Integrand size = 16, antiderivative size = 118 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {\text {arctanh}(\cos (x))}{2 a^2}-\frac {2 b^2 \text {arctanh}(\cos (x))}{a^4}-\frac {\left (a^2+b^2\right ) \text {arctanh}(\cos (x))}{a^4}+\frac {3 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))} \]
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Time = 0.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3184, 3172, 3855, 3153, 212, 3853, 3182} \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {2 b^2 \text {arctanh}(\cos (x))}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\text {arctanh}(\cos (x))}{2 a^2}-\frac {\cot (x) \csc (x)}{2 a^2}-\frac {\left (a^2+b^2\right ) \text {arctanh}(\cos (x))}{a^4}+\frac {3 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))} \]
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Rule 212
Rule 3153
Rule 3172
Rule 3182
Rule 3184
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^3(x) \, dx}{a^2}-\frac {(2 b) \int \frac {\csc ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {\csc (x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2} \\ & = \frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}+\frac {\int \csc (x) \, dx}{2 a^2}+\frac {\left (2 b^2\right ) \int \csc (x) \, dx}{a^4}+\frac {\left (a^2+b^2\right ) \int \csc (x) \, dx}{a^4}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{a^4}-\frac {\left (2 b \left (a^2+b^2\right )\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{a^4} \\ & = -\frac {\text {arctanh}(\cos (x))}{2 a^2}-\frac {2 b^2 \text {arctanh}(\cos (x))}{a^4}-\frac {\left (a^2+b^2\right ) \text {arctanh}(\cos (x))}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}+\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^4}+\frac {\left (2 b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^4} \\ & = -\frac {\text {arctanh}(\cos (x))}{2 a^2}-\frac {2 b^2 \text {arctanh}(\cos (x))}{a^4}-\frac {\left (a^2+b^2\right ) \text {arctanh}(\cos (x))}{a^4}+\frac {3 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{a^4}+\frac {2 b \csc (x)}{a^3}-\frac {\cot (x) \csc (x)}{2 a^2}+\frac {a^2+b^2}{a^3 (a \cos (x)+b \sin (x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(118)=236\).
Time = 2.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.29 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {-48 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right ) (b+a \cot (x))+8 a^3 \csc (x)+8 a b^2 \csc (x)-12 a^2 b \log \left (\cos \left (\frac {x}{2}\right )\right )-24 b^3 \log \left (\cos \left (\frac {x}{2}\right )\right )-12 a^3 \cot (x) \log \left (\cos \left (\frac {x}{2}\right )\right )-24 a b^2 \cot (x) \log \left (\cos \left (\frac {x}{2}\right )\right )+12 a^2 b \log \left (\sin \left (\frac {x}{2}\right )\right )+24 b^3 \log \left (\sin \left (\frac {x}{2}\right )\right )+12 a^3 \cot (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+24 a b^2 \cot (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+a^2 b \sec ^2\left (\frac {x}{2}\right )+a^3 \cot (x) \sec ^2\left (\frac {x}{2}\right )-a \csc ^2\left (\frac {x}{2}\right ) \left (-4 a b \cos (x)+a^2 \cot (x)+b (a-4 b \sin (x))\right )+8 a b^2 \tan \left (\frac {x}{2}\right )+8 a^2 b \cot (x) \tan \left (\frac {x}{2}\right )}{8 a^4 (b+a \cot (x))} \]
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Time = 0.71 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.34
method | result | size |
default | \(-\frac {1}{8 a^{2} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {x}{2}\right )}+\frac {\frac {\tan \left (\frac {x}{2}\right )^{2} a}{2}+4 b \tan \left (\frac {x}{2}\right )}{4 a^{3}}+\frac {\frac {4 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tan \left (\frac {x}{2}\right )-\frac {\left (a^{2}+b^{2}\right ) a}{2}\right )}{\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a}-6 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4}}\) | \(158\) |
risch | \(\frac {{\mathrm e}^{i x} \left (3 i a b \,{\mathrm e}^{4 i x}+3 a^{2} {\mathrm e}^{4 i x}+6 b^{2} {\mathrm e}^{4 i x}-2 a^{2} {\mathrm e}^{2 i x}-12 b^{2} {\mathrm e}^{2 i x}-3 i b a +3 a^{2}+6 b^{2}\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right ) a^{3}}-\frac {3 i \sqrt {-a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i x}-\frac {\sqrt {-a^{2}-b^{2}}\, \left (i b +a \right )}{a^{2}+b^{2}}\right )}{a^{4}}+\frac {3 i \sqrt {-a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i x}+\frac {\sqrt {-a^{2}-b^{2}}\, \left (i b +a \right )}{a^{2}+b^{2}}\right )}{a^{4}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) b^{2}}{a^{4}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) b^{2}}{a^{4}}\) | \(297\) |
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Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (110) = 220\).
