Optimal. Leaf size=156 \[ \frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{93 a^5 \tan (c+d x) \sec (c+d x)}{16 d} \]
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Rubi [A] time = 0.197865, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3791, 3768, 3770, 3767} \[ \frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{93 a^5 \tan (c+d x) \sec (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 3791
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx &=\int \left (a^5 \sec ^3(c+d x)+5 a^5 \sec ^4(c+d x)+10 a^5 \sec ^5(c+d x)+10 a^5 \sec ^6(c+d x)+5 a^5 \sec ^7(c+d x)+a^5 \sec ^8(c+d x)\right ) \, dx\\ &=a^5 \int \sec ^3(c+d x) \, dx+a^5 \int \sec ^8(c+d x) \, dx+\left (5 a^5\right ) \int \sec ^4(c+d x) \, dx+\left (5 a^5\right ) \int \sec ^7(c+d x) \, dx+\left (10 a^5\right ) \int \sec ^5(c+d x) \, dx+\left (10 a^5\right ) \int \sec ^6(c+d x) \, dx\\ &=\frac{a^5 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{5 a^5 \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{2} a^5 \int \sec (c+d x) \, dx+\frac{1}{6} \left (25 a^5\right ) \int \sec ^5(c+d x) \, dx+\frac{1}{2} \left (15 a^5\right ) \int \sec ^3(c+d x) \, dx-\frac{a^5 \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac{\left (5 a^5\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac{\left (10 a^5\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{17 a^5 \sec (c+d x) \tan (c+d x)}{4 d}+\frac{85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{1}{8} \left (25 a^5\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{4} \left (15 a^5\right ) \int \sec (c+d x) \, dx\\ &=\frac{17 a^5 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{1}{16} \left (25 a^5\right ) \int \sec (c+d x) \, dx\\ &=\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{a^5 \tan ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 1.34252, size = 229, normalized size = 1.47 \[ -\frac{a^5 (\cos (c+d x)+1)^5 \sec ^{10}\left (\frac{1}{2} (c+d x)\right ) \sec ^7(c+d x) \left (624960 \cos ^7(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-\sec (c) (-162400 \sin (2 c+d x)+118825 \sin (c+2 d x)+118825 \sin (3 c+2 d x)+305088 \sin (2 c+3 d x)-16800 \sin (4 c+3 d x)+62860 \sin (3 c+4 d x)+62860 \sin (5 c+4 d x)+107296 \sin (4 c+5 d x)+9765 \sin (5 c+6 d x)+9765 \sin (7 c+6 d x)+15328 \sin (6 c+7 d x)+374080 \sin (d x))\right )}{3440640 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 168, normalized size = 1.1 \begin{align*}{\frac{93\,{a}^{5}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{93\,{a}^{5}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{958\,{a}^{5}\tan \left ( dx+c \right ) }{105\,d}}+{\frac{479\,{a}^{5}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{105\,d}}+{\frac{85\,{a}^{5} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{24\,d}}+{\frac{76\,{a}^{5}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35\,d}}+{\frac{5\,{a}^{5} \left ( \sec \left ( dx+c \right ) \right ) ^{5}\tan \left ( dx+c \right ) }{6\,d}}+{\frac{{a}^{5}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12852, size = 424, normalized size = 2.72 \begin{align*} \frac{96 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{5} + 2240 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{5} + 5600 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{5} - 175 \, a^{5}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2100 \, a^{5}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{5}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{3360 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80474, size = 413, normalized size = 2.65 \begin{align*} \frac{9765 \, a^{5} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9765 \, a^{5} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (15328 \, a^{5} \cos \left (d x + c\right )^{6} + 9765 \, a^{5} \cos \left (d x + c\right )^{5} + 7664 \, a^{5} \cos \left (d x + c\right )^{4} + 5950 \, a^{5} \cos \left (d x + c\right )^{3} + 3648 \, a^{5} \cos \left (d x + c\right )^{2} + 1400 \, a^{5} \cos \left (d x + c\right ) + 240 \, a^{5}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{5} \left (\int \sec ^{3}{\left (c + d x \right )}\, dx + \int 5 \sec ^{4}{\left (c + d x \right )}\, dx + \int 10 \sec ^{5}{\left (c + d x \right )}\, dx + \int 10 \sec ^{6}{\left (c + d x \right )}\, dx + \int 5 \sec ^{7}{\left (c + d x \right )}\, dx + \int \sec ^{8}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47336, size = 230, normalized size = 1.47 \begin{align*} \frac{9765 \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 9765 \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (9765 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 65100 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 184233 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 285696 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 260183 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 132020 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 43995 \, a^{5} \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 )}^{7}}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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