Optimal. Leaf size=175 \[ \frac{5 \left (a^2-6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot (c+d x)}{d}-2 a b x+\frac{b^2 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.26146, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2911, 3473, 8, 4366, 455, 1814, 1157, 388, 206} \[ \frac{5 \left (a^2-6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{\left (13 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot (c+d x)}{d}-2 a b x+\frac{b^2 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2911
Rule 3473
Rule 8
Rule 4366
Rule 455
Rule 1814
Rule 1157
Rule 388
Rule 206
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cot ^6(c+d x) \, dx+\int \cot ^6(c+d x) \csc (c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{2 a b \cot ^5(c+d x)}{5 d}-(2 a b) \int \cot ^4(c+d x) \, dx-\frac{\operatorname{Subst}\left (\int \frac{x^6 \left (a^2+b^2-b^2 x^2\right )}{\left (1-x^2\right )^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+(2 a b) \int \cot ^2(c+d x) \, dx+\frac{\operatorname{Subst}\left (\int \frac{a^2+6 a^2 x^2+6 a^2 x^4-6 b^2 x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{6 d}\\ &=-\frac{2 a b \cot (c+d x)}{d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}+\frac{\left (13 a^2-6 b^2\right ) \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}-(2 a b) \int 1 \, dx-\frac{\operatorname{Subst}\left (\int \frac{3 \left (3 a^2-2 b^2\right )+24 \left (a^2-b^2\right ) x^2-24 b^2 x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{24 d}\\ &=-2 a b x-\frac{2 a b \cot (c+d x)}{d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}-\frac{\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac{\left (13 a^2-6 b^2\right ) \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{\operatorname{Subst}\left (\int \frac{3 \left (5 a^2-14 b^2\right )-48 b^2 x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{48 d}\\ &=-2 a b x+\frac{b^2 \cos (c+d x)}{d}-\frac{2 a b \cot (c+d x)}{d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}-\frac{\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac{\left (13 a^2-6 b^2\right ) \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{\left (5 \left (a^2-6 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{16 d}\\ &=-2 a b x+\frac{5 \left (a^2-6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{b^2 \cos (c+d x)}{d}-\frac{2 a b \cot (c+d x)}{d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}-\frac{\left (11 a^2-18 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac{\left (13 a^2-6 b^2\right ) \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 [B] time = 1.02215, size = 384, normalized size = 2.19 \[ \frac{-5 a^2 \csc ^6\left (\frac{1}{2} (c+d x)\right )+60 a^2 \csc ^4\left (\frac{1}{2} (c+d x)\right )-330 a^2 \csc ^2\left (\frac{1}{2} (c+d x)\right )+5 a^2 \sec ^6\left (\frac{1}{2} (c+d x)\right )-60 a^2 \sec ^4\left (\frac{1}{2} (c+d x)\right )+330 a^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )-600 a^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+600 a^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2944 a b \tan \left (\frac{1}{2} (c+d x)\right )-2944 a b \cot \left (\frac{1}{2} (c+d x)\right )+768 a b \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)-2624 a b \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-12 a b \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )+164 a b \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )-3840 a b c-3840 a b d x+1920 b^2 \cos (c+d x)-30 b^2 \csc ^4\left (\frac{1}{2} (c+d x)\right )+540 b^2 \csc ^2\left (\frac{1}{2} (c+d x)\right )+30 b^2 \sec ^4\left (\frac{1}{2} (c+d x)\right )-540 b^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )+3600 b^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-3600 b^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{1920 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 318, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}-{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}-{\frac{5\,{a}^{2}\cos \left ( dx+c \right ) }{16\,d}}-{\frac{5\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{2\,ab \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{2\,ab \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{ab\cot \left ( dx+c \right ) }{d}}-2\,abx-2\,{\frac{abc}{d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d}}+{\frac{5\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{15\,{b}^{2}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{15\,{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47612, size = 296, normalized size = 1.69 \begin{align*} -\frac{64 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a b - 5 \, a^{2}{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \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} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, b^{2}{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88429, size = 919, normalized size = 5.25 \begin{align*} -\frac{960 \, a b d x \cos \left (d x + c\right )^{6} - 480 \, b^{2} \cos \left (d x + c\right )^{7} - 2880 \, a b d x \cos \left (d x + c\right )^{4} + 2880 \, a b d x \cos \left (d x + c\right )^{2} - 330 \,{\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 960 \, a b d x + 400 \,{\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 150 \,{\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right ) - 75 \,{\left ({\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 6 \, b^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 75 \,{\left ({\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 6 \, b^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 64 \,{\left (23 \, a b \cos \left (d x + c\right )^{5} - 35 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\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 = 1.29188, size = 455, normalized size = 2.6 \begin{align*} \frac{5 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 24 \, a b \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} + 30 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 280 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 225 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 480 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3840 \,{\left (d x + c\right )} a b + 2640 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 600 \,{\left (a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{3840 \, b^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + \frac{1470 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 8820 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 2640 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 225 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 480 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 280 \, a b \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} - 30 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, a b \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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