Integrand size = 27, antiderivative size = 158 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {6 b \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^4 d}-\frac {\cos (c+d x)}{2 a^2 d \left (1-\cos ^2(c+d x)\right )}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a^3 d (a+b \sin (c+d x))} \]
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Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2969, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}+\frac {6 b \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^4 d}-\frac {\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))} \]
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Rule 210
Rule 632
Rule 2739
Rule 2969
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (2 \left (a^2-3 b^2\right )-a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 b} \\ & = -\frac {\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-3 b \left (a^2-2 b^2\right )+3 a b^2 \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^3 b} \\ & = -\frac {\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}-\frac {\left (3 \left (a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx}{2 a^4}+\frac {\left (3 b \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^4} \\ & = \frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^4 d}-\frac {\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}+\frac {\left (6 b \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 d} \\ & = \frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^4 d}-\frac {\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}-\frac {\left (12 b \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 d} \\ & = \frac {6 b \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^4 d}-\frac {\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac {\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))} \\ \end{align*}
Time = 3.01 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.21 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {48 b \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+8 a b \cot \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+12 \left (a^2-2 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-12 \left (a^2-2 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {8 a \left (-a^2+b^2\right ) \cos (c+d x)}{a+b \sin (c+d x)}-8 a b \tan \left (\frac {1}{2} (c+d x)\right )}{8 a^4 d} \]
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Time = 0.55 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {\frac {4 \left (-\frac {b \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3}}{2}+\frac {a \,b^{2}}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+6 b \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(209\) |
default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {\frac {4 \left (-\frac {b \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3}}{2}+\frac {a \,b^{2}}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+6 b \sqrt {a^{2}-b^{2}}\, \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(209\) |
risch | \(\frac {i \left (-3 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+6 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-12 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{3}+2 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-4 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+2 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-2 a^{2} b \right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} b \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{3} d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{4} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}-\frac {3 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{4}}+\frac {3 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{4}}\) | \(395\) |
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Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (147) = 294\).
Time = 0.38 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.09 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {6 \, a^{2} b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (a b \cos \left (d x + c\right )^{2} - a b + {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{5} d \cos \left (d x + c\right )^{2} - a^{5} d + {\left (a^{4} b d \cos \left (d x + c\right )^{2} - a^{4} b d\right )} \sin \left (d x + c\right )\right )}}, -\frac {6 \, a^{2} b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left (a b \cos \left (d x + c\right )^{2} - a b + {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{5} d \cos \left (d x + c\right )^{2} - a^{5} d + {\left (a^{4} b d \cos \left (d x + c\right )^{2} - a^{4} b d\right )} \sin \left (d x + c\right )\right )}}\right ] \]
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\[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.44 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.74 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {12 \, {\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {48 \, {\left (a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} - \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {16 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} - a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{4}} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 10.32 (sec) , antiderivative size = 675, normalized size of antiderivative = 4.27 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {17\,a^2}{2}-16\,b^2\right )+\frac {a^2}{2}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (a^2\,b-2\,b^3\right )}{a}-3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,a^2-6\,b^2\right )}{2\,a^4\,d}-\frac {6\,b\,\mathrm {atanh}\left (\frac {72\,b^4\,\sqrt {b^2-a^2}}{18\,a^4\,b+72\,b^5-90\,a^2\,b^3-216\,a\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+72\,a^3\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a}}-\frac {54\,b^2\,\sqrt {b^2-a^2}}{18\,a^2\,b-90\,b^3+\frac {72\,b^5}{a^2}+72\,a\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {216\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3}}+\frac {18\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{18\,a\,b-\frac {90\,b^3}{a}+\frac {72\,b^5}{a^3}+72\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {216\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4}}-\frac {144\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{18\,a^3\,b-90\,a\,b^3+\frac {72\,b^5}{a}-216\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+72\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2}}+\frac {144\,b^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{18\,a^5\,b+72\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b^2-90\,a^3\,b^3-216\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^4+72\,a\,b^5+144\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}\right )\,\sqrt {b^2-a^2}}{a^4\,d} \]
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