Optimal. Leaf size=33 \[ \frac {2 \log (\cos (c+d x)+1)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 72} \[ \frac {2 \log (\cos (c+d x)+1)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 72
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {a-a x}{x (a+a x)} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x}-\frac {2}{1+x}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\log (\cos (c+d x))}{a^2 d}+\frac {2 \log (1+\cos (c+d x))}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 30, normalized size = 0.91 \[ \frac {4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 31, normalized size = 0.94 \[ -\frac {\log \left (-\cos \left (d x + c\right )\right ) - 2 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 33, normalized size = 1.00 \[ -\frac {\log \left ({\left | \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1 \right |}\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 34, normalized size = 1.03 \[ -\frac {\ln \left (\sec \left (d x +c \right )\right )}{a^{2} d}+\frac {2 \ln \left (1+\sec \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 31, normalized size = 0.94 \[ \frac {\frac {2 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {\log \left (\cos \left (d x + c\right )\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 22, normalized size = 0.67 \[ -\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )}{a^2\,d} \]
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 \[ \frac {\int \frac {\tan ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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