Integrand size = 21, antiderivative size = 106 \[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {x}{a}-\frac {2 b^3 \text {arctanh}\left (\frac {\sqrt {a^2-b^2} \tan \left (\frac {1}{2} (c+d x)\right )}{a+b}\right )}{a \left (a^2-b^2\right )^{3/2} d}-\frac {a \cot (c+d x)}{\left (a^2-b^2\right ) d}+\frac {b \csc (c+d x)}{\left (a^2-b^2\right ) d} \]
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Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3983, 2981, 2686, 8, 3554, 2814, 2738, 214} \[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {a \cot (c+d x)}{d \left (a^2-b^2\right )}+\frac {b \csc (c+d x)}{d \left (a^2-b^2\right )}+\frac {b^2 x}{a \left (a^2-b^2\right )}-\frac {a x}{a^2-b^2}-\frac {2 b^3 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d (a-b)^{3/2} (a+b)^{3/2}} \]
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Rule 8
Rule 214
Rule 2686
Rule 2738
Rule 2814
Rule 2981
Rule 3554
Rule 3983
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (c+d x) \cot ^2(c+d x)}{b+a \cos (c+d x)} \, dx \\ & = \frac {a \int \cot ^2(c+d x) \, dx}{a^2-b^2}-\frac {b \int \cot (c+d x) \csc (c+d x) \, dx}{a^2-b^2}+\frac {b^2 \int \frac {\cos (c+d x)}{b+a \cos (c+d x)} \, dx}{a^2-b^2} \\ & = \frac {b^2 x}{a \left (a^2-b^2\right )}-\frac {a \cot (c+d x)}{\left (a^2-b^2\right ) d}-\frac {a \int 1 \, dx}{a^2-b^2}-\frac {b^3 \int \frac {1}{b+a \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )}+\frac {b \text {Subst}(\int 1 \, dx,x,\csc (c+d x))}{\left (a^2-b^2\right ) d} \\ & = -\frac {a x}{a^2-b^2}+\frac {b^2 x}{a \left (a^2-b^2\right )}-\frac {a \cot (c+d x)}{\left (a^2-b^2\right ) d}+\frac {b \csc (c+d x)}{\left (a^2-b^2\right ) d}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a \left (a^2-b^2\right ) d} \\ & = -\frac {a x}{a^2-b^2}+\frac {b^2 x}{a \left (a^2-b^2\right )}-\frac {2 b^3 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{3/2} (a+b)^{3/2} d}-\frac {a \cot (c+d x)}{\left (a^2-b^2\right ) d}+\frac {b \csc (c+d x)}{\left (a^2-b^2\right ) d} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-2 b^3 \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) \sin (c+d x)+\sqrt {a^2-b^2} \left (-a b+a^2 \cos (c+d x)+\left (a^2-b^2\right ) (c+d x) \sin (c+d x)\right )\right )}{2 a (a-b) (a+b) \sqrt {a^2-b^2} d} \]
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Time = 0.76 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 b^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) a \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{2 \left (a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(115\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 b^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) a \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{2 \left (a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(115\) |
risch | \(-\frac {x}{a}-\frac {2 i \left (-b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{d \left (a^{2}-b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}\) | \(223\) |
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Time = 0.30 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.42 \[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=\left [-\frac {\sqrt {a^{2} - b^{2}} b^{3} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) \sin \left (d x + c\right ) - 2 \, a^{3} b + 2 \, a b^{3} + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d x \sin \left (d x + c\right ) + 2 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )}{2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sin \left (d x + c\right )}, -\frac {\sqrt {-a^{2} + b^{2}} b^{3} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - a^{3} b + a b^{3} + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d x \sin \left (d x + c\right ) + {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 582 vs. \(2 (101) = 202\).
Time = 0.38 (sec) , antiderivative size = 582, normalized size of antiderivative = 5.49 \[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {2 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4} - 2 \, b^{5} - a^{2} {\left | -a^{3} + a b^{2} \right |} + a b {\left | -a^{3} + a b^{2} \right |} + b^{2} {\left | -a^{3} + a b^{2} \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-\frac {a^{2} b - b^{3} + \sqrt {{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} + {\left (a^{2} b - b^{3}\right )}^{2}}}{a^{3} - a^{2} b - a b^{2} + b^{3}}}}\right )\right )}}{a^{2} b {\left | -a^{3} + a b^{2} \right |} - b^{3} {\left | -a^{3} + a b^{2} \right |} + {\left (a^{3} - a b^{2}\right )}^{2}} + \frac {2 \, {\left ({\left (a^{2} - a b - b^{2}\right )} \sqrt {-a^{2} + b^{2}} {\left | -a^{3} + a b^{2} \right |} {\left | -a + b \right |} + {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4} - 2 \, b^{5}\right )} \sqrt {-a^{2} + b^{2}} {\left | -a + b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-\frac {a^{2} b - b^{3} - \sqrt {{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} + {\left (a^{2} b - b^{3}\right )}^{2}}}{a^{3} - a^{2} b - a b^{2} + b^{3}}}}\right )\right )}}{{\left (a^{3} - a b^{2}\right )}^{2} {\left (a^{2} - 2 \, a b + b^{2}\right )} - {\left (a^{4} b - 2 \, a^{3} b^{2} + 2 \, a b^{4} - b^{5}\right )} {\left | -a^{3} + a b^{2} \right |}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a - b} + \frac {1}{{\left (a + b\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
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Time = 16.55 (sec) , antiderivative size = 1002, normalized size of antiderivative = 9.45 \[ \int \frac {\cot ^2(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {1{}\mathrm {i}\,\cos \left (c+d\,x\right )\,a^6-a^5\,b\,1{}\mathrm {i}-2{}\mathrm {i}\,\cos \left (c+d\,x\right )\,a^4\,b^2+a^3\,b^3\,2{}\mathrm {i}+1{}\mathrm {i}\,\cos \left (c+d\,x\right )\,a^2\,b^4-a\,b^5\,1{}\mathrm {i}}{1{}\mathrm {i}\,d\,\sin \left (c+d\,x\right )\,a^7-3{}\mathrm {i}\,d\,\sin \left (c+d\,x\right )\,a^5\,b^2+3{}\mathrm {i}\,d\,\sin \left (c+d\,x\right )\,a^3\,b^4-1{}\mathrm {i}\,d\,\sin \left (c+d\,x\right )\,a\,b^6}+\frac {-a^6\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}+b^6\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}-a^2\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}+a^4\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}+b^3\,\mathrm {atanh}\left (\frac {2\,b^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}^{3/2}-a^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+2\,b^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-9\,a^2\,b^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+3\,a^3\,b^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+18\,a^4\,b^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-12\,a^5\,b^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-21\,a^6\,b^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+19\,a^7\,b^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+15\,a^8\,b^5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-15\,a^9\,b^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-6\,a^{10}\,b^3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+6\,a^{11}\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+a^{12}\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^{16}-8\,a^{14}\,b^2+28\,a^{12}\,b^4-55\,a^{10}\,b^6+65\,a^8\,b^8-46\,a^6\,b^{10}+18\,a^4\,b^{12}-3\,a^2\,b^{14}\right )}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}\,2{}\mathrm {i}}{1{}\mathrm {i}\,d\,a^7-3{}\mathrm {i}\,d\,a^5\,b^2+3{}\mathrm {i}\,d\,a^3\,b^4-1{}\mathrm {i}\,d\,a\,b^6} \]
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