Optimal. Leaf size=177 \[ \frac{\left (2 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{\left (4 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} x \left (4 a^2-3 b^2\right )-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{5 a b \cos ^3(c+d x)}{3 d}-\frac{5 a b \cos (c+d x)}{d}-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.439628, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2911, 2592, 288, 302, 206, 456, 1259, 1261, 203} \[ \frac{\left (2 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{\left (4 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} x \left (4 a^2-3 b^2\right )-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{5 a b \cos ^3(c+d x)}{3 d}-\frac{5 a b \cos (c+d x)}{d}-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2911
Rule 2592
Rule 288
Rule 302
Rule 206
Rule 456
Rule 1259
Rule 1261
Rule 203
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^3(c+d x) \cot ^3(c+d x) \, dx+\int \cos ^2(c+d x) \cot ^4(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^2+\left (a^2+b^2\right ) x^2}{x^4 \left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{d}-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}-\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-4 a^2-4 b^2 x^2+3 b^2 x^4}{x^4 \left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 d}+\frac{(5 a b) \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}+\frac{\left (4 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-8 a^2+8 \left (a^2-b^2\right ) x^2+\left (-4 a^2+7 b^2\right ) x^4}{x^4 \left (1+x^2\right )} \, dx,x,\tan (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,\cos (c+d x)\right )}{d}\\ &=-\frac{5 a b \cos (c+d x)}{d}-\frac{5 a b \cos ^3(c+d x)}{3 d}-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}+\frac{\left (4 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{8 a^2}{x^4}+\frac{8 \left (2 a^2-b^2\right )}{x^2}-\frac{5 \left (4 a^2-3 b^2\right )}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{8 d}+\frac{(5 a b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{5 a b \cos (c+d x)}{d}-\frac{5 a b \cos ^3(c+d x)}{3 d}+\frac{\left (2 a^2-b^2\right ) \cot (c+d x)}{d}-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{\left (4 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{\left (5 \left (4 a^2-3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=\frac{5}{8} \left (4 a^2-3 b^2\right ) x+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{5 a b \cos (c+d x)}{d}-\frac{5 a b \cos ^3(c+d x)}{3 d}+\frac{\left (2 a^2-b^2\right ) \cot (c+d x)}{d}-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{\left (4 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac{b^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 6.25002, size = 336, normalized size = 1.9 \[ \frac{5 \left (4 a^2-3 b^2\right ) (c+d x)}{8 d}+\frac{\left (a^2-2 b^2\right ) \sin (2 (c+d x))}{4 d}+\frac{\csc \left (\frac{1}{2} (c+d x)\right ) \left (7 a^2 \cos \left (\frac{1}{2} (c+d x)\right )-3 b^2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{6 d}+\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (3 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-7 a^2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{6 d}-\frac{a^2 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{24 d}+\frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{24 d}-\frac{9 a b \cos (c+d x)}{2 d}-\frac{a b \cos (3 (c+d x))}{6 d}-\frac{a b \csc ^2\left (\frac{1}{2} (c+d x)\right )}{4 d}+\frac{a b \sec ^2\left (\frac{1}{2} (c+d x)\right )}{4 d}-\frac{5 a b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{5 a b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}-\frac{b^2 \sin (4 (c+d x))}{32 d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.089, size = 321, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{4\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d\sin \left ( dx+c \right ) }}+{\frac{4\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{5\,{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{5\,{a}^{2}x}{2}}+{\frac{5\,{a}^{2}c}{2\,d}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{5\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-5\,{\frac{ab\cos \left ( dx+c \right ) }{d}}-5\,{\frac{ab\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d\sin \left ( dx+c \right ) }}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{d}}-{\frac{5\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}-{\frac{15\,{b}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{15\,{b}^{2}x}{8}}-{\frac{15\,{b}^{2}c}{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52359, size = 255, normalized size = 1.44 \begin{align*} \frac{4 \,{\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^{2} - 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 )} a b - 3 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} b^{2}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90171, size = 635, normalized size = 3.59 \begin{align*} \frac{6 \, b^{2} \cos \left (d x + c\right )^{7} - 3 \,{\left (4 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 20 \,{\left (4 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 60 \,{\left (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 (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 ) - 15 \,{\left (4 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right ) -{\left (16 \, a b \cos \left (d x + c\right )^{5} - 15 \,{\left (4 \, a^{2} - 3 \, b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 80 \, a b \cos \left (d x + c\right )^{3} + 15 \,{\left (4 \, a^{2} - 3 \, b^{2}\right )} d x - 120 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (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 [B] time = 1.25046, size = 494, normalized size = 2.79 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 120 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 27 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \,{\left (4 \, a^{2} - 3 \, b^{2}\right )}{\left (d x + c\right )} + \frac{220 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 27 \, 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} - 6 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}} - \frac{2 \,{\left (12 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 27 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 144 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 12 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 336 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 12 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 304 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 27 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 112 \, a b\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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