Optimal. Leaf size=519 \[ \frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 e^{3/2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}}-\frac {b \tan ^{-1}\left (\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}\right )}{c e} \]
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Rubi [A] time = 1.27, antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6303, 5792, 5662, 92, 205, 5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac {b \sqrt {-d} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 e^{3/2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \tan ^{-1}\left (\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}\right )}{c e} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5662
Rule 5707
Rule 5792
Rule 5800
Rule 6303
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{x^2 \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{e x^2}-\frac {d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{x^2} \, dx,x,\frac {1}{x}\right )}{e}+\frac {d \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{c e}+\frac {d \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 )}{e}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}+\frac {d \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 )}{2 e^{3/2}}+\frac {d \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 )}{2 e^{3/2}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}\right )}{c^2 e}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \tan ^{-1}\left (\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}\right )}{c e}+\frac {d \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 )}{2 e^{3/2}}+\frac {d \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 )}{2 e^{3/2}}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \tan ^{-1}\left (\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}\right )}{c e}+\frac {d \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 )}{2 e^{3/2}}+\frac {d \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 )}{2 e^{3/2}}+\frac {d \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 )}{2 e^{3/2}}+\frac {d \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 )}{2 e^{3/2}}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \tan ^{-1}\left (\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}\right )}{c e}+\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\sqrt {-d} \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 )}{2 e^{3/2}}+\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \tan ^{-1}\left (\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}\right )}{c e}+\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\sqrt {-d} \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 )}{2 e^{3/2}}+\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\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 )}{2 e^{3/2}}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \tan ^{-1}\left (\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}\right )}{c e}+\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\sqrt {-d} \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 )}{2 e^{3/2}}+\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {\sqrt {-d} \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 )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.75, size = 921, normalized size = 1.77 \[ \frac {4 a c \sqrt {e} x-4 a c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (4 \sqrt {e} \left (c x \text {sech}^{-1}(c x)-2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )\right )\right )-2 i c \sqrt {d} \left (-4 i \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {e}}{c \sqrt {d}}+1}}{\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {\left (i \sqrt {d} c+\sqrt {e}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {d c^2+e}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \log \left (\frac {i e^{-\text {sech}^{-1}(c x)} \left (\sqrt {e}-\sqrt {d c^2+e}\right )}{c \sqrt {d}}+1\right )+2 i \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {e}}{c \sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (\frac {i e^{-\text {sech}^{-1}(c x)} \left (\sqrt {e}-\sqrt {d c^2+e}\right )}{c \sqrt {d}}+1\right )-\text {sech}^{-1}(c x) \log \left (\frac {i e^{-\text {sech}^{-1}(c x)} \left (\sqrt {e}+\sqrt {d c^2+e}\right )}{c \sqrt {d}}+1\right )-2 i \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {e}}{c \sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (\frac {i e^{-\text {sech}^{-1}(c x)} \left (\sqrt {e}+\sqrt {d c^2+e}\right )}{c \sqrt {d}}+1\right )+\text {Li}_2\left (\frac {i \left (\sqrt {d c^2+e}-\sqrt {e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\text {Li}_2\left (-\frac {i \left (\sqrt {e}+\sqrt {d c^2+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )\right )+2 i c \sqrt {d} \left (-4 i \sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {e}-i c \sqrt {d}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {d c^2+e}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \log \left (\frac {i e^{-\text {sech}^{-1}(c x)} \left (\sqrt {d c^2+e}-\sqrt {e}\right )}{c \sqrt {d}}+1\right )+2 i \sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (\frac {i e^{-\text {sech}^{-1}(c x)} \left (\sqrt {d c^2+e}-\sqrt {e}\right )}{c \sqrt {d}}+1\right )-\text {sech}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {d c^2+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i \sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {d c^2+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\text {Li}_2\left (-\frac {i \left (\sqrt {d c^2+e}-\sqrt {e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\text {Li}_2\left (\frac {i \left (\sqrt {e}+\sqrt {d c^2+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )\right )\right )}{4 c e^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {arsech}\left (c x\right ) + a x^{2}}{e x^{2} + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.53, size = 411, normalized size = 0.79 \[ \frac {a x}{e}-\frac {a d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{e \sqrt {d e}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) x}{e}+\frac {c b d \left (\munderset {\textit {\_R1} =\RootOf \left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e +c^{2} d \right ) \left (\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{8 e^{2}}-\frac {2 b \arctan \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{c e}-\frac {c b d \left (\munderset {\textit {\_R1} =\RootOf \left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{8 e^{2}} \]
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 \[ -a {\left (\frac {d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e} - \frac {x}{e}\right )} + b \int \frac {x^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{e x^{2} + d}\,{d x} \]
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 \[ \int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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