Optimal. Leaf size=1276 \[ \frac {b \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2}}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {sech}^{-1}(c x)} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {sech}^{-1}(c x)} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}} \]
[Out]
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Rubi [A] time = 4.04, antiderivative size = 1276, normalized size of antiderivative = 1.00, number of steps used = 63, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6303, 5792, 5707, 5802, 96, 93, 205, 5800, 5562, 2190, 2279, 2391} \[ \frac {b \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2}}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {sech}^{-1}(c x)} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {sech}^{-1}(c x)} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 96
Rule 205
Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5707
Rule 5792
Rule 5800
Rule 5802
Rule 6303
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {e \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{d \left (e+d x^2\right )^3}+\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{d \left (e+d x^2\right )^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{d}+\frac {e \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d}+\frac {e \operatorname {Subst}\left (\int \left (-\frac {d^3 \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}-d x\right )^3}-\frac {3 d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d^3 \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}+d x\right )^3}-\frac {3 d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {3 d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{8 e^2 \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d}\\ &=-\frac {3 \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 e}-\frac {3 \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 e}+\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e}+\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e}-\frac {3 \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{8 e}+\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{2 e}-\frac {\sqrt {-d} \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {e}}-\frac {\sqrt {-d} \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {e}}\\ &=\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 \operatorname {Subst}\left (\int \left (-\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{8 e}+\frac {\operatorname {Subst}\left (\int \left (-\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{2 e}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}-d x\right )} \, dx,x,\frac {1}{x}\right )}{16 c d e}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}+d x\right )} \, dx,x,\frac {1}{x}\right )}{16 c d e}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}-d x\right )} \, dx,x,\frac {1}{x}\right )}{4 c d e}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}+d x\right )} \, dx,x,\frac {1}{x}\right )}{4 c d e}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 c \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 c \sqrt {-d} \sqrt {e}}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{16 d e^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{16 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{4 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{4 d e^{3/2}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (-d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{8 c d e}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{8 c d e}-\frac {b \operatorname {Subst}\left (\int \frac {1}{d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (-d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{2 c d e}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{2 c d e}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}-d x\right )} \, dx,x,\frac {1}{x}\right )}{16 c d \left (c^2 d+e\right )}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (\sqrt {-d} \sqrt {e}+d x\right )} \, dx,x,\frac {1}{x}\right )}{16 c d \left (c^2 d+e\right )}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {3 \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{16 d e^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{16 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 d e^{3/2}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (-d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{8 c d \left (c^2 d+e\right )}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-d+\frac {\sqrt {-d} \sqrt {e}}{c}-\left (d+\frac {\sqrt {-d} \sqrt {e}}{c}\right ) x^2} \, dx,x,\frac {\sqrt {1+\frac {1}{c x}}}{\sqrt {-1+\frac {1}{c x}}}\right )}{8 c d \left (c^2 d+e\right )}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{16 d e^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{16 d e^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{16 d e^{3/2}}+\frac {3 \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{16 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 d e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 d e^{3/2}}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{4 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{4 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{4 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{4 (-d)^{3/2} e^{3/2}}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} e^{3/2}}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 \sqrt {-d} \sqrt {e} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{16 d e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \tan ^{-1}\left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} \left (c^2 d+e\right )}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{3/2} e^{3/2}}\\ \end {align*}
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Mathematica [C] time = 6.14, size = 2030, normalized size = 1.59 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {arsech}\left (c x\right ) + a x^{2}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.28, size = 2537, normalized size = 1.99 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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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