Time = 0.32 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.92 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {6 \, a^{2} b \cos \left (x\right ) \sin \left (x\right ) + 4 \, a^{3} + 12 \, a b^{2} - 6 \, {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right )^{2} - 6 \, {\left (a b \cos \left (x\right )^{3} - a b \cos \left (x\right ) + {\left (b^{2} \cos \left (x\right )^{2} - b^{2}\right )} \sin \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) + 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right )^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right ) - {\left (a^{2} b + 2 \, b^{3} - {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right )^{3} - {\left (a^{3} + 2 \, a b^{2}\right )} \cos \left (x\right ) - {\left (a^{2} b + 2 \, b^{3} - {\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{5} \cos \left (x\right )^{3} - a^{5} \cos \left (x\right ) + {\left (a^{4} b \cos \left (x\right )^{2} - a^{4} b\right )} \sin \left (x\right )\right )}} \]
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\[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{\left (a \cos {\left (x \right )} + b \sin {\left (x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (110) = 220\).
Time = 0.31 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.05 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {a^{3} - \frac {6 \, a^{2} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {{\left (17 \, a^{3} + 32 \, a b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{8 \, {\left (\frac {a^{5} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {2 \, a^{4} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {a^{5} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}\right )}} + \frac {\frac {8 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{3}} + \frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{4}} + \frac {3 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.82 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {3 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{4}} + \frac {3 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{4}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {2 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, x\right ) + b^{3} \tan \left (\frac {1}{2} \, x\right ) + a^{3} + a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )} a^{4}} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 36 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}}{8 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2}} \]
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Time = 22.09 (sec) , antiderivative size = 511, normalized size of antiderivative = 4.33 \[ \int \frac {\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (\frac {17\,a^2}{2}+16\,b^2\right )-\frac {a^2}{2}+3\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (a^2\,b+2\,b^3\right )}{a}}{-4\,a^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^2}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (3\,a^2+6\,b^2\right )}{2\,a^4}+\frac {b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^3}-\frac {6\,b\,\mathrm {atanh}\left (\frac {54\,b^2\,\sqrt {a^2+b^2}}{18\,a^2\,b+90\,b^3+\frac {72\,b^5}{a^2}+\frac {216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^3}+72\,a\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )}+\frac {72\,b^4\,\sqrt {a^2+b^2}}{18\,a^4\,b+72\,b^5+90\,a^2\,b^3+72\,a^3\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a}+216\,a\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}+\frac {144\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )+90\,a\,b^3+18\,a^3\,b+\frac {72\,b^5}{a}+72\,a^2\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}}+\frac {144\,b^5\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{18\,a^5\,b+72\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^4\,b^2+90\,a^3\,b^3+216\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2\,b^4+72\,a\,b^5+144\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^6}+\frac {18\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{18\,a\,b+72\,b^2\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {90\,b^3}{a}+\frac {72\,b^5}{a^3}+\frac {216\,b^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}+\frac {144\,b^6\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^4}}\right )\,\sqrt {a^2+b^2}}{a^4} \]
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