Optimal. Leaf size=192 \[ \frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{13 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \csc ^7(c+d x)}{14 d}-\frac{3 a^3 \csc ^5(c+d x)}{2 d}-\frac{5 a^3 \csc ^3(c+d x)}{2 d}-\frac{15 a^3 \csc (c+d x)}{2 d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.314372, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3872, 2873, 3767, 2621, 302, 207, 2620, 270, 288} \[ \frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{13 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \csc ^7(c+d x)}{14 d}-\frac{3 a^3 \csc ^5(c+d x)}{2 d}-\frac{5 a^3 \csc ^3(c+d x)}{2 d}-\frac{15 a^3 \csc (c+d x)}{2 d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2873
Rule 3767
Rule 2621
Rule 302
Rule 207
Rule 2620
Rule 270
Rule 288
Rubi steps
\begin{align*} \int \csc ^8(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\int (-a-a \cos (c+d x))^3 \csc ^8(c+d x) \sec ^3(c+d x) \, dx\\ &=\int \left (a^3 \csc ^8(c+d x)+3 a^3 \csc ^8(c+d x) \sec (c+d x)+3 a^3 \csc ^8(c+d x) \sec ^2(c+d x)+a^3 \csc ^8(c+d x) \sec ^3(c+d x)\right ) \, dx\\ &=a^3 \int \csc ^8(c+d x) \, dx+a^3 \int \csc ^8(c+d x) \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^8(c+d x) \sec (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^8(c+d x) \sec ^2(c+d x) \, dx\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{x^{10}}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac{a^3 \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{x^8} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1+\frac{1}{x^8}+\frac{4}{x^6}+\frac{6}{x^4}+\frac{4}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1+x^2+x^4+x^6+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (9 a^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=-\frac{13 a^3 \cot (c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \csc (c+d x)}{d}-\frac{a^3 \csc ^3(c+d x)}{d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^7(c+d x)}{7 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \tan (c+d x)}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (9 a^3\right ) \operatorname{Subst}\left (\int \left (1+x^2+x^4+x^6+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{13 a^3 \cot (c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{15 a^3 \csc (c+d x)}{2 d}-\frac{5 a^3 \csc ^3(c+d x)}{2 d}-\frac{3 a^3 \csc ^5(c+d x)}{2 d}-\frac{15 a^3 \csc ^7(c+d x)}{14 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \tan (c+d x)}{d}-\frac{\left (9 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{13 a^3 \cot (c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{15 a^3 \csc (c+d x)}{2 d}-\frac{5 a^3 \csc ^3(c+d x)}{2 d}-\frac{3 a^3 \csc ^5(c+d x)}{2 d}-\frac{15 a^3 \csc ^7(c+d x)}{14 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 1.18217, size = 430, normalized size = 2.24 \[ \frac{a^3 \cos (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^3 \left (-8 \csc (2 c) (2776 \sin (c-d x)-6080 \sin (c+d x)+8816 \sin (2 (c+d x))-7904 \sin (3 (c+d x))+4864 \sin (4 (c+d x))-1824 \sin (5 (c+d x))+304 \sin (6 (c+d x))-9580 \sin (2 c+d x)-10024 \sin (3 c+d x)+13891 \sin (c+2 d x)+7720 \sin (2 (c+2 d x))+13891 \sin (3 c+2 d x)+10080 \sin (4 c+2 d x)-10060 \sin (c+3 d x)-12454 \sin (2 c+3 d x)-12454 \sin (4 c+3 d x)-6580 \sin (5 c+3 d x)+7664 \sin (3 c+4 d x)+7664 \sin (5 c+4 d x)+2520 \sin (6 c+4 d x)-3420 \sin (3 c+5 d x)-2874 \sin (4 c+5 d x)-2874 \sin (6 c+5 d x)-420 \sin (7 c+5 d x)+640 \sin (4 c+6 d x)+479 \sin (5 c+6 d x)+479 \sin (7 c+6 d x)+5264 \sin (2 c)-9580 \sin (d x)+8480 \sin (2 d x)) \csc (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )-860160 \cos ^2(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+860160 \cos ^2(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{917504 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.076, size = 360, normalized size = 1.9 \begin{align*} -{\frac{80\,{a}^{3}\cot \left ( dx+c \right ) }{7\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{6}}{7\,d}}-{\frac{6\,{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{35\,d}}-{\frac{8\,{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{35\,d}}-{\frac{3\,{a}^{3}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{3\,{a}^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{3}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{15\,{a}^{3}}{2\,d\sin \left ( dx+c \right ) }}+{\frac{15\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}-{\frac{3\,{a}^{3}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}\cos \left ( dx+c \right ) }}-{\frac{24\,{a}^{3}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) }}-{\frac{48\,{a}^{3}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) }}+{\frac{192\,{a}^{3}}{35\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-{\frac{{a}^{3}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{9\,{a}^{3}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{3}}{2\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01563, size = 362, normalized size = 1.89 \begin{align*} -\frac{a^{3}{\left (\frac{2 \,{\left (315 \, \sin \left (d x + c\right )^{8} - 210 \, \sin \left (d x + c\right )^{6} - 42 \, \sin \left (d x + c\right )^{4} - 18 \, \sin \left (d x + c\right )^{2} - 10\right )}}{\sin \left (d x + c\right )^{9} - \sin \left (d x + c\right )^{7}} - 315 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{3}{\left (\frac{2 \,{\left (105 \, \sin \left (d x + c\right )^{6} + 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} + 15\right )}}{\sin \left (d x + c\right )^{7}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{3}{\left (\frac{140 \, \tan \left (d x + c\right )^{6} + 70 \, \tan \left (d x + c\right )^{4} + 28 \, \tan \left (d x + c\right )^{2} + 5}{\tan \left (d x + c\right )^{7}} - 35 \, \tan \left (d x + c\right )\right )} + \frac{4 \,{\left (35 \, \tan \left (d x + c\right )^{6} + 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{140 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80772, size = 698, normalized size = 3.64 \begin{align*} -\frac{320 \, a^{3} \cos \left (d x + c\right )^{6} - 750 \, a^{3} \cos \left (d x + c\right )^{5} + 170 \, a^{3} \cos \left (d x + c\right )^{4} + 720 \, a^{3} \cos \left (d x + c\right )^{3} - 520 \, a^{3} \cos \left (d x + c\right )^{2} + 42 \, a^{3} \cos \left (d x + c\right ) + 14 \, a^{3} - 105 \,{\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \,{\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{28 \,{\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \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 = 1.39616, size = 228, normalized size = 1.19 \begin{align*} \frac{840 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 840 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - 7 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{112 \,{\left (5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 7 \, a^{3} \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 )}^{2}} - \frac{1050 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 112 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 14 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{112 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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