Optimal. Leaf size=202 \[ -\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac{15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2} \]
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Rubi [A] time = 0.169485, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {2722, 2591, 288, 302, 203, 2592, 321, 206, 3473, 8} \[ -\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac{15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 2722
Rule 2591
Rule 288
Rule 302
Rule 203
Rule 2592
Rule 321
Rule 206
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \left (b^2 \cos ^2(c+d x) \cot ^4(c+d x)+2 a b \cos (c+d x) \cot ^5(c+d x)+a^2 \cot ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \, dx+(2 a b) \int \cos (c+d x) \cot ^5(c+d x) \, dx+b^2 \int \cos ^2(c+d x) \cot ^4(c+d x) \, dx\\ &=-\frac{a^2 \cot ^5(c+d x)}{5 d}-a^2 \int \cot ^4(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+a^2 \int \cot ^2(c+d x) \, dx+\frac{(5 a b) \operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{2 d}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac{a^2 \cot (c+d x)}{d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-a^2 \int 1 \, dx-\frac{(15 a b) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-a^2 x+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{(15 a b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-a^2 x+\frac{5 b^2 x}{2}-\frac{15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 1.09302, size = 351, normalized size = 1.74 \[ \frac{\left (560 b^2-368 a^2\right ) \cot \left (\frac{1}{2} (c+d x)\right )+368 a^2 \tan \left (\frac{1}{2} (c+d x)\right )+96 a^2 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)-328 a^2 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-\frac{3}{2} a^2 \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )+\frac{41}{2} a^2 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )-480 a^2 c-480 a^2 d x+960 a b \cos (c+d x)-15 a b \csc ^4\left (\frac{1}{2} (c+d x)\right )+270 a b \csc ^2\left (\frac{1}{2} (c+d x)\right )+15 a b \sec ^4\left (\frac{1}{2} (c+d x)\right )-270 a b \sec ^2\left (\frac{1}{2} (c+d x)\right )+1800 a b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-1800 a b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+120 b^2 \sin (2 (c+d x))-560 b^2 \tan \left (\frac{1}{2} (c+d x)\right )+160 b^2 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-10 b^2 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+1200 b^2 c+1200 b^2 d x}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 302, normalized size = 1.5 \begin{align*} -{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}}-{a}^{2}x-{\frac{{a}^{2}c}{d}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d}}+{\frac{5\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{15\,ab\cos \left ( dx+c \right ) }{4\,d}}+{\frac{15\,ab\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{4\,d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{4\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d\sin \left ( dx+c \right ) }}+{\frac{4\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{3\,d}}+{\frac{5\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{5\,{b}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{5\,{b}^{2}x}{2}}+{\frac{5\,{b}^{2}c}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67638, size = 247, normalized size = 1.22 \begin{align*} -\frac{8 \,{\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^{2} - 20 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} b^{2} + 15 \, a b{\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 )}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63935, size = 786, normalized size = 3.89 \begin{align*} -\frac{60 \, b^{2} \cos \left (d x + c\right )^{7} + 92 \,{\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 140 \,{\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 225 \,{\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 225 \,{\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 60 \,{\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right ) + 30 \,{\left (2 \,{\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{4} - 8 \, a b \cos \left (d x + c\right )^{5} - 4 \,{\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 25 \, a b \cos \left (d x + c\right )^{3} + 2 \,{\left (2 \, a^{2} - 5 \, b^{2}\right )} d x - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + 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 = 2.77852, size = 455, normalized size = 2.25 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 240 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1800 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 330 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 540 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 240 \,{\left (2 \, a^{2} - 5 \, b^{2}\right )}{\left (d x + c\right )} - \frac{480 \,{\left (b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, a b\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac{4110 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 330 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 540 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 240 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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