Optimal. Leaf size=844 \[ -\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {b c \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}{d^2} \]
[Out]
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Rubi [A] time = 2.95, antiderivative size = 844, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {6303, 5792, 5654, 74, 5707, 5802, 93, 205, 5800, 5562, 2190, 2279, 2391} \[ -\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {b c \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 74
Rule 93
Rule 205
Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5654
Rule 5707
Rule 5792
Rule 5800
Rule 5802
Rule 6303
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {x^4 \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{d^2}+\frac {e^2 \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}-\frac {2 e \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right ) \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \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^2}\\ &=-\frac {a}{d^2 x}-\frac {b \operatorname {Subst}\left (\int \cosh ^{-1}\left (\frac {x}{c}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \operatorname {Subst}\left (\int \left (\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \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^2}\\ &=-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {b \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{c d^2}+\frac {\sqrt {e} \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 )}{d^2}+\frac {\sqrt {e} \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 )}{d^2}+\frac {e \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 d}+\frac {e \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 d}+\frac {e \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 d}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \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 )}{d^2}+\frac {\sqrt {e} \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 )}{d^2}-\frac {(b e) \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^2}+\frac {(b e) \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^2}+\frac {e \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 d}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {\sqrt {e} \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^2}-\frac {\sqrt {e} \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^2}+\frac {\sqrt {e} \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 )}{d^2}+\frac {\sqrt {e} \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 )}{d^2}+\frac {\sqrt {e} \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 )}{d^2}+\frac {\sqrt {e} \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 )}{d^2}-\frac {(b e) \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^2}+\frac {(b e) \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^2}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {\sqrt {e} \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 )}{(-d)^{5/2}}+\frac {\sqrt {e} \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 )}{(-d)^{5/2}}-\frac {\sqrt {e} \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 )}{(-d)^{5/2}}+\frac {\sqrt {e} \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 )}{(-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}-\frac {\sqrt {e} \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^2}-\frac {\sqrt {e} \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^2}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {\sqrt {e} \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 )}{(-d)^{5/2}}+\frac {\sqrt {e} \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 )}{(-d)^{5/2}}-\frac {\sqrt {e} \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 )}{(-d)^{5/2}}+\frac {\sqrt {e} \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 )}{(-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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 )}{(-d)^{5/2}}-\frac {\sqrt {e} \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^2}-\frac {\sqrt {e} \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^2}-\frac {\sqrt {e} \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^2}-\frac {\sqrt {e} \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^2}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \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)^{5/2}}\\ &=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d^2}-\frac {a}{d^2 x}-\frac {b \text {sech}^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {b e \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 )}{2 d^2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.39, size = 1305, normalized size = 1.55 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsech}\left (c x\right ) + a}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 12.21, size = 1952, normalized size = 2.31 \[ \text {result too large to display} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \]
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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