Integrand size = 21, antiderivative size = 632 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3} \]
[Out]
Time = 0.73 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {4817, 4723, 270, 4721, 3798, 2221, 2317, 2438, 4813, 385, 211, 4825, 4617} \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}-\frac {2 e \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {b c e \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}-\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x} \]
[In]
[Out]
Rule 211
Rule 270
Rule 385
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4617
Rule 4721
Rule 4723
Rule 4813
Rule 4817
Rule 4825
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \arcsin (c x)}{d^2 x^3}-\frac {2 e (a+b \arcsin (c x))}{d^3 x}+\frac {e^2 x (a+b \arcsin (c x))}{d^2 \left (d+e x^2\right )^2}+\frac {2 e^2 x (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arcsin (c x)}{x^3} \, dx}{d^2}-\frac {(2 e) \int \frac {a+b \arcsin (c x)}{x} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \arcsin (c x))}{d+e x^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx}{d^2} \\ & = -\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{2 d^2}-\frac {(2 e) \text {Subst}(\int (a+b x) \cot (x) \, dx,x,\arcsin (c x))}{d^3}+\frac {(b c e) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 d^2}+\frac {\left (2 e^2\right ) \int \left (-\frac {a+b \arcsin (c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^3} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {i e (a+b \arcsin (c x))^2}{b d^3}+\frac {(4 i e) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{d^3}+\frac {(b c e) \text {Subst}\left (\int \frac {1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-c^2 x^2}}\right )}{2 d^2}-\frac {e^{3/2} \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{d^3}+\frac {e^{3/2} \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{d^3} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {i e (a+b \arcsin (c x))^2}{b d^3}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {(2 b e) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3}-\frac {e^{3/2} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}-\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{d^3}+\frac {e^{3/2} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{c \sqrt {-d}+\sqrt {e} \sin (x)} \, dx,x,\arcsin (c x)\right )}{d^3} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {(i b e) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {\left (i e^{3/2}\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 )}{d^3}-\frac {\left (i e^{3/2}\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 )}{d^3}+\frac {\left (i e^{3/2}\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 )}{d^3}+\frac {\left (i e^{3/2}\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 )}{d^3} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3}-\frac {(b e) \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 )}{d^3} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {(i b e) \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 )}{d^3}+\frac {(i b e) \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 )}{d^3}+\frac {(i b e) \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 )}{d^3}+\frac {(i b e) \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 )}{d^3} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3} \\ \end{align*}
Time = 1.73 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a d}{x^2}+\frac {2 a d e}{d+e x^2}+8 a e \log (x)-4 a e \log \left (d+e x^2\right )+b \left (\frac {2 d \left (c x \sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{x^2}+\sqrt {d} e \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-i \sqrt {d} e \left (-\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+2 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+2 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-4 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )\right )}{4 d^3} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.96 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.93
method | result | size |
parts | \(-\frac {a}{2 d^{2} x^{2}}-\frac {2 a e \ln \left (x \right )}{d^{3}}+\frac {a e \ln \left (e \,x^{2}+d \right )}{d^{3}}-\frac {a e}{2 d^{2} \left (e \,x^{2}+d \right )}+b \,c^{2} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 c^{2} x^{2} d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}-\frac {i \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-4 c^{2} d -2 e}{4 \sqrt {d^{2} c^{4}+c^{2} e d}}\right ) e}{2 d^{3} c^{2} \left (c^{2} d +e \right )}-\frac {i e \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}^{2} e -2 c^{2} d -e}\right )}{2 d^{3} c^{2}}-\frac {i e^{2} \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}-1\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}^{2} e -2 c^{2} d -e}\right )}{2 d^{3} c^{2}}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}\right )\) | \(589\) |
derivativedivides | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}+b \,c^{4} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {i e^{2} \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}-1\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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}-\frac {i e \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {i \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {d^{2} c^{4}+c^{2} e d}}\right ) e}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}\right )\right )\) | \(615\) |
default | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}+b \,c^{4} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {i e^{2} \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}-1\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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}-\frac {i e \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {i \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {d^{2} c^{4}+c^{2} e d}}\right ) e}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}\right )\right )\) | \(615\) |
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\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{3} \left (d + e x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]
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