Optimal. Leaf size=72 \[ \frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan (c+d x)}{a^2 d}+\frac {\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac {\tan (c+d x) \sec (c+d x)}{a^2 d}-\frac {x}{a^2} \]
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Rubi [A] time = 0.15, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3888, 3886, 3473, 8, 2611, 3770, 2607, 30} \[ \frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan (c+d x)}{a^2 d}+\frac {\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac {\tan (c+d x) \sec (c+d x)}{a^2 d}-\frac {x}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 3473
Rule 3770
Rule 3886
Rule 3888
Rubi steps
\begin {align*} \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\frac {\int (-a+a \sec (c+d x))^2 \tan ^2(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \tan ^2(c+d x)-2 a^2 \sec (c+d x) \tan ^2(c+d x)+a^2 \sec ^2(c+d x) \tan ^2(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \tan ^2(c+d x) \, dx}{a^2}+\frac {\int \sec ^2(c+d x) \tan ^2(c+d x) \, dx}{a^2}-\frac {2 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{a^2}\\ &=\frac {\tan (c+d x)}{a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a^2 d}-\frac {\int 1 \, dx}{a^2}+\frac {\int \sec (c+d x) \, dx}{a^2}+\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {x}{a^2}+\frac {\tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac {\tan (c+d x)}{a^2 d}-\frac {\sec (c+d x) \tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] time = 6.31, size = 767, normalized size = 10.65 \[ -\frac {4 x \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{(a \sec (c+d x)+a)^2}+\frac {8 \sin \left (\frac {d x}{2}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{3 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (a \sec (c+d x)+a)^2 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {8 \sin \left (\frac {d x}{2}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) (a \sec (c+d x)+a)^2 \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {\left (7 \sin \left (\frac {c}{2}\right )-5 \cos \left (\frac {c}{2}\right )\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{3 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (a \sec (c+d x)+a)^2 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {\left (7 \sin \left (\frac {c}{2}\right )+5 \cos \left (\frac {c}{2}\right )\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) (a \sec (c+d x)+a)^2 \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {2 \sin \left (\frac {d x}{2}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{3 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (a \sec (c+d x)+a)^2 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {2 \sin \left (\frac {d x}{2}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x)}{3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) (a \sec (c+d x)+a)^2 \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}-\frac {4 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a \sec (c+d x)+a)^2}+\frac {4 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 97, normalized size = 1.35 \[ -\frac {6 \, d x \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{6 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.13, size = 99, normalized size = 1.38 \[ -\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} - \frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} + \frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {4 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.60, size = 185, normalized size = 2.57 \[ -\frac {1}{3 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2} d}-\frac {1}{3 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2} d}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.82, size = 196, normalized size = 2.72 \[ -\frac {\frac {4 \, {\left (\frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {6 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 111, normalized size = 1.54 \[ \frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {x}{a^2}+\frac {\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2\right )} \]
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 \[ \frac {\int \frac {\tan ^{6}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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