Integrand size = 18, antiderivative size = 893 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=-\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}} \]
[Out]
Time = 0.60 (sec) , antiderivative size = 893, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {205, 211, 5032, 6857, 585, 78, 5028, 2456, 2441, 2440, 2438} \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\frac {3 i b \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \log \left (-\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b \log \left (\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c}{8 d \left (c^2 d-e\right ) \left (e x^2+d\right )}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (e x^2+d\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (e x^2+d\right )^2}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}} \]
[In]
[Out]
Rule 78
Rule 205
Rule 211
Rule 585
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 5028
Rule 5032
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-(b c) \int \frac {\frac {x}{4 d \left (d+e x^2\right )^2}+\frac {3 x}{8 d^2 \left (d+e x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}}{1+c^2 x^2} \, dx \\ & = \frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-(b c) \int \left (\frac {x \left (5 d+3 e x^2\right )}{8 d^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx \\ & = \frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {(b c) \int \frac {x \left (5 d+3 e x^2\right )}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{8 d^2}-\frac {(3 b c) \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{8 d^{5/2} \sqrt {e}} \\ & = \frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {(b c) \text {Subst}\left (\int \frac {5 d+3 e x}{\left (1+c^2 x\right ) (d+e x)^2} \, dx,x,x^2\right )}{16 d^2}-\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{5/2} \sqrt {e}} \\ & = \frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {(b c) \text {Subst}\left (\int \left (\frac {5 c^4 d-3 c^2 e}{\left (c^2 d-e\right )^2 \left (1+c^2 x\right )}-\frac {2 d e}{\left (c^2 d-e\right ) (d+e x)^2}+\frac {e \left (-5 c^2 d+3 e\right )}{\left (-c^2 d+e\right )^2 (d+e x)}\right ) \, dx,x,x^2\right )}{16 d^2}-\frac {(3 i b c) \int \left (\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{16 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \left (\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{16 d^{5/2} \sqrt {e}} \\ & = -\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}-\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}}-\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}} \\ & = -\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}-\frac {(3 b c) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3}-\frac {(3 b c) \int \frac {\log \left (\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3}+\frac {(3 b c) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3}+\frac {(3 b c) \int \frac {\log \left (\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3} \\ & = -\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {(3 i b c) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {(3 i b c) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {(3 i b c) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {(3 i b c) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}} \\ & = -\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x (a+b \arctan (c x))}{4 d \left (d+e x^2\right )^2}+\frac {3 x (a+b \arctan (c x))}{8 d^2 \left (d+e x^2\right )}+\frac {3 (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}} \\ \end{align*}
Time = 10.77 (sec) , antiderivative size = 1745, normalized size of antiderivative = 1.95 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\frac {a x}{4 d \left (d+e x^2\right )^2}+\frac {3 a x}{8 d^2 \left (d+e x^2\right )}+\frac {3 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {b c \left (10 c^2 d \log \left (1+\frac {\left (c^2 d-e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )-6 e \log \left (1+\frac {\left (c^2 d-e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )+\frac {3 c^2 d \left (c^2 d-e\right ) \left (-4 \arctan (c x) \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )+2 \arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (-i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )\right )\right )}{\sqrt {-c^2 d e}}-\frac {3 \left (c^2 d-e\right ) e \left (-4 \arctan (c x) \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )+2 \arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (-i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )\right )\right )}{\sqrt {-c^2 d e}}-\frac {16 c^2 d \left (c^2 d-e\right ) e \arctan (c x) \sin (2 \arctan (c x))}{\left (c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))\right )^2}+\frac {8 c^2 d e+4 \left (5 c^4 d^2-8 c^2 d e+3 e^2\right ) \arctan (c x) \sin (2 \arctan (c x))}{c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}\right )}{32 d^2 \left (-c^2 d+e\right )^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4006 vs. \(2 (681 ) = 1362\).
Time = 1.99 (sec) , antiderivative size = 4007, normalized size of antiderivative = 4.49
method | result | size |
parts | \(\text {Expression too large to display}\) | \(4007\) |
derivativedivides | \(\text {Expression too large to display}\) | \(4032\) |
default | \(\text {Expression too large to display}\) | \(4032\) |
risch | \(\text {Expression too large to display}\) | \(5059\) |
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\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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