Integrand size = 21, antiderivative size = 214 \[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f} \]
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Time = 0.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {16, 3554, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f} \]
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Rule 16
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {\int (d \cot (e+f x))^{5/2} \, dx}{d^4} \\ & = -\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}-\frac {\int \sqrt {d \cot (e+f x)} \, dx}{d^2} \\ & = -\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{d f} \\ & = -\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {2 \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f} \\ & = -\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}-\frac {\text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {\text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f} \\ & = -\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 d f}+\frac {\text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 d f} \\ & = -\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.47 \[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=-\frac {\cot ^{\frac {5}{4}}(e+f x) \left (-3 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot (e+f x)}+3 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot (e+f x)}+2 \cot ^{\frac {7}{4}}(e+f x)\right )}{3 f (d \cot (e+f x))^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\cot \left (f x +e \right ) d \right )^{\frac {3}{2}}}{3}-\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {\cot \left (f x +e \right ) d -\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}{\cot \left (f x +e \right ) d +\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f \,d^{3}}\) | \(156\) |
default | \(-\frac {2 \left (\frac {\left (\cot \left (f x +e \right ) d \right )^{\frac {3}{2}}}{3}-\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {\cot \left (f x +e \right ) d -\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}{\cot \left (f x +e \right ) d +\left (d^{2}\right )^{\frac {1}{4}} \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {\cot \left (f x +e \right ) d}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f \,d^{3}}\) | \(156\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.62 \[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {3 \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}}\right ) \sin \left (2 \, f x + 2 \, e\right ) - 3 i \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}}\right ) \sin \left (2 \, f x + 2 \, e\right ) + 3 i \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}}\right ) \sin \left (2 \, f x + 2 \, e\right ) - 3 \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}}\right ) \sin \left (2 \, f x + 2 \, e\right ) - 4 \, \sqrt {\frac {d \cos \left (2 \, f x + 2 \, e\right ) + d}{\sin \left (2 \, f x + 2 \, e\right )}} {\left (\cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}{6 \, d^{2} f \sin \left (2 \, f x + 2 \, e\right )} \]
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\[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {3 \, d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )} - 8 \, \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {3}{2}}}{12 \, d^{3} f} \]
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\[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{4}}{\left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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Time = 3.55 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.36 \[ \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {cot}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{3/2}\,f}-\frac {2\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^{3/2}}{3\,d^3\,f}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {cot}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{3/2}\,f} \]
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