Integrand size = 23, antiderivative size = 268 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
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Time = 0.36 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3754, 3624, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {4 a b \sqrt {\cot (c+d x)}}{d} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3624
Rule 3754
Rubi steps \begin{align*} \text {integral}& = \int \cot ^{\frac {5}{2}}(c+d x) (b+a \cot (c+d x))^2 \, dx \\ & = -\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}+\int \cot ^{\frac {5}{2}}(c+d x) \left (-a^2+b^2+2 a b \cot (c+d x)\right ) \, dx \\ & = -\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}+\int \cot ^{\frac {3}{2}}(c+d x) \left (-2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right ) \, dx \\ & = \frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}+\int \sqrt {\cot (c+d x)} \left (a^2-b^2-2 a b \cot (c+d x)\right ) \, dx \\ & = \frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}+\int \frac {2 a b+\left (a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}+\frac {2 \text {Subst}\left (\int \frac {-2 a b+\left (-a^2+b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d} \\ & = \frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d} \\ & = \frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {4 a b \sqrt {\cot (c+d x)}}{d}+\frac {2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {4 a b \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x)}{7 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.59 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.80 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {\frac {2}{7} a^2 \cot ^{\frac {7}{2}}(c+d x)+\frac {2}{3} \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )+\frac {1}{10} a b \left (-10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-40 \sqrt {\cot (c+d x)}+8 \cot ^{\frac {5}{2}}(c+d x)-5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{d} \]
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Time = 1.55 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {\frac {2 a^{2} \left (\cot ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 a b \left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 a^{2} \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {2 b^{2} \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 a b \left (\sqrt {\cot }\left (d x +c \right )\right )+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(251\) |
default | \(-\frac {\frac {2 a^{2} \left (\cot ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 a b \left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 a^{2} \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {2 b^{2} \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 a b \left (\sqrt {\cot }\left (d x +c \right )\right )+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(251\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (226) = 452\).
Time = 0.29 (sec) , antiderivative size = 1083, normalized size of antiderivative = 4.04 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {105 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{3} - 105 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{3} - 105 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{3} + 105 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{3} + \frac {4 \, {\left (210 \, a b \tan \left (d x + c\right )^{3} - 42 \, a b \tan \left (d x + c\right ) + 35 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 15 \, a^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}}}{210 \, d \tan \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.82 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {210 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {1680 \, a b}{\sqrt {\tan \left (d x + c\right )}} + \frac {336 \, a b}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {280 \, {\left (a^{2} - b^{2}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {120 \, a^{2}}{\tan \left (d x + c\right )^{\frac {7}{2}}}}{420 \, d} \]
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\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2 \,d x \]
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