Optimal. Leaf size=138 \[ \frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d} \]
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Rubi [A] time = 0.248182, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2869, 2757, 3768, 3770, 3767} \[ \frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2869
Rule 2757
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \frac{\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \csc ^7(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \csc ^5(c+d x)-2 a^2 \csc ^6(c+d x)+a^2 \csc ^7(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \csc ^5(c+d x) \, dx}{a^2}+\frac{\int \csc ^7(c+d x) \, dx}{a^2}-\frac{2 \int \csc ^6(c+d x) \, dx}{a^2}\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{3 \int \csc ^3(c+d x) \, dx}{4 a^2}+\frac{5 \int \csc ^5(c+d x) \, dx}{6 a^2}+\frac{2 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=\frac{2 \cot (c+d x)}{a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{3 \int \csc (c+d x) \, dx}{8 a^2}+\frac{5 \int \csc ^3(c+d x) \, dx}{8 a^2}\\ &=-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{5 \int \csc (c+d x) \, dx}{16 a^2}\\ &=-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.806337, size = 229, normalized size = 1.66 \[ \frac{\csc ^6(c+d x) \left (3840 \sin (2 (c+d x))-1536 \sin (4 (c+d x))+256 \sin (6 (c+d x))-2820 \cos (c+d x)+1870 \cos (3 (c+d x))-330 \cos (5 (c+d x))+1650 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2475 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-990 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+165 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-1650 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2475 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+990 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-165 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{7680 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 246, normalized size = 1.8 \begin{align*}{\frac{1}{384\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{1}{80\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{5}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{5}{48\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{31}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{5}{8\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{5}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{80\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{5}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{11}{16\,d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{384\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{5}{48\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{31}{128\,d{a}^{2}} \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 = 1.1117, size = 371, normalized size = 2.69 \begin{align*} -\frac{\frac{\frac{1200 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{465 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{75 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{24 \, \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^{2}} - \frac{1320 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac{{\left (\frac{24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{75 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{465 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1200 \, \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^{2} \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 = 1.19363, size = 555, normalized size = 4.02 \begin{align*} \frac{330 \, \cos \left (d x + c\right )^{5} - 880 \, \cos \left (d x + c\right )^{3} - 165 \,{\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 ) + 165 \,{\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 ) - 64 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 630 \, \cos \left (d x + c\right )}{480 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} 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 = 1.39306, size = 290, normalized size = 2.1 \begin{align*} \frac{\frac{1320 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{3234 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1200 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 465 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 200 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 75 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}} + \frac{5 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 24 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 75 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 200 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 465 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1200 \, a^{10} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{12}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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