Optimal. Leaf size=174 \[ \frac{\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac{5 \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{5 a b \cot ^3(c+d x)}{3 d}+\frac{5 a b \cot (c+d x)}{d}+\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+5 a b x-\frac{b^2 \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.278132, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2911, 2591, 288, 302, 203, 455, 1814, 1153, 206} \[ \frac{\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac{5 \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{5 a b \cot ^3(c+d x)}{3 d}+\frac{5 a b \cot (c+d x)}{d}+\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+5 a b x-\frac{b^2 \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2911
Rule 2591
Rule 288
Rule 302
Rule 203
Rule 455
Rule 1814
Rule 1153
Rule 206
Rubi steps
\begin{align*} \int \cos (c+d x) \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^2(c+d x) \cot ^4(c+d x) \, dx+\int \cos (c+d x) \cot ^5(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^6 \left (a^2+b^2-b^2 x^2\right )}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{d}-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{a^2+4 a^2 x^2+4 a^2 x^4-4 b^2 x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac{(5 a b) \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac{\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{7 a^2-4 b^2+8 \left (a^2-b^2\right ) x^2-8 b^2 x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}-\frac{(5 a b) \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{5 a b \cot (c+d x)}{d}-\frac{5 a b \cot ^3(c+d x)}{3 d}+\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac{\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \left (-8 \left (a^2-2 b^2\right )+8 b^2 x^2+\frac{5 \left (3 a^2-4 b^2\right )}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{8 d}-\frac{(5 a b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=5 a b x+\frac{\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac{b^2 \cos ^3(c+d x)}{3 d}+\frac{5 a b \cot (c+d x)}{d}-\frac{5 a b \cot ^3(c+d x)}{3 d}+\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac{\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{\left (5 \left (3 a^2-4 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}\\ &=5 a b x-\frac{5 \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac{b^2 \cos ^3(c+d x)}{3 d}+\frac{5 a b \cot (c+d x)}{d}-\frac{5 a b \cot ^3(c+d x)}{3 d}+\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac{\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 6.18144, size = 337, normalized size = 1.94 \[ \frac{\left (9 a^2-4 b^2\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{\left (4 b^2-9 a^2\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{5 \left (3 a^2-4 b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{5 \left (3 a^2-4 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{a^2 \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{a^2 \sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{5 a b (c+d x)}{d}+\frac{a b \sin (2 (c+d x))}{2 d}+\frac{(2 a-3 b) (2 a+3 b) \cos (c+d x)}{4 d}-\frac{7 a b \tan \left (\frac{1}{2} (c+d x)\right )}{3 d}+\frac{7 a b \cot \left (\frac{1}{2} (c+d x)\right )}{3 d}-\frac{a b \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{12 d}+\frac{a b \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{12 d}-\frac{b^2 \cos (3 (c+d x))}{12 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.092, size = 334, normalized size = 1.9 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d}}+{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{15\,{a}^{2}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{15\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{2\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{8\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d\sin \left ( dx+c \right ) }}+{\frac{8\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{10\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{3\,d}}+5\,{\frac{ab\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+5\,abx+5\,{\frac{abc}{d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d}}-{\frac{5\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{6\,d}}-{\frac{5\,{b}^{2}\cos \left ( dx+c \right ) }{2\,d}}-{\frac{5\,{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52289, size = 277, normalized size = 1.59 \begin{align*} \frac{16 \,{\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 )} a b - 4 \,{\left (4 \, \cos \left (d x + c\right )^{3} - \frac{6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \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 )} b^{2} - 3 \, a^{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 )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85872, size = 776, normalized size = 4.46 \begin{align*} -\frac{16 \, b^{2} \cos \left (d x + c\right )^{7} - 240 \, a b d x \cos \left (d x + c\right )^{4} + 480 \, a b d x \cos \left (d x + c\right )^{2} - 16 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 240 \, a b d x + 50 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right ) + 15 \,{\left ({\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, b^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 15 \,{\left ({\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, b^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 16 \,{\left (3 \, a b \cos \left (d x + c\right )^{5} - 20 \, 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 )}{48 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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 = 1.26861, size = 467, normalized size = 2.68 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 16 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 48 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 960 \,{\left (d x + c\right )} a b - 432 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 120 \,{\left (3 \, a^{2} - 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{128 \,{\left (3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 9 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a^{2} + 7 \, b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}} - \frac{750 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1000 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 432 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 48 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 16 \, 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 )^{4}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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