Optimal. Leaf size=66 \[ \frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec ^2(c+d x)}{2 a d}-\frac {\sec (c+d x)}{a d}-\frac {\log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.06, antiderivative size = 66, 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, 75} \[ \frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec ^2(c+d x)}{2 a d}-\frac {\sec (c+d x)}{a d}-\frac {\log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 75
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^2 (a+a x)}{x^4} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {a^3}{x^3}-\frac {a^3}{x^2}+\frac {a^3}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\log (\cos (c+d x))}{a d}-\frac {\sec (c+d x)}{a d}-\frac {\sec ^2(c+d x)}{2 a d}+\frac {\sec ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 65, normalized size = 0.98 \[ -\frac {\sec ^3(c+d x) (6 \cos (2 (c+d x))+3 \cos (3 (c+d x)) \log (\cos (c+d x))+\cos (c+d x) (9 \log (\cos (c+d x))+6)+2)}{12 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 55, normalized size = 0.83 \[ -\frac {6 \, \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) + 6 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 2}{6 \, a d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.20, size = 157, normalized size = 2.38 \[ \frac {\frac {6 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac {6 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a} + \frac {\frac {21 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 3}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 62, normalized size = 0.94 \[ \frac {\sec ^{3}\left (d x +c \right )}{3 d a}-\frac {\sec ^{2}\left (d x +c \right )}{2 d a}-\frac {\sec \left (d x +c \right )}{d a}+\frac {\ln \left (\sec \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 50, normalized size = 0.76 \[ -\frac {\frac {6 \, \log \left (\cos \left (d x + c\right )\right )}{a} + \frac {6 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 2}{a \cos \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.19, size = 99, normalized size = 1.50 \[ \frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {4}{3}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \]
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 \[ \frac {\int \frac {\tan ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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