Integrand size = 21, antiderivative size = 704 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3} \]
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Time = 0.93 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5349, 4817, 4813, 390, 385, 211, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^3}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (\frac {d}{x^2}+e\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (\frac {d}{x^2}+e\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{d^3 \sqrt {c^2 d+e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 x \left (c^2 d+e\right ) \left (\frac {d}{x^2}+e\right )} \]
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Rule 211
Rule 385
Rule 390
Rule 2221
Rule 2317
Rule 2438
Rule 4615
Rule 4813
Rule 4817
Rule 4825
Rule 5349
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^5 \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {e^2 x \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^3}-\frac {2 e x \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}+\frac {x \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {x \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \text {Subst}\left (\int \frac {x \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \text {Subst}\left (\int \frac {x \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )}{d^2} \\ & = \frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}-\frac {\text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{c d^3}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}} \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{4 c d^3} \\ & = -\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}-\frac {\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 (-d)^{5/2}}+\frac {\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 (-d)^{5/2}}+\frac {(b e) \text {Subst}\left (\int \frac {1}{e-\left (-d-\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{c d^3}-\frac {\left (b e \left (c^2 d+2 e\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{8 c d^3 \left (c^2 d+e\right )} \\ & = -\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac {\left (b e \left (c^2 d+2 e\right )\right ) \text {Subst}\left (\int \frac {1}{e-\left (-d-\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 c d^3 \left (c^2 d+e\right )} \\ & = -\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2}}+\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2}} \\ & = -\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 d^3}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 d^3}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 d^3}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 d^3} \\ & = -\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 d^3}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 d^3}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 d^3}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 d^3} \\ & = -\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2114\) vs. \(2(704)=1408\).
Time = 6.06 (sec) , antiderivative size = 2114, normalized size of antiderivative = 3.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.31 (sec) , antiderivative size = 3479, normalized size of antiderivative = 4.94
method | result | size |
parts | \(\text {Expression too large to display}\) | \(3479\) |
derivativedivides | \(\text {Expression too large to display}\) | \(3553\) |
default | \(\text {Expression too large to display}\) | \(3553\) |
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\[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
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Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]
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