Integrand size = 21, antiderivative size = 565 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}} \]
[Out]
Time = 0.90 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5349, 4817, 4723, 272, 65, 214, 4757, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}} \]
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 2221
Rule 2317
Rule 2438
Rule 4615
Rule 4723
Rule 4757
Rule 4817
Rule 4825
Rule 5349
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{x^2 \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {a+b \arcsin \left (\frac {x}{c}\right )}{e x^2}-\frac {d \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{x^2} \, dx,x,\frac {1}{x}\right )}{e}+\frac {d \text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c e}+\frac {d \text {Subst}\left (\int \left (\frac {a+b \arcsin \left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \arcsin \left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}}+\frac {(b c) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {\sqrt {-d} \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 e^{3/2}}+\frac {\sqrt {-d} \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 e^{3/2}}-\frac {\sqrt {-d} \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 e^{3/2}}+\frac {\sqrt {-d} \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 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {\sqrt {-d} \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 e^{3/2}}+\frac {\sqrt {-d} \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 e^{3/2}}-\frac {\sqrt {-d} \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 e^{3/2}}+\frac {\sqrt {-d} \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 e^{3/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {\sqrt {-d} \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 e^{3/2}}+\frac {\sqrt {-d} \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 e^{3/2}}-\frac {\sqrt {-d} \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 e^{3/2}}+\frac {\sqrt {-d} \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 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1260\) vs. \(2(565)=1130\).
Time = 1.82 (sec) , antiderivative size = 1260, normalized size of antiderivative = 2.23 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {i \left (-4 i a c \sqrt {e} x-4 i b c \sqrt {e} x \csc ^{-1}(c x)+4 i a c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+8 i b c \sqrt {d} \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 )-8 i b c \sqrt {d} \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 )+b c \sqrt {d} \pi \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 b c \sqrt {d} \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 )+4 b c \sqrt {d} \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 )-b c \sqrt {d} \pi \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 b c \sqrt {d} \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 )-4 b c \sqrt {d} \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 )-b c \sqrt {d} \pi \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 b c \sqrt {d} \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 )+4 b c \sqrt {d} \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 )+b c \sqrt {d} \pi \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 b c \sqrt {d} \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 )-4 b c \sqrt {d} \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 )+b c \sqrt {d} \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )-b c \sqrt {d} \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )-4 i b \sqrt {e} \log \left (\cos \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )+4 i b \sqrt {e} \log \left (\sin \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )+2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )\right )}{4 c e^{3/2}} \]
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[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 44.19 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.73
method | result | size |
parts | \(\frac {a x}{e}-\frac {a d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arccsc}\left (c x \right ) x}{e}-\frac {c^{2} \ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {c^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {c^{4} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}+\frac {c^{4} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\) | \(415\) |
derivativedivides | \(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+b \,c^{2} \left (\frac {c x \,\operatorname {arccsc}\left (c x \right )}{e}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}-\frac {\ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\) | \(417\) |
default | \(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+b \,c^{2} \left (\frac {c x \,\operatorname {arccsc}\left (c x \right )}{e}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}-\frac {\ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\) | \(417\) |
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\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{e x^{2} + d} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \]
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