Optimal. Leaf size=28 \[ \frac {\sec (c+d x)}{a d}+\frac {\log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.05, antiderivative size = 28, 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} \[ \frac {\sec (c+d x)}{a d}+\frac {\log (\cos (c+d x))}{a d} \]
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)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {a-a x}{x^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a}{x^2}-\frac {a}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {\log (\cos (c+d x))}{a d}+\frac {\sec (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 21, normalized size = 0.75 \[ \frac {\sec (c+d x)+\log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 33, normalized size = 1.18 \[ \frac {\cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 1}{a d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.12, size = 111, normalized size = 3.96 \[ -\frac {\frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a} + \frac {\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 30, normalized size = 1.07 \[ \frac {\sec \left (d x +c \right )}{d a}-\frac {\ln \left (\sec \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 28, normalized size = 1.00 \[ \frac {\frac {\log \left (\cos \left (d x + c\right )\right )}{a} + \frac {1}{a \cos \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 44, normalized size = 1.57 \[ \frac {2}{d\,\left (a-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d} \]
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 \[ \frac {\int \frac {\tan ^{3}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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