Optimal. Leaf size=169 \[ -\frac{3 \tan ^5(c+d x)}{5 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{\tan (c+d x)}{a^3 d}+\frac{19 \tanh ^{-1}(\sin (c+d x))}{16 a^3 d}+\frac{\tan ^3(c+d x) \sec ^3(c+d x)}{6 a^3 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{8 a^3 d}+\frac{3 \tan ^3(c+d x) \sec (c+d x)}{4 a^3 d}-\frac{17 \tan (c+d x) \sec (c+d x)}{16 a^3 d}-\frac{x}{a^3} \]
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Rubi [A] time = 0.268065, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3888, 3886, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac{3 \tan ^5(c+d x)}{5 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{\tan (c+d x)}{a^3 d}+\frac{19 \tanh ^{-1}(\sin (c+d x))}{16 a^3 d}+\frac{\tan ^3(c+d x) \sec ^3(c+d x)}{6 a^3 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{8 a^3 d}+\frac{3 \tan ^3(c+d x) \sec (c+d x)}{4 a^3 d}-\frac{17 \tan (c+d x) \sec (c+d x)}{16 a^3 d}-\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3886
Rule 3473
Rule 8
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rubi steps
\begin{align*} \int \frac{\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\frac{\int (-a+a \sec (c+d x))^3 \tan ^4(c+d x) \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \tan ^4(c+d x)+3 a^3 \sec (c+d x) \tan ^4(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^4(c+d x)+a^3 \sec ^3(c+d x) \tan ^4(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \tan ^4(c+d x) \, dx}{a^3}+\frac{\int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{a^3}+\frac{3 \int \sec (c+d x) \tan ^4(c+d x) \, dx}{a^3}-\frac{3 \int \sec ^2(c+d x) \tan ^4(c+d x) \, dx}{a^3}\\ &=-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x) \tan ^3(c+d x)}{4 a^3 d}+\frac{\sec ^3(c+d x) \tan ^3(c+d x)}{6 a^3 d}-\frac{\int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{2 a^3}+\frac{\int \tan ^2(c+d x) \, dx}{a^3}-\frac{9 \int \sec (c+d x) \tan ^2(c+d x) \, dx}{4 a^3}-\frac{3 \operatorname{Subst}\left (\int x^4 \, dx,x,\tan (c+d x)\right )}{a^3 d}\\ &=\frac{\tan (c+d x)}{a^3 d}-\frac{9 \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{8 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x) \tan ^3(c+d x)}{4 a^3 d}+\frac{\sec ^3(c+d x) \tan ^3(c+d x)}{6 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{\int \sec ^3(c+d x) \, dx}{8 a^3}-\frac{\int 1 \, dx}{a^3}+\frac{9 \int \sec (c+d x) \, dx}{8 a^3}\\ &=-\frac{x}{a^3}+\frac{9 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac{\tan (c+d x)}{a^3 d}-\frac{17 \sec (c+d x) \tan (c+d x)}{16 a^3 d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{8 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x) \tan ^3(c+d x)}{4 a^3 d}+\frac{\sec ^3(c+d x) \tan ^3(c+d x)}{6 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{\int \sec (c+d x) \, dx}{16 a^3}\\ &=-\frac{x}{a^3}+\frac{19 \tanh ^{-1}(\sin (c+d x))}{16 a^3 d}+\frac{\tan (c+d x)}{a^3 d}-\frac{17 \sec (c+d x) \tan (c+d x)}{16 a^3 d}-\frac{\sec ^3(c+d x) \tan (c+d x)}{8 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x) \tan ^3(c+d x)}{4 a^3 d}+\frac{\sec ^3(c+d x) \tan ^3(c+d x)}{6 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.852181, size = 303, normalized size = 1.79 \[ -\frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (9120 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec (c) \sec ^6(c+d x) (-210 \sin (2 c+d x)-1440 \sin (c+2 d x)+1200 \sin (3 c+2 d x)+865 \sin (2 c+3 d x)+865 \sin (4 c+3 d x)-1296 \sin (3 c+4 d x)-240 \sin (5 c+4 d x)+435 \sin (4 c+5 d x)+435 \sin (6 c+5 d x)-176 \sin (5 c+6 d x)+2400 d x \cos (c)+1800 d x \cos (c+2 d x)+1800 d x \cos (3 c+2 d x)+720 d x \cos (3 c+4 d x)+720 d x \cos (5 c+4 d x)+120 d x \cos (5 c+6 d x)+120 d x \cos (7 c+6 d x)+1760 \sin (c)-210 \sin (d x))\right )}{960 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 312, normalized size = 1.9 \begin{align*} -2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{1}{6\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-6}}+{\frac{11}{10\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-{\frac{11}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}+{\frac{11}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{5}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{35}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{19}{16\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{6\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-6}}+{\frac{11}{10\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-5}}+{\frac{11}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-4}}+{\frac{11}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{5}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{35}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{19}{16\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.71528, size = 463, normalized size = 2.74 \begin{align*} -\frac{\frac{2 \,{\left (\frac{45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{95 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{366 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{1746 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{3135 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{525 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )}}{a^{3} - \frac{6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac{480 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{285 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{285 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25241, size = 362, normalized size = 2.14 \begin{align*} -\frac{480 \, d x \cos \left (d x + c\right )^{6} - 285 \, \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) + 285 \, \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (176 \, \cos \left (d x + c\right )^{5} - 435 \, \cos \left (d x + c\right )^{4} + 208 \, \cos \left (d x + c\right )^{3} + 110 \, \cos \left (d x + c\right )^{2} - 144 \, \cos \left (d x + c\right ) + 40\right )} \sin \left (d x + c\right )}{480 \, a^{3} d \cos \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 42.3985, size = 201, normalized size = 1.19 \begin{align*} -\frac{\frac{240 \,{\left (d x + c\right )}}{a^{3}} - \frac{285 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac{285 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{2 \,{\left (525 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 3135 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1746 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 366 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 95 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 45 \, \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 )}^{6} a^{3}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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