Optimal. Leaf size=124 \[ \frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac{7 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]
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Rubi [A] time = 0.363856, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2875, 2873, 2607, 30, 2611, 3768, 3770, 14} \[ \frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac{7 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2607
Rule 30
Rule 2611
Rule 3768
Rule 3770
Rule 14
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \cot ^2(c+d x) \csc ^5(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \cot ^2(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^4(c+d x)+a^3 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \cot ^2(c+d x) \csc ^2(c+d x) \, dx}{a^3}+\frac{\int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a^3}+\frac{3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a^3}-\frac{3 \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3}\\ &=-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{\int \csc ^5(c+d x) \, dx}{6 a^3}-\frac{3 \int \csc ^3(c+d x) \, dx}{4 a^3}-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac{\cot ^3(c+d x)}{3 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{\int \csc ^3(c+d x) \, dx}{8 a^3}-\frac{3 \int \csc (c+d x) \, dx}{8 a^3}-\frac{3 \operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{\int \csc (c+d x) \, dx}{16 a^3}\\ &=\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.971315, size = 242, normalized size = 1.95 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6 \left (704 \tan \left (\frac{1}{2} (c+d x)\right )-704 \cot \left (\frac{1}{2} (c+d x)\right )+210 \csc ^2\left (\frac{1}{2} (c+d x)\right )+5 \sec ^6\left (\frac{1}{2} (c+d x)\right )+90 \sec ^4\left (\frac{1}{2} (c+d x)\right )-210 \sec ^2\left (\frac{1}{2} (c+d x)\right )-840 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+840 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-544 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)+(18 \sin (c+d x)-5) \csc ^6\left (\frac{1}{2} (c+d x)\right )+(34 \sin (c+d x)-90) \csc ^4\left (\frac{1}{2} (c+d x)\right )-36 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )\right )}{1920 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.187, size = 246, normalized size = 2. \begin{align*}{\frac{1}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{3}{160\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{7}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{5}{16\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{5}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{3}{160\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{7}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{7}{16\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{7}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{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 = 1.06775, size = 370, normalized size = 2.98 \begin{align*} \frac{\frac{\frac{600 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{140 \, \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{36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{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{36 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{105 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{600 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{3} \sin \left (d x + c\right )^{6}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13023, size = 532, normalized size = 4.29 \begin{align*} -\frac{210 \, \cos \left (d x + c\right )^{5} - 80 \, \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 ) + 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 ) - 32 \,{\left (11 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )}{480 \,{\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 )}} \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 = 1.34516, size = 292, 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{2058 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 600 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 5}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}} - \frac{5 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 36 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 105 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 140 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 600 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{18}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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