Optimal. Leaf size=48 \[ \frac {\sec ^2(c+d x)}{2 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 48, 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} \[ \frac {\sec ^2(c+d x)}{2 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^2}{x^3} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^3}-\frac {2 a^2}{x^2}+\frac {a^2}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\log (\cos (c+d x))}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}+\frac {\sec ^2(c+d x)}{2 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 51, normalized size = 1.06 \[ -\frac {\sec ^2(c+d x) (4 \cos (c+d x)+\cos (2 (c+d x)) \log (\cos (c+d x))+\log (\cos (c+d x))-1)}{2 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 45, normalized size = 0.94 \[ -\frac {2 \, \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 4 \, \cos \left (d x + c\right ) - 1}{2 \, a^{2} d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 8.11, size = 136, normalized size = 2.83 \[ \frac {\frac {2 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac {2 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{2}} - \frac {\frac {6 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 46, normalized size = 0.96 \[ \frac {\sec ^{2}\left (d x +c \right )}{2 a^{2} d}-\frac {2 \sec \left (d x +c \right )}{a^{2} d}+\frac {\ln \left (\sec \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 40, normalized size = 0.83 \[ -\frac {\frac {2 \, \log \left (\cos \left (d x + c\right )\right )}{a^{2}} + \frac {4 \, \cos \left (d x + c\right ) - 1}{a^{2} \cos \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 77, normalized size = 1.60 \[ \frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}+\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^2\,d} \]
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 \[ \frac {\int \frac {\tan ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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