Integrand size = 18, antiderivative size = 762 \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {-a-b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt {c^2 d+e}}+\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt {c^2 d+e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \]
[Out]
Time = 1.67 (sec) , antiderivative size = 759, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5339, 4817, 4757, 4827, 739, 212, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d+e}}\right )}{4 d^{3/2} \sqrt {c^2 d+e}}+\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d+e}}\right )}{4 d^{3/2} \sqrt {c^2 d+e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \]
[In]
[Out]
Rule 212
Rule 739
Rule 2221
Rule 2317
Rule 2438
Rule 4615
Rule 4757
Rule 4817
Rule 4825
Rule 4827
Rule 5339
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2 \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {e \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{d \left (e+d x^2\right )^2}+\frac {a+b \arcsin \left (\frac {x}{c}\right )}{d \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 )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d}+\frac {e \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 )}{d} \\ & = -\frac {\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 )}{d}+\frac {e \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 )}{d} \\ & = -\left (\frac {1}{4} \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 )\right )-\frac {1}{4} \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 )-\frac {1}{2} \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 )-\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 \sqrt {e}}-\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 \sqrt {e}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {1}{2} \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 )+\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 d}-\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 d}-\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 \sqrt {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 \sqrt {e}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\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 d}+\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 d}+\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 d \sqrt {e}}+\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 d \sqrt {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 )}{2 d \sqrt {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 )}{2 d \sqrt {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 )}{2 d \sqrt {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 )}{2 d \sqrt {e}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \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 d^{3/2} \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 d^{3/2} \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 )}{2 (-d)^{3/2} \sqrt {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 )}{2 (-d)^{3/2} \sqrt {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 )}{2 (-d)^{3/2} \sqrt {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 )}{2 (-d)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {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 d \sqrt {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 d \sqrt {e}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \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 d^{3/2} \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 d^{3/2} \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 )}{2 (-d)^{3/2} \sqrt {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 )}{2 (-d)^{3/2} \sqrt {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 )}{2 (-d)^{3/2} \sqrt {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 )}{2 (-d)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {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 d \sqrt {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 d \sqrt {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 d \sqrt {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 d \sqrt {e}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \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 d^{3/2} \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 d^{3/2} \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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {e}}+\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 (-d)^{3/2} \sqrt {e}}-\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 (-d)^{3/2} \sqrt {e}}+\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 (-d)^{3/2} \sqrt {e}}-\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 (-d)^{3/2} \sqrt {e}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \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 d^{3/2} \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 d^{3/2} \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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {e}}+\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)^{3/2} \sqrt {e}}-\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)^{3/2} \sqrt {e}}-\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 (-d)^{3/2} \sqrt {e}}+\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 (-d)^{3/2} \sqrt {e}}-\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 (-d)^{3/2} \sqrt {e}}+\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 (-d)^{3/2} \sqrt {e}} \\ & = -\frac {a+b \csc ^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \csc ^{-1}(c x)}{4 d \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 d^{3/2} \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 d^{3/2} \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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {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 (-d)^{3/2} \sqrt {e}}+\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 (-d)^{3/2} \sqrt {e}}-\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 (-d)^{3/2} \sqrt {e}}+\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 (-d)^{3/2} \sqrt {e}}-\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 (-d)^{3/2} \sqrt {e}} \\ \end{align*}
Time = 2.06 (sec) , antiderivative size = 1477, normalized size of antiderivative = 1.94 \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {a x}{d^2+d e x^2}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}+\frac {b \left (\frac {2 \sqrt {d} \csc ^{-1}(c x)}{-i \sqrt {d} \sqrt {e}+e x}+\frac {2 \sqrt {d} \csc ^{-1}(c x)}{i \sqrt {d} \sqrt {e}+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 {e}}-\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 {e}}-\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 {e}}+\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 {e}}-\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 {e}}+\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 {e}}-\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 {e}}+\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 {e}}+\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 {e}}-\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 {e}}-\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 {e}}-\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 {e}}+\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 {e}}+\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 {e}}-\frac {i \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )}{\sqrt {e}}+\frac {i \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )}{\sqrt {e}}+\frac {2 i \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 {-c^2 d-e}}-\frac {2 i \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 {-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 {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 {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 {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 {e}}\right )}{4 d^{3/2}}\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 33.95 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.09
method | result | size |
parts | \(\frac {a x}{2 d \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arccsc}\left (c x \right ) x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c^{2} \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 d}-\frac {c^{2} \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 d}+\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 d^{4} c^{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 d^{4} \left (c^{2} d +e \right ) c^{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 d^{4} c^{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 d^{4} \left (c^{2} d +e \right ) c^{3}}\right )}{c}\) | \(832\) |
derivativedivides | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arccsc}\left (c x \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\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 d \,c^{2}}-\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 d \,c^{2}}+\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^{7} d^{4}}-\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^{7} d^{4} \left (c^{2} d +e \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 ) \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^{7} d^{4}}-\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^{7} d^{4} \left (c^{2} d +e \right )}\right )}{c}\) | \(847\) |
default | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arccsc}\left (c x \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\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 d \,c^{2}}-\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 d \,c^{2}}+\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^{7} d^{4}}-\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^{7} d^{4} \left (c^{2} d +e \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 ) \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^{7} d^{4}}-\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^{7} d^{4} \left (c^{2} d +e \right )}\right )}{c}\) | \(847\) |
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\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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