Integrand size = 21, antiderivative size = 765 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}} \]
[Out]
Time = 0.92 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5349, 4757, 4827, 739, 212, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d+e}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d+e}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}} \]
[In]
[Out]
Rule 212
Rule 739
Rule 2221
Rule 2317
Rule 2438
Rule 4615
Rule 4757
Rule 4825
Rule 4827
Rule 5349
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {d \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {d \text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e}+\frac {d \text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e}+\frac {d \text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{2 e} \\ & = \frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}+\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}+\frac {d \text {Subst}\left (\int \left (-\frac {a+b \arcsin \left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \arcsin \left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{2 e} \\ & = \frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\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 )}{4 e^{3/2}}-\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 )}{4 e^{3/2}}+\frac {b \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 c e}-\frac {b \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 c e} \\ & = \frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \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 )}{4 e^{3/2}}-\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 )}{4 e^{3/2}} \\ & = \frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\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 )}{4 e^{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 )}{4 e^{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 )}{4 e^{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 )}{4 e^{3/2}} \\ & = \frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}} \\ & = \frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}} \\ & = \frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} e \sqrt {c^2 d+e}}-\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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}} \\ \end{align*}
Time = 1.70 (sec) , antiderivative size = 1482, normalized size of antiderivative = 1.94 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {-\frac {4 a \sqrt {e} x}{d+e x^2}+\frac {4 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+b \left (\frac {2 \csc ^{-1}(c x)}{i \sqrt {d}-\sqrt {e} x}-\frac {2 \csc ^{-1}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {8 \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )}{\sqrt {d}}-\frac {8 \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )}{\sqrt {d}}-\frac {i \pi \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}+\frac {2 i \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {4 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}+\frac {i \pi \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {2 i \csc ^{-1}(c x) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}+\frac {4 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}+\frac {i \pi \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {2 i \csc ^{-1}(c x) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {4 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {i \pi \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}+\frac {2 i \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}+\frac {4 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {i \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )}{\sqrt {d}}+\frac {i \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )}{\sqrt {d}}-\frac {2 i \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (\sqrt {e}+c \left (-i c \sqrt {d}-\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {d} \sqrt {-c^2 d-e}}+\frac {2 i \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (-\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{\sqrt {d} \sqrt {-c^2 d-e}}+\frac {2 \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {2 \operatorname {PolyLog}\left (2,\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}-\frac {2 \operatorname {PolyLog}\left (2,-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}+\frac {2 \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )}{\sqrt {d}}\right )}{8 e^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 39.56 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.10
method | result | size |
parts | \(-\frac {a x}{2 e \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+\frac {b \left (-\frac {c^{5} \operatorname {arccsc}\left (c x \right ) x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c \,d^{3} e}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 \left (c^{2} d +e \right ) e \,d^{3} c}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c \,d^{3} e}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e -2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 \left (c^{2} d +e \right ) e \,d^{3} c}-\frac {c^{4} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{4 e}-\frac {c^{4} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}\right )}{4 e}\right )}{c^{3}}\) | \(844\) |
derivativedivides | \(\frac {-\frac {a \,c^{5} x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {c x \,\operatorname {arccsc}\left (c x \right )}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) e}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c^{5} e \left (c^{2} d +e \right ) d^{3}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e -2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c^{5} e \left (c^{2} d +e \right ) d^{3}}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}}{4 e}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}}{4 e}\right )}{c^{3}}\) | \(853\) |
default | \(\frac {-\frac {a \,c^{5} x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {c x \,\operatorname {arccsc}\left (c x \right )}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) e}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \arctan \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c^{5} e \left (c^{2} d +e \right ) d^{3}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e -2 \sqrt {e \left (c^{2} d +e \right )}\, e +2 e^{2}\right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c^{5} e \left (c^{2} d +e \right ) d^{3}}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}}{4 e}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}}{4 e}\right )}{c^{3}}\) | \(853\) |
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\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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