Integrand size = 21, antiderivative size = 1335 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}} \]
[Out]
Time = 1.37 (sec) , antiderivative size = 1335, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5100, 205, 211, 5032, 6857, 455, 36, 31, 5028, 2456, 2441, 2440, 2438, 5030} \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 \sqrt {d} e^{3/2}}-\frac {x (a+b \arctan (c x))}{2 e \left (e x^2+d\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {i b \log (i c x+1) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (i c x+1) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {i b c \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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {i b c \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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (c^2 x^2+1\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (e x^2+d\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {i b c \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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {i b c \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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}} \]
[In]
[Out]
Rule 31
Rule 36
Rule 205
Rule 211
Rule 455
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 5028
Rule 5030
Rule 5032
Rule 5100
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d (a+b \arctan (c x))}{e \left (d+e x^2\right )^2}+\frac {a+b \arctan (c x)}{e \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{d+e x^2} \, dx}{e}-\frac {d \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx}{e} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {a \int \frac {1}{d+e x^2} \, dx}{e}+\frac {b \int \frac {\arctan (c x)}{d+e x^2} \, dx}{e}+\frac {(b c d) \int \frac {\frac {x}{2 d \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{e} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {(i b) \int \frac {\log (1-i c x)}{d+e x^2} \, dx}{2 e}-\frac {(i b) \int \frac {\log (1+i c x)}{d+e x^2} \, dx}{2 e}+\frac {(b c d) \int \left (\frac {x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{e} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {(b c) \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{2 \sqrt {d} e^{3/2}}+\frac {(i b) \int \left (\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 e}-\frac {(i b) \int \left (\frac {\sqrt {-d} \log (1+i c x)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (1+i c x)}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 e}+\frac {(b c) \int \frac {x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 e} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {(i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 \sqrt {d} e^{3/2}}-\frac {(i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 \sqrt {d} e^{3/2}}+\frac {(b c) \text {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 e}-\frac {(i b) \int \frac {\log (1-i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 \sqrt {-d} e}-\frac {(i b) \int \frac {\log (1-i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 \sqrt {-d} e}+\frac {(i b) \int \frac {\log (1+i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 \sqrt {-d} e}+\frac {(i b) \int \frac {\log (1+i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 \sqrt {-d} e} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(b c) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{4 \left (c^2 d-e\right )}-\frac {(b c) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{-i c \sqrt {-d}+\sqrt {e}}\right )}{1-i c x} \, dx}{4 \sqrt {-d} e^{3/2}}-\frac {(b c) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{i c \sqrt {-d}+\sqrt {e}}\right )}{1+i c x} \, dx}{4 \sqrt {-d} e^{3/2}}+\frac {(b c) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{-i c \sqrt {-d}-\sqrt {e}}\right )}{1-i c x} \, dx}{4 \sqrt {-d} e^{3/2}}+\frac {(b c) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{i c \sqrt {-d}-\sqrt {e}}\right )}{1+i c x} \, dx}{4 \sqrt {-d} e^{3/2}}+\frac {(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}{4 \sqrt {d} e^{3/2}}-\frac {(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}{4 \sqrt {d} e^{3/2}}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{4 \left (c^2 d-e\right ) e} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{-i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{-i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 \sqrt {-d} e^{3/2}}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 \sqrt {-d} e^{3/2}}+\frac {(i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 \sqrt {d} e^{3/2}}+\frac {(i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 \sqrt {d} e^{3/2}}-\frac {(i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 \sqrt {d} e^{3/2}}-\frac {(i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 \sqrt {d} e^{3/2}} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {(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}{8 \sqrt {-c^2} d e}+\frac {(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}{8 \sqrt {-c^2} d e}-\frac {(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}{8 \sqrt {-c^2} d e}-\frac {(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}{8 \sqrt {-c^2} d e} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {(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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {(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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {(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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}} \\ & = -\frac {x (a+b \arctan (c x))}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}-\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}} \\ \end{align*}
Time = 9.21 (sec) , antiderivative size = 877, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {a x}{2 e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {b c \left (-\frac {2 \log \left (\frac {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )}{c^2 d-e}+\frac {-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 i c^2 d \left (e+i \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 {\sqrt {-c^2 d e}}{c e x}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\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 \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\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 )}{\sqrt {-c^2 d e}}-\frac {4 \arctan (c x) \sin (2 \arctan (c x))}{c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}\right )}{8 e} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2304 vs. \(2 (991 ) = 1982\).
Time = 1.86 (sec) , antiderivative size = 2305, normalized size of antiderivative = 1.73
method | result | size |
parts | \(\text {Expression too large to display}\) | \(2305\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2344\) |
default | \(\text {Expression too large to display}\) | \(2344\) |
risch | \(\text {Expression too large to display}\) | \(2391\) |
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\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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