Integrand size = 21, antiderivative size = 649 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}} \]
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Time = 0.68 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4817, 4723, 272, 44, 65, 214, 4757, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {e (a+b \arcsin (c x))}{d^2 x}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}-\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2} \]
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Rule 44
Rule 65
Rule 214
Rule 272
Rule 2221
Rule 2317
Rule 2438
Rule 4617
Rule 4723
Rule 4757
Rule 4817
Rule 4825
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \arcsin (c x)}{d x^4}-\frac {e (a+b \arcsin (c x))}{d^2 x^2}+\frac {e^2 (a+b \arcsin (c x))}{d^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arcsin (c x)}{x^4} \, dx}{d}-\frac {e \int \frac {a+b \arcsin (c x)}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \arcsin (c x)}{d+e x^2} \, dx}{d^2} \\ & = -\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}+\frac {(b c) \int \frac {1}{x^3 \sqrt {1-c^2 x^2}} \, dx}{3 d}-\frac {(b c e) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{d^2}+\frac {e^2 \int \left (\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \arcsin (c x))}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^2} \\ & = -\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{6 d}-\frac {(b c e) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d^2}-\frac {e^2 \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 (-d)^{5/2}}-\frac {e^2 \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{5/2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{12 d}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c d^2}-\frac {e^2 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}}-\frac {e^2 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {(b c) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{6 d}-\frac {\left (i e^2\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}-\sqrt {c^2 d+e}-\sqrt {e} e^{i x}} \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}}-\frac {\left (i e^2\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}+\sqrt {c^2 d+e}-\sqrt {e} e^{i x}} \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}}-\frac {\left (i e^2\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}-\sqrt {c^2 d+e}+\sqrt {e} e^{i x}} \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}}-\frac {\left (i e^2\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i c \sqrt {-d}+\sqrt {c^2 d+e}+\sqrt {e} e^{i x}} \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}}-\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^{i x}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right ) \, dx,x,\arcsin (c x)\right )}{2 (-d)^{5/2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {\left (i b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{2 (-d)^{5/2}}-\frac {\left (i b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{2 (-d)^{5/2}}+\frac {\left (i b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{2 (-d)^{5/2}}-\frac {\left (i b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{2 (-d)^{5/2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 531, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=-\frac {a}{3 d x^3}+\frac {a e}{d^2 x}+\frac {a e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}+b \left (-\frac {e \left (-\frac {\arcsin (c x)}{x}-c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d^2}-\frac {c x \sqrt {1-c^2 x^2}+2 \arcsin (c x)+c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d x^3}-\frac {e^{3/2} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}+\frac {e^{3/2} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.28 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.76
method | result | size |
parts | \(a \left (-\frac {1}{3 d \,x^{3}}+\frac {e}{d^{2} x}+\frac {e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}\right )-\frac {b \left (-4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}+4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x +24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}-24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}-24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}+8 c^{4} d^{2} \arcsin \left (c x \right )+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e -4 c^{2} d -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right ) c^{3} x^{3} e^{2}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right ) c^{3} x^{3} e^{2}\right )}{24 c^{4} x^{3} d^{3}}\) | \(491\) |
derivativedivides | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}+\frac {b \left (4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}-4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}+24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}-8 c^{4} d^{2} \arcsin \left (c x \right )+24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\) | \(501\) |
default | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}+\frac {b \left (4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}-4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}+24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}-8 c^{4} d^{2} \arcsin \left (c x \right )+24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\) | \(501\) |
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\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]
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\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \]
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Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^4\,\left (e\,x^2+d\right )} \,d x \]
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