Optimal. Leaf size=35 \[ \frac {2}{a^3 d (\cos (c+d x)+1)}+\frac {\log (\cos (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 35, 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, 43} \[ \frac {2}{a^3 d (\cos (c+d x)+1)}+\frac {\log (\cos (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {a-a x}{(a+a x)^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {2}{a (1+x)^2}-\frac {1}{a (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {2}{a^3 d (1+\cos (c+d x))}+\frac {\log (1+\cos (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 33, normalized size = 0.94 \[ \frac {\tan ^2\left (\frac {1}{2} (c+d x)\right )+2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 42, normalized size = 1.20 \[ \frac {{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2}{a^{3} d \cos \left (d x + c\right ) + a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.90, size = 56, normalized size = 1.60 \[ -\frac {\frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\cos \left (d x + c\right ) - 1}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 51, normalized size = 1.46 \[ -\frac {\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}-\frac {2}{d \,a^{3} \left (1+\sec \left (d x +c \right )\right )}+\frac {\ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 36, normalized size = 1.03 \[ \frac {\frac {2}{a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 36, normalized size = 1.03 \[ -\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.81, size = 457, normalized size = 13.06 \[ \begin {cases} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} - \frac {2 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec {\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} + \frac {2 \log {\left (\sec {\left (c + d x \right )} + 1 \right )} \sec ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} + \frac {4 \log {\left (\sec {\left (c + d x \right )} + 1 \right )} \sec {\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} + \frac {2 \log {\left (\sec {\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} + \frac {\tan ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} - \frac {2 \sec {\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} - \frac {2}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec {\left (c + d x \right )} + 2 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \tan ^{3}{\relax (c )}}{\left (a \sec {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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