Integrand size = 25, antiderivative size = 141 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=-\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{7/2}}+\frac {8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac {4 a^3}{d e^3 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{5 d e (e \cot (c+d x))^{5/2}} \]
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Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3646, 3709, 3610, 3613, 211} \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=-\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{7/2}}+\frac {4 a^3}{d e^3 \sqrt {e \cot (c+d x)}}+\frac {8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}} \]
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Rule 211
Rule 3610
Rule 3613
Rule 3646
Rule 3709
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{5 d e (e \cot (c+d x))^{5/2}}-\frac {2 \int \frac {-6 a^3 e^2-5 a^3 e^2 \cot (c+d x)-a^3 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{5/2}} \, dx}{5 e^3} \\ & = \frac {8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{5 d e (e \cot (c+d x))^{5/2}}-\frac {2 \int \frac {-5 a^3 e^3+5 a^3 e^3 \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{5 e^5} \\ & = \frac {8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac {4 a^3}{d e^3 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{5 d e (e \cot (c+d x))^{5/2}}-\frac {2 \int \frac {5 a^3 e^4+5 a^3 e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{5 e^7} \\ & = \frac {8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac {4 a^3}{d e^3 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{5 d e (e \cot (c+d x))^{5/2}}+\frac {\left (20 a^6 e\right ) \text {Subst}\left (\int \frac {1}{-50 a^6 e^8-e x^2} \, dx,x,\frac {5 a^3 e^4-5 a^3 e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d} \\ & = -\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{7/2}}+\frac {8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac {4 a^3}{d e^3 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{5 d e (e \cot (c+d x))^{5/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(415\) vs. \(2(141)=282\).
Time = 5.75 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.94 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\frac {a^3 \left (80 \cos ^3(c+d x)+40 \cos ^2(c+d x) \sin (c+d x)+8 \cos (c+d x) \sin ^2(c+d x)+10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {7}{2}}(c+d x) \sin ^3(c+d x)-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {7}{2}}(c+d x) \sin ^3(c+d x)-20 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \left (2 \sqrt [4]{-\cot (c+d x)} \cot ^{\frac {13}{4}}(c+d x)-3 \left (-\cot ^2(c+d x)\right )^{7/4}\right ) \sin ^3(c+d x)+20 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \left (2 \sqrt [4]{-\cot (c+d x)} \cot ^{\frac {13}{4}}(c+d x)+3 \left (-\cot ^2(c+d x)\right )^{7/4}\right ) \sin ^3(c+d x)+5 \sqrt {2} \cot ^{\frac {7}{2}}(c+d x) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^3(c+d x)-5 \sqrt {2} \cot ^{\frac {7}{2}}(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^3(c+d x)\right ) (1+\tan (c+d x))^3}{20 d e^3 \sqrt {e \cot (c+d x)} (\cos (c+d x)+\sin (c+d x))^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs. \(2(120)=240\).
Time = 0.05 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.29
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}-\frac {e}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(323\) |
default | \(-\frac {2 a^{3} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}-\frac {e}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(323\) |
parts | \(-\frac {2 a^{3} e \left (-\frac {1}{5 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}+\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 e^{4} \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}-\frac {a^{3} \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 )}{4 d \,e^{4}}+\frac {3 a^{3} \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 )}{4 e^{4}}+\frac {2}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}-\frac {6 a^{3} \left (-\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 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d e}\) | \(634\) |
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Time = 0.28 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.44 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\left [\frac {5 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \sqrt {-\frac {1}{e}} \log \left (\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 2 \, {\left (5 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} - 5 \, a^{3} - {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 11 \, a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{5 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )}}, -\frac {2 \, {\left (\frac {5 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right )}{\sqrt {e}} + {\left (5 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} - 5 \, a^{3} - {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 11 \, a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{5 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )}}\right ] \]
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\[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=a^{3} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx\right ) \]
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Exception generated. \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 13.54 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.89 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\frac {4\,e\,a^3\,{\mathrm {cot}\left (c+d\,x\right )}^2+2\,e\,a^3\,\mathrm {cot}\left (c+d\,x\right )+\frac {2\,e\,a^3}{5}}{d\,e^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}+\frac {\sqrt {2}\,a^3\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d\,e^{7/2}} \]
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