Integrand size = 23, antiderivative size = 208 \[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\frac {(a+b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a+b) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}+\frac {(a-b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {(a-b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}} \]
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Time = 0.18 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\frac {(a+b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a+b) \arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \sqrt {e}}+\frac {(a-b) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {(a-b) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e}} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {-a e-b x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d} \\ & = -\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}-\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} \\ & = -\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}-\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}+\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 \sqrt {e}}+\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 \sqrt {e}} \\ & = \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 \sqrt {e}}-\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 \sqrt {e}}-\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 \sqrt {e}}+\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 \sqrt {e}} \\ & = \frac {(a+b) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a+b) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}+\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 \sqrt {e}}-\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 \sqrt {e}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\frac {3 \sqrt {2} b \left (-2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )+8 a \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.31
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{d}\) | \(273\) |
default | \(\frac {-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{d}\) | \(273\) |
parts | \(-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d e}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}\) | \(275\) |
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Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (161) = 322\).
Time = 0.27 (sec) , antiderivative size = 721, normalized size of antiderivative = 3.47 \[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\frac {1}{2} \, \sqrt {-\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} + 2 \, a b}{d^{2} e}} \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 (b d^{3} e^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} + {\left (a^{3} - a b^{2}\right )} d e\right )} \sqrt {-\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} + 2 \, a b}{d^{2} e}}\right ) - \frac {1}{2} \, \sqrt {-\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} + 2 \, a b}{d^{2} e}} \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 (b d^{3} e^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} + {\left (a^{3} - a b^{2}\right )} d e\right )} \sqrt {-\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} + 2 \, a b}{d^{2} e}}\right ) - \frac {1}{2} \, \sqrt {\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} - 2 \, a b}{d^{2} e}} \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 (b d^{3} e^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} - {\left (a^{3} - a b^{2}\right )} d e\right )} \sqrt {\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} - 2 \, a b}{d^{2} e}}\right ) + \frac {1}{2} \, \sqrt {\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} - 2 \, a b}{d^{2} e}} \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 (b d^{3} e^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} - {\left (a^{3} - a b^{2}\right )} d e\right )} \sqrt {\frac {d^{2} e \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4} e^{2}}} - 2 \, a b}{d^{2} e}}\right ) \]
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\[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\int \frac {a + b \cot {\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\int { \frac {b \cot \left (d x + c\right ) + a}{\sqrt {e \cot \left (d x + c\right )}} \,d x } \]
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Time = 13.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx=\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,\sqrt {e}}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,\sqrt {e}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d\,\sqrt {e}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,1{}\mathrm {i}}{d\,\sqrt {e}} \]
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