Integrand size = 36, antiderivative size = 156 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {B \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {B \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {B \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {B \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
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Time = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {21, 3554, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {B \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {B \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {B \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {B \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d} \]
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Rule 21
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rubi steps \begin{align*} \text {integral}& = B \int \cot ^{\frac {5}{2}}(c+d x) \, dx \\ & = -\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}-B \int \sqrt {\cot (c+d x)} \, dx \\ & = -\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {B \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {(2 B) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = -\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {B \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}+\frac {B \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = -\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {B \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}+\frac {B \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}+\frac {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} d}+\frac {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} d} \\ & = -\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {B \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {B \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {B \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {B \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d} \\ & = -\frac {B \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {B \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 B \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {B \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {B \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.57 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {B \left (-3 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)}+3 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)}+2 \cot ^{\frac {7}{4}}(c+d x)\right )}{3 d \sqrt [4]{\cot (c+d x)}} \]
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Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.65
method | result | size |
default | \(\frac {B \left (-\frac {2 \cot \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(102\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.94 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {3 \, d \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {B \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + {\left (d \cos \left (2 \, d x + 2 \, c\right ) + d\right )} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}}}{\cos \left (2 \, d x + 2 \, c\right ) + 1}\right ) \sin \left (2 \, d x + 2 \, c\right ) - 3 \, d \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {B \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) - {\left (d \cos \left (2 \, d x + 2 \, c\right ) + d\right )} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}}}{\cos \left (2 \, d x + 2 \, c\right ) + 1}\right ) \sin \left (2 \, d x + 2 \, c\right ) - 3 i \, d \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {B \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) - {\left (i \, d \cos \left (2 \, d x + 2 \, c\right ) + i \, d\right )} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}}}{\cos \left (2 \, d x + 2 \, c\right ) + 1}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 3 i \, d \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {B \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) - {\left (-i \, d \cos \left (2 \, d x + 2 \, c\right ) - i \, d\right )} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}}}{\cos \left (2 \, d x + 2 \, c\right ) + 1}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, {\left (B \cos \left (2 \, d x + 2 \, c\right ) + B\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, d \sin \left (2 \, d x + 2 \, c\right )} \]
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\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=B \int \cot ^{\frac {5}{2}}{\left (c + d x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {3 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} B - \frac {8 \, B}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \]
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\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\int { \frac {{\left (B b \tan \left (d x + c\right ) + B a\right )} \cot \left (d x + c\right )^{\frac {5}{2}}}{b \tan \left (d x + c\right ) + a} \,d x } \]
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Time = 9.93 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.41 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {{\left (-1\right )}^{1/4}\,B\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\frac {1}{\mathrm {tan}\left (c+d\,x\right )}}\right )}{d}-\frac {2\,B\,{\left (\frac {1}{\mathrm {tan}\left (c+d\,x\right )}\right )}^{3/2}}{3\,d}-\frac {{\left (-1\right )}^{1/4}\,B\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\frac {1}{\mathrm {tan}\left (c+d\,x\right )}}\right )}{d} \]
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