Integrand size = 23, antiderivative size = 271 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \]
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Time = 0.79 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3754, 3647, 3728, 3735, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a-b) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a+b) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} d \left (a^2+b^2\right )}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d} \]
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Rule 65
Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3647
Rule 3715
Rule 3728
Rule 3735
Rule 3754
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{\frac {7}{2}}(c+d x)}{b+a \cot (c+d x)} \, dx \\ & = -\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {2 \int \frac {\sqrt {\cot (c+d x)} \left (\frac {3 b}{2}+\frac {3}{2} a \cot (c+d x)+\frac {3}{2} b \cot ^2(c+d x)\right )}{b+a \cot (c+d x)} \, dx}{3 a} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {4 \int \frac {\frac {3 b^2}{4}-\frac {3}{4} \left (a^2-b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{3 a^2} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {4 \int \frac {\frac {3 a^2 b}{4}-\frac {3}{4} a^3 \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{3 a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {8 \text {Subst}\left (\int \frac {-\frac {3 a^2 b}{4}+\frac {3 a^3 x^2}{4}}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{3 a^2 \left (a^2+b^2\right ) d}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a-b) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {(a+b) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a+b) \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} \left (a^2+b^2\right ) d}+\frac {(a+b) \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} \left (a^2+b^2\right ) d} \\ & = -\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d} \\ & = -\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right ) d}+\frac {2 b \sqrt {\cot (c+d x)}}{a^2 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.42 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {6 \sqrt {2} a^{5/2} b \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-6 \sqrt {2} a^{5/2} b \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-24 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )+24 a^{5/2} b \sqrt {\cot (c+d x)}+24 \sqrt {a} b^3 \sqrt {\cot (c+d x)}-8 a^{7/2} \cot ^{\frac {3}{2}}(c+d x)-8 a^{3/2} b^2 \cot ^{\frac {3}{2}}(c+d x)+8 a^{7/2} \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )+3 \sqrt {2} a^{5/2} b \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-3 \sqrt {2} a^{5/2} b \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{12 a^{5/2} \left (a^2+b^2\right ) d} \]
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Time = 1.40 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.50
method | result | size |
derivativedivides | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}+6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b +6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -3 \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -24 b^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-24 \sqrt {a b}\, \tan \left (d x +c \right ) a^{2} b -24 \sqrt {a b}\, \tan \left (d x +c \right ) b^{3}+8 \sqrt {a b}\, a^{3}+8 \sqrt {a b}\, a \,b^{2}\right )}{12 d \,a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}\) | \(406\) |
default | \(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 \sqrt {a b}\, \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}+6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b +6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{3}-6 \sqrt {a b}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -3 \sqrt {a b}\, \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a^{2} b -24 b^{4} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-24 \sqrt {a b}\, \tan \left (d x +c \right ) a^{2} b -24 \sqrt {a b}\, \tan \left (d x +c \right ) b^{3}+8 \sqrt {a b}\, a^{3}+8 \sqrt {a b}\, a \,b^{2}\right )}{12 d \,a^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}\) | \(406\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1446 vs. \(2 (227) = 454\).
Time = 0.39 (sec) , antiderivative size = 2922, normalized size of antiderivative = 10.78 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\cot ^{\frac {5}{2}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {24 \, b^{4} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a b}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a + b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a + b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )}}{a^{2} + b^{2}} - \frac {8 \, {\left (\frac {3 \, b}{\sqrt {\tan \left (d x + c\right )}} - \frac {a}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )}}{a^{2}}}{12 \, d} \]
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\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {5}{2}}}{b \tan \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}}{a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \]
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