Integrand size = 23, antiderivative size = 94 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=-\frac {\sqrt {2} a e^{3/2} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d} \]
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Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3609, 3613, 211} \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=-\frac {\sqrt {2} a e^{3/2} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d} \]
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Rule 211
Rule 3609
Rule 3613
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}+\int \sqrt {e \cot (c+d x)} (-a e+a e \cot (c+d x)) \, dx \\ & = -\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}+\int \frac {-a e^2-a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx \\ & = -\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d}-\frac {\left (2 a^2 e^4\right ) \text {Subst}\left (\int \frac {1}{-2 a^2 e^4-e x^2} \, dx,x,\frac {-a e^2+a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d} \\ & = -\frac {\sqrt {2} a e^{3/2} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {2 a e \sqrt {e \cot (c+d x)}}{d}-\frac {2 a (e \cot (c+d x))^{3/2}}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=-\frac {2 a e \sqrt {e \cot (c+d x)} \left (\cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right )+3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\tan ^2(c+d x)\right )\right )}{3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(302\) vs. \(2(77)=154\).
Time = 0.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.22
method | result | size |
parts | \(-\frac {2 a e \left (\sqrt {e \cot \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d}+\frac {a \left (-\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(303\) |
derivativedivides | \(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(304\) |
default | \(-\frac {a \left (\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(304\) |
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.55 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=\left [\frac {3 \, \sqrt {2} a \sqrt {-e} e \log \left (\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) \sin \left (2 \, d x + 2 \, c\right ) - 4 \, {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, d \sin \left (2 \, d x + 2 \, c\right )}, -\frac {3 \, \sqrt {2} a e^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 2 \, {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, d \sin \left (2 \, d x + 2 \, c\right )}\right ] \]
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\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=a \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx\right ) \]
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Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=\text {Exception raised: ValueError} \]
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\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
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Time = 13.51 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.04 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x)) \, dx=-\frac {2\,a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {2\,a\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1-\mathrm {i}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (-1-\mathrm {i}\right )}{d} \]
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