Optimal. Leaf size=124 \[ -\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{16 d} \]
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Rubi [A] time = 0.255639, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2611, 3768, 3770, 2607, 14} \[ -\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^2(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^2(c+d x) \csc ^4(c+d x)+a^2 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+a^2 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{1}{6} a^2 \int \csc ^5(c+d x) \, dx-\frac{1}{4} a^2 \int \csc ^3(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{1}{8} a^2 \int \csc (c+d x) \, dx-\frac{1}{8} a^2 \int \csc ^3(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{1}{16} a^2 \int \csc (c+d x) \, dx\\ &=\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.68412, size = 229, normalized size = 1.85 \[ \frac{a^2 \csc ^6(c+d x) \left (-960 \sin (2 (c+d x))-384 \sin (4 (c+d x))+64 \sin (6 (c+d x))-1500 \cos (c+d x)+130 \cos (3 (c+d x))+90 \cos (5 (c+d x))-450 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-675 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+270 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-45 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+450 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+675 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-270 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+45 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{7680 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 160, normalized size = 1.3 \begin{align*} -{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2}\cos \left ( dx+c \right ) }{16\,d}}-{\frac{3\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{4\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39654, size = 252, normalized size = 2.03 \begin{align*} -\frac{5 \, a^{2}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, a^{2}{\left (\frac{2 \,{\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{64 \,{\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{2}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74528, size = 567, normalized size = 4.57 \begin{align*} -\frac{90 \, a^{2} \cos \left (d x + c\right )^{5} - 80 \, a^{2} \cos \left (d x + c\right )^{3} - 90 \, a^{2} \cos \left (d x + c\right ) - 45 \,{\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 45 \,{\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 64 \,{\left (2 \, a^{2} \cos \left (d x + c\right )^{5} - 5 \, a^{2} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{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 [B] time = 1.45891, size = 308, normalized size = 2.48 \begin{align*} \frac{5 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 45 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 40 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 360 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{882 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 240 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 40 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 45 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, a^{2} \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}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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