Integrand size = 21, antiderivative size = 783 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4} \]
[Out]
Time = 0.81 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4817, 4723, 270, 4721, 3798, 2221, 2317, 2438, 4813, 390, 385, 211, 4825, 4617} \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}-\frac {3 e \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^4}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c e \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x} \]
[In]
[Out]
Rule 211
Rule 270
Rule 385
Rule 390
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^3 x^3}-\frac {3 e (a+b \arcsin (c x))}{d^4 x}+\frac {e^2 x (a+b \arcsin (c x))}{d^2 \left (d+e x^2\right )^3}+\frac {2 e^2 x (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )^2}+\frac {3 e^2 x (a+b \arcsin (c x))}{d^4 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arcsin (c x)}{x^3} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \arcsin (c x)}{x} \, dx}{d^4}+\frac {\left (3 e^2\right ) \int \frac {x (a+b \arcsin (c x))}{d+e x^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \arcsin (c x))}{\left (d+e x^2\right )^3} \, dx}{d^2} \\ & = -\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{2 d^3}-\frac {(3 e) \text {Subst}(\int (a+b x) \cot (x) \, dx,x,\arcsin (c x))}{d^4}+\frac {(b c e) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{d^3}+\frac {(b c e) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d^2}+\frac {\left (3 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^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {3 i e (a+b \arcsin (c x))^2}{2 b d^4}+\frac {(6 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^4}+\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 )}{d^3}-\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (b c e \left (2 c^2 d+e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d^3 \left (c^2 d+e\right )} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {3 i e (a+b \arcsin (c x))^2}{2 b d^4}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}+\frac {(3 b e) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^4}-\frac {\left (3 e^{3/2}\right ) \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^4}+\frac {\left (3 e^{3/2}\right ) \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^4}+\frac {\left (b c e \left (2 c^2 d+e\right )\right ) \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 )}{8 d^3 \left (c^2 d+e\right )} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \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 )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}-\frac {(3 i b e) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{2 d^4}-\frac {\left (3 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 )}{2 d^4}-\frac {\left (3 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 )}{2 d^4}+\frac {\left (3 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 )}{2 d^4}+\frac {\left (3 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 )}{2 d^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \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 )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4}-\frac {(3 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 )}{2 d^4}-\frac {(3 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 )}{2 d^4}-\frac {(3 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 )}{2 d^4}-\frac {(3 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 )}{2 d^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \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 )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4}+\frac {(3 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 )}{2 d^4}+\frac {(3 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 )}{2 d^4}+\frac {(3 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 )}{2 d^4}+\frac {(3 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 )}{2 d^4} \\ & = -\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \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 )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}+\frac {3 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 )}{2 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}-\frac {3 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 )}{2 d^4}-\frac {3 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 )}{2 d^4}-\frac {3 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 )}{2 d^4}-\frac {3 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 )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4} \\ \end{align*}
Time = 5.65 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.36 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {-\frac {8 a d}{x^2}-\frac {4 a d^2 e}{\left (d+e x^2\right )^2}-\frac {16 a d e}{d+e x^2}-48 a e \log (x)+24 a e \log \left (d+e x^2\right )+b \left (-\frac {8 c d \sqrt {1-c^2 x^2}}{x}+\frac {c d e^{3/2} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}+\frac {c d e^{3/2} \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {8 d \arcsin (c x)}{x^2}-\frac {9 \sqrt {d} e \arcsin (c x)}{\sqrt {d}-i \sqrt {e} x}-\frac {d e \arcsin (c x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}-\frac {9 \sqrt {d} e \arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {d e \arcsin (c x)}{\left (i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {9 c \sqrt {d} e \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}}-\frac {9 i c \sqrt {d} e \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}}+24 e \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+24 e \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+24 e \arcsin (c x) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+24 e \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-48 e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+\frac {i c^3 d^{3/2} e \log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}-\frac {i c^3 d^{3/2} e \log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}-24 i e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-24 i e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-24 i e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-24 i e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+24 i e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{16 d^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.52 (sec) , antiderivative size = 1344, normalized size of antiderivative = 1.72
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1344\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1395\) |
default | \(\text {Expression too large to display}\) | \(1395\) |
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\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^3} \,d x \]
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