Optimal. Leaf size=417 \[ -\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 d}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d} \]
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Rubi [A] time = 0.99, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6303, 5792, 5800, 5562, 2190, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 d}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5792
Rule 5800
Rule 6303
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )} \, dx &=-\operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\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 )}{2 \sqrt {-d}}+\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 )}{2 \sqrt {-d}}\\ &=-\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 )}{2 \sqrt {-d}}+\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 )}{2 \sqrt {-d}}\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d}-\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 )}{2 \sqrt {-d}}-\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 )}{2 \sqrt {-d}}+\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 )}{2 \sqrt {-d}}+\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 )}{2 \sqrt {-d}}\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d}-\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 )}{2 d}-\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 )}{2 d}-\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 )}{2 d}-\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 )}{2 d}+\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 )}{2 d}+\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 )}{2 d}+\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 )}{2 d}+\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 )}{2 d}\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d}-\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 )}{2 d}-\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 )}{2 d}-\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 )}{2 d}-\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 )}{2 d}+\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 )}{2 d}+\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 )}{2 d}+\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 )}{2 d}+\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 )}{2 d}\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d}-\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 )}{2 d}-\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 )}{2 d}-\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 )}{2 d}-\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 )}{2 d}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}\\ \end {align*}
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Mathematica [C] time = 0.93, size = 386, normalized size = 0.93 \[ \frac {-2 a \log \left (d+e x^2\right )+4 a \log (x)+b \left (\text {Li}_2\left (-\frac {\left (d c^2+2 e-2 \sqrt {e \left (d c^2+e\right )}\right ) e^{-2 \text {sech}^{-1}(c x)}}{c^2 d}\right )+\text {Li}_2\left (-\frac {\left (d c^2+2 \left (e+\sqrt {e \left (d c^2+e\right )}\right )\right ) e^{-2 \text {sech}^{-1}(c x)}}{c^2 d}\right )-2 \left (\text {sech}^{-1}(c x) \left (\log \left (\frac {e^{-2 \text {sech}^{-1}(c x)} \left (-2 \sqrt {e \left (c^2 d+e\right )}+c^2 d \left (e^{2 \text {sech}^{-1}(c x)}+1\right )+2 e\right )}{c^2 d}\right )+\log \left (\frac {e^{-2 \text {sech}^{-1}(c x)} \left (2 \left (\sqrt {e \left (c^2 d+e\right )}+e\right )+c^2 d \left (e^{2 \text {sech}^{-1}(c x)}+1\right )\right )}{c^2 d}\right )\right )+i \sin ^{-1}\left (\sqrt {\frac {e}{c^2 d}+1}\right ) \left (2 \tanh ^{-1}\left (\frac {e \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{\sqrt {e \left (c^2 d+e\right )}}\right )-\log \left (\frac {e^{-2 \text {sech}^{-1}(c x)} \left (-2 \sqrt {e \left (c^2 d+e\right )}+c^2 d \left (e^{2 \text {sech}^{-1}(c x)}+1\right )+2 e\right )}{c^2 d}\right )+\log \left (\frac {e^{-2 \text {sech}^{-1}(c x)} \left (2 \left (\sqrt {e \left (c^2 d+e\right )}+e\right )+c^2 d \left (e^{2 \text {sech}^{-1}(c x)}+1\right )\right )}{c^2 d}\right )\right )+\text {sech}^{-1}(c x)^2\right )\right )}{4 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsech}\left (c x\right ) + a}{e x^{3} + d x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.89, size = 3157, normalized size = 7.57 \[ \text {output too large to display} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {\log \left (e x^{2} + d\right )}{d} - \frac {2 \, \log \relax (x)}{d}\right )} + b \int \frac {\log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{e x^{3} + d x}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x\,\left (e\,x^2+d\right )} \,d x \]
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 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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