Optimal. Leaf size=1518 \[ -\frac {7 x \left (a+b \tan ^{-1}(c x)\right ) e}{8 d^3 \left (e x^2+d\right )}-\frac {x \left (a+b \tan ^{-1}(c x)\right ) e}{4 d^2 \left (e x^2+d\right )^2}+\frac {b c \log \left (c^2 x^2+1\right ) e}{4 d^3 \left (c^2 d-e\right )}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) e}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c \log \left (e x^2+d\right ) e}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) e}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (e x^2+d\right )}-\frac {7 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \sqrt {e}}{8 d^{7/2}}-\frac {a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \sqrt {e}}{d^{7/2}}-\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 ) \sqrt {e}}{4 (-d)^{7/2}}+\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 ) \sqrt {e}}{4 (-d)^{7/2}}-\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 ) \sqrt {e}}{4 (-d)^{7/2}}+\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 ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {7 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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}+\frac {b c \log (x)}{d^3}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d^3} \]
[Out]
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Rubi [A] time = 2.64, antiderivative size = 1518, normalized size of antiderivative = 1.00, number of steps used = 73, number of rules used = 19, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {4980, 4852, 266, 36, 29, 31, 199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391, 444, 4910} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 77
Rule 199
Rule 205
Rule 266
Rule 444
Rule 571
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 4852
Rule 4908
Rule 4910
Rule 4912
Rule 4980
Rule 6725
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (d+e x^2\right )^3} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d^3 x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d \left (d+e x^2\right )^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx}{d^3}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{d+e x^2} \, dx}{d^3}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{d^2}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^3}-\frac {(a e) \int \frac {1}{d+e x^2} \, dx}{d^3}-\frac {(b e) \int \frac {\tan ^{-1}(c x)}{d+e x^2} \, dx}{d^3}+\frac {(b c e) \int \frac {\frac {x}{2 d \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{d^2}+\frac {(b c e) \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 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^3}-\frac {(i b e) \int \frac {\log (1-i c x)}{d+e x^2} \, dx}{2 d^3}+\frac {(i b e) \int \frac {\log (1+i c x)}{d+e x^2} \, dx}{2 d^3}+\frac {(b c e) \int \left (\frac {x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}+\frac {(b c e) \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 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^3}+\frac {\left (3 b c \sqrt {e}\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{8 d^{7/2}}+\frac {\left (b c \sqrt {e}\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{2 d^{7/2}}-\frac {(i b e) \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 d^3}+\frac {(i b e) \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 d^3}+\frac {(b c e) \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^3}+\frac {(b c e) \int \frac {x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^3}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {\left (3 i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 d^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 d^{7/2}}-\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{7/2}}-\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{7/2}}+\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{7/2}}+\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{7/2}}+\frac {(b c e) \operatorname {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^3}+\frac {(b c e) \operatorname {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 d^3}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}-\frac {\left (b c \sqrt {e}\right ) \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 (-d)^{7/2}}-\frac {\left (b c \sqrt {e}\right ) \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 (-d)^{7/2}}+\frac {\left (b c \sqrt {e}\right ) \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 (-d)^{7/2}}+\frac {\left (b c \sqrt {e}\right ) \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 (-d)^{7/2}}+\frac {\left (3 i b c \sqrt {e}\right ) \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^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \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^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \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 d^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \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 d^{7/2}}+\frac {(b c e) \operatorname {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^3}+\frac {\left (b c^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {\left (b c e^2\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{4 d^3 \left (c^2 d-e\right )}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {\left (i b \sqrt {e}\right ) \operatorname {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 (-d)^{7/2}}-\frac {\left (i b \sqrt {e}\right ) \operatorname {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 (-d)^{7/2}}-\frac {\left (i b \sqrt {e}\right ) \operatorname {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 (-d)^{7/2}}+\frac {\left (i b \sqrt {e}\right ) \operatorname {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 (-d)^{7/2}}+\frac {\left (3 i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{7/2}}+\frac {\left (3 i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 d^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 d^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 d^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 d^{7/2}}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {7 i b c \sqrt {e} \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^{7/2}}+\frac {7 i b c \sqrt {e} \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^{7/2}}+\frac {7 i b c \sqrt {e} \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^{7/2}}-\frac {7 i b c \sqrt {e} \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^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {(3 b c e) \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^4}+\frac {(3 b c e) \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^4}-\frac {(3 b c e) \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^4}-\frac {(3 b c e) \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^4}+\frac {(b c e) \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^4}+\frac {(b c e) \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^4}-\frac {(b c e) \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^4}-\frac {(b c e) \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^4}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {7 i b c \sqrt {e} \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^{7/2}}+\frac {7 i b c \sqrt {e} \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^{7/2}}+\frac {7 i b c \sqrt {e} \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^{7/2}}-\frac {7 i b c \sqrt {e} \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^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {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^{7/2}}+\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {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^{7/2}}+\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {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^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {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^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \operatorname {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} d^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \operatorname {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} d^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \operatorname {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} d^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \operatorname {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} d^{7/2}}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {7 i b c \sqrt {e} \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^{7/2}}+\frac {7 i b c \sqrt {e} \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^{7/2}}+\frac {7 i b c \sqrt {e} \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^{7/2}}-\frac {7 i b c \sqrt {e} \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^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {7 i b c \sqrt {e} \text {Li}_2\left (\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^{7/2}}+\frac {7 i b c \sqrt {e} \text {Li}_2\left (\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^{7/2}}-\frac {7 i b c \sqrt {e} \text {Li}_2\left (\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^{7/2}}+\frac {7 i b c \sqrt {e} \text {Li}_2\left (\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^{7/2}}\\ \end {align*}
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Mathematica [A] time = 13.43, size = 2005, normalized size = 1.32 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e^{3} x^{8} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{4} + d^{3} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.38, size = 6655, normalized size = 4.38 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, a {\left (\frac {15 \, e^{2} x^{4} + 25 \, d e x^{2} + 8 \, d^{2}}{d^{3} e^{2} x^{5} + 2 \, d^{4} e x^{3} + d^{5} x} + \frac {15 \, e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d^{3}}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e^{3} x^{8} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{4} + d^{3} x^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,{\left (e\,x^2+d\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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