Integrand size = 23, antiderivative size = 229 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=-\frac {(a-b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {(a-b) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {2 a}{d e \sqrt {e \cot (c+d x)}}+\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}} \]
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Time = 0.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3610, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=-\frac {(a-b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {(a-b) \arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2}}+\frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{3/2}}+\frac {2 a}{d e \sqrt {e \cot (c+d x)}} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3610
Rule 3615
Rubi steps \begin{align*} \text {integral}& = \frac {2 a}{d e \sqrt {e \cot (c+d x)}}+\frac {\int \frac {b e-a e \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{e^2} \\ & = \frac {2 a}{d e \sqrt {e \cot (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {-b e^2+a e x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2} \\ & = \frac {2 a}{d e \sqrt {e \cot (c+d x)}}+\frac {(a-b) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}-\frac {(a+b) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e} \\ & = \frac {2 a}{d e \sqrt {e \cot (c+d x)}}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e} \\ & = \frac {2 a}{d e \sqrt {e \cot (c+d x)}}+\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}} \\ & = -\frac {(a-b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {(a-b) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {2 a}{d e \sqrt {e \cot (c+d x)}}+\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\frac {3 a \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+8 \sqrt {\tan (c+d x)}\right )+8 b \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.03 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{e}+\frac {2 a}{e \sqrt {e \cot \left (d x +c \right )}}}{d}\) | \(295\) |
default | \(\frac {-\frac {2 \left (\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{e}+\frac {2 a}{e \sqrt {e \cot \left (d x +c \right )}}}{d}\) | \(295\) |
parts | \(-\frac {2 a e \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}-\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \,e^{2}}\) | \(297\) |
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Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (180) = 360\).
Time = 0.29 (sec) , antiderivative size = 890, normalized size of antiderivative = 3.89 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=-\frac {{\left (d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}\right )} \sqrt {\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} + 2 \, a b}{d^{2} e^{3}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left (a d^{3} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} - {\left (a^{2} b - b^{3}\right )} d e^{2}\right )} \sqrt {\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} + 2 \, a b}{d^{2} e^{3}}}\right ) - {\left (d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}\right )} \sqrt {\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} + 2 \, a b}{d^{2} e^{3}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left (a d^{3} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} - {\left (a^{2} b - b^{3}\right )} d e^{2}\right )} \sqrt {\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} + 2 \, a b}{d^{2} e^{3}}}\right ) - {\left (d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}\right )} \sqrt {-\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} - 2 \, a b}{d^{2} e^{3}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + {\left (a d^{3} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} + {\left (a^{2} b - b^{3}\right )} d e^{2}\right )} \sqrt {-\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} - 2 \, a b}{d^{2} e^{3}}}\right ) + {\left (d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}\right )} \sqrt {-\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} - 2 \, a b}{d^{2} e^{3}}} \log \left (-{\left (a^{4} - b^{4}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - {\left (a d^{3} e^{5} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} + {\left (a^{2} b - b^{3}\right )} d e^{2}\right )} \sqrt {-\frac {d^{2} e^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{6}}} - 2 \, a b}{d^{2} e^{3}}}\right ) - 4 \, a \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{2 \, {\left (d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}\right )}} \]
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\[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\int \frac {a + b \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\int { \frac {b \cot \left (d x + c\right ) + a}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Time = 13.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.60 \[ \int \frac {a+b \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\frac {2\,a}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,e^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,e^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d\,e^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d\,e^{3/2}} \]
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