Integrand size = 25, antiderivative size = 160 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=-\frac {2 \sqrt {2} a^3 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e} \]
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Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3647, 3711, 3609, 3613, 214} \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=-\frac {2 \sqrt {2} a^3 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}+\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e} \]
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Rule 214
Rule 3609
Rule 3613
Rule 3647
Rule 3711
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int (e \cot (c+d x))^{3/2} \left (-a^3 e-7 a^3 e \cot (c+d x)-8 a^3 e \cot ^2(c+d x)\right ) \, dx}{7 e} \\ & = -\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int (e \cot (c+d x))^{3/2} \left (7 a^3 e-7 a^3 e \cot (c+d x)\right ) \, dx}{7 e} \\ & = -\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (7 a^3 e^2+7 a^3 e^2 \cot (c+d x)\right ) \, dx}{7 e} \\ & = \frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}-\frac {2 \int \frac {-7 a^3 e^3+7 a^3 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{7 e} \\ & = \frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e}+\frac {\left (28 a^6 e^5\right ) \text {Subst}\left (\int \frac {1}{98 a^6 e^6-e x^2} \, dx,x,\frac {-7 a^3 e^3-7 a^3 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d} \\ & = -\frac {2 \sqrt {2} a^3 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(380\) vs. \(2(160)=320\).
Time = 5.18 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.38 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=-\frac {a^3 (e \cot (c+d x))^{3/2} (1+\cot (c+d x))^3 \left (-420 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+420 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+\sqrt [4]{\cot (c+d x)} \sin (c+d x) \left (60 \cos ^2(c+d x) \cot ^{\frac {3}{2}}(c+d x)-210 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)+210 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)-840 \sqrt {\cot (c+d x)} \sin ^2(c+d x)+280 \cot ^{\frac {3}{2}}(c+d x) \sin ^2(c+d x)-105 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+105 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+126 \cot ^{\frac {3}{2}}(c+d x) \sin (2 (c+d x))\right )\right )}{210 d \cot ^{\frac {7}{4}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(135)=270\).
Time = 0.04 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.12
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {3 e \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {e \cot \left (d x +c \right )}\, e^{3}+2 e^{4} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(339\) |
default | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {3 e \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {e \cot \left (d x +c \right )}\, e^{3}+2 e^{4} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(339\) |
parts | \(-\frac {2 a^{3} 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 {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{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 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d \,e^{2}}+\frac {3 a^{3} \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}-\frac {6 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\sqrt {e \cot \left (d x +c \right )}\, e^{2}+\frac {e^{2} \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 e}\) | \(654\) |
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Time = 0.28 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.04 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\left [\frac {105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\right )} \sqrt {e} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \, {\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \, {\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{105 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}, \frac {2 \, {\left (105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \, {\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{105 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}\right ] \]
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\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=a^{3} \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\text {Exception raised: ValueError} \]
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\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
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Time = 14.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\frac {4\,a^3\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {6\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}{7\,d\,e^2}-\frac {4\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}+\frac {\sqrt {2}\,a^3\,e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^6\,e^{9/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,32{}\mathrm {i}}{32\,a^6\,e^5+32\,a^6\,e^5\,\mathrm {cot}\left (c+d\,x\right )}\right )\,2{}\mathrm {i}}{d} \]
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