Optimal. Leaf size=140 \[ -\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]
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Rubi [A] time = 0.231049, antiderivative size = 140, 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 = {2709, 3768, 3770, 3767} \[ -\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2709
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \frac{\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \left (a^5 \csc ^3(c+d x)-3 a^5 \csc ^4(c+d x)+2 a^5 \csc ^5(c+d x)+2 a^5 \csc ^6(c+d x)-3 a^5 \csc ^7(c+d x)+a^5 \csc ^8(c+d x)\right ) \, dx}{a^8}\\ &=\frac{\int \csc ^3(c+d x) \, dx}{a^3}+\frac{\int \csc ^8(c+d x) \, dx}{a^3}+\frac{2 \int \csc ^5(c+d x) \, dx}{a^3}+\frac{2 \int \csc ^6(c+d x) \, dx}{a^3}-\frac{3 \int \csc ^4(c+d x) \, dx}{a^3}-\frac{3 \int \csc ^7(c+d x) \, dx}{a^3}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\int \csc (c+d x) \, dx}{2 a^3}+\frac{3 \int \csc ^3(c+d x) \, dx}{2 a^3}-\frac{5 \int \csc ^5(c+d x) \, dx}{2 a^3}-\frac{\operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac{2 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}+\frac{3 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{4 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{3 \int \csc (c+d x) \, dx}{4 a^3}-\frac{15 \int \csc ^3(c+d x) \, dx}{8 a^3}\\ &=-\frac{5 \tanh ^{-1}(\cos (c+d x))}{4 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}-\frac{15 \int \csc (c+d x) \, dx}{16 a^3}\\ &=-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.962836, size = 251, normalized size = 1.79 \[ \frac{\csc ^7(c+d x) \left (4998 \sin (2 (c+d x))+504 \sin (4 (c+d x))-210 \sin (6 (c+d x))-4704 \cos (c+d x)+672 \cos (3 (c+d x))+1120 \cos (5 (c+d x))-160 \cos (7 (c+d x))+3675 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2205 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+735 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-105 \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-3675 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2205 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-735 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+105 \sin (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{21504 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.198, size = 284, normalized size = 2. \begin{align*}{\frac{1}{896\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}+{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{5}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{13}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{29}{128\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{896\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}+{\frac{29}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{5}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{5}{16\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}-{\frac{13}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01462, size = 425, normalized size = 3.04 \begin{align*} -\frac{\frac{\frac{609 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{91 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{21 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} - \frac{840 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{{\left (\frac{21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{105 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{91 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{609 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{3} \sin \left (d x + c\right )^{7}}}{2688 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.17624, size = 612, normalized size = 4.37 \begin{align*} \frac{320 \, \cos \left (d x + c\right )^{7} - 1120 \, \cos \left (d x + c\right )^{5} + 896 \, \cos \left (d x + c\right )^{3} - 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 42 \,{\left (5 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \,{\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\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 = 1.37335, size = 329, normalized size = 2.35 \begin{align*} \frac{\frac{840 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{2178 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 609 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 91 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}} + \frac{3 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 21 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 63 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 105 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 91 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 63 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 609 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{21}}}{2688 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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