}{a + b \tan {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 19, normalized size = 1.06 \begin {gather*} \frac {\log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.59, size = 18, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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