Optimal. Leaf size=119 \[ \frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan (c+d x)}{a^2 d}-\frac {3 \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {\tan ^3(c+d x) \sec (c+d x)}{2 a^2 d}+\frac {3 \tan (c+d x) \sec (c+d x)}{4 a^2 d}+\frac {x}{a^2} \]
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Rubi [A] time = 0.19, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan (c+d x)}{a^2 d}-\frac {3 \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {\tan ^3(c+d x) \sec (c+d x)}{2 a^2 d}+\frac {3 \tan (c+d x) \sec (c+d x)}{4 a^2 d}+\frac {x}{a^2} \]
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 ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\frac {\int (-a+a \sec (c+d x))^2 \tan ^4(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \tan ^4(c+d x)-2 a^2 \sec (c+d x) \tan ^4(c+d x)+a^2 \sec ^2(c+d x) \tan ^4(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \tan ^4(c+d x) \, dx}{a^2}+\frac {\int \sec ^2(c+d x) \tan ^4(c+d x) \, dx}{a^2}-\frac {2 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{a^2}\\ &=\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\sec (c+d x) \tan ^3(c+d x)}{2 a^2 d}-\frac {\int \tan ^2(c+d x) \, dx}{a^2}+\frac {3 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{2 a^2}+\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {\tan (c+d x)}{a^2 d}+\frac {3 \sec (c+d x) \tan (c+d x)}{4 a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\sec (c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac {\tan ^5(c+d x)}{5 a^2 d}-\frac {3 \int \sec (c+d x) \, dx}{4 a^2}+\frac {\int 1 \, dx}{a^2}\\ &=\frac {x}{a^2}-\frac {3 \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {\tan (c+d x)}{a^2 d}+\frac {3 \sec (c+d x) \tan (c+d x)}{4 a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\sec (c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac {\tan ^5(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [B] time = 5.91, size = 495, normalized size = 4.16 \[ \frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (-\frac {151 \sin \left (\frac {c}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {151 \sin \left (\frac {c}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {36 \sin \left (\frac {c}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {36 \sin \left (\frac {c}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {180 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {180 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {\sec (c) \sin \left (\frac {d x}{2}\right ) \left (333 \cos \left (2 c+\frac {3 d x}{2}\right )+287 \cos \left (2 c+\frac {5 d x}{2}\right )+67 \cos \left (4 c+\frac {7 d x}{2}\right )+68 \cos \left (4 c+\frac {9 d x}{2}\right )+293 \cos \left (\frac {d x}{2}\right )\right ) \sec ^5(c+d x)}{2 d}+\frac {\cos \left (\frac {c}{2}\right ) \sec (c) \left (-43 \sin \left (\frac {c}{2}+d x\right )-43 \sin \left (\frac {3 c}{2}+d x\right )-346 \sin \left (\frac {3 c}{2}+2 d x\right )+346 \sin \left (\frac {5 c}{2}+2 d x\right )+149 \sin \left (\frac {5 c}{2}+3 d x\right )+149 \sin \left (\frac {7 c}{2}+3 d x\right )+308 \sin \left (\frac {c}{2}\right )\right ) \sec ^4(c+d x)}{4 d}+240 x\right )}{60 a^2 (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 117, normalized size = 0.98 \[ \frac {120 \, d x \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) + 45 \, \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (68 \, \cos \left (d x + c\right )^{4} - 75 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) - 12\right )} \sin \left (d x + c\right )}{120 \, a^{2} d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 13.72, size = 136, normalized size = 1.14 \[ \frac {\frac {60 \, {\left (d x + c\right )}}{a^{2}} - \frac {45 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} + \frac {45 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 530 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 328 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} a^{2}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.66, size = 269, normalized size = 2.26 \[ -\frac {1}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {1}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {19}{12 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 a^{2} d}-\frac {1}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {19}{12 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {7}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 a^{2} d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 301, normalized size = 2.53 \[ -\frac {\frac {2 \, {\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {110 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {328 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {530 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {105 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )}}{a^{2} - \frac {5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {10 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {45 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} - \frac {45 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 179, normalized size = 1.50 \[ \frac {x}{a^2}-\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^2\,d}+\frac {\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2}-\frac {53\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {164\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2\right )} \]
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 \[ \frac {\int \frac {\tan ^{8}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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