Optimal. Leaf size=229 \[ \frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}-\frac {\log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \text {Li}_2\left (-\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \text {Li}_2\left (-\frac {\left (e+\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e} \]
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Rubi [A] time = 0.93, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6287, 2518} \[ -\frac {b \text {PolyLog}\left (2,-\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \text {PolyLog}\left (2,-\frac {\left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}-\frac {\log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e} \]
Antiderivative was successfully verified.
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Rule 2518
Rule 6287
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \int \frac {\sqrt {\frac {1-c x}{1+c x}} \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{x (1-c x)} \, dx}{e}+\frac {b \int \frac {\sqrt {\frac {1-c x}{1+c x}} \log \left (1+\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{x (1-c x)} \, dx}{e}+\frac {b \int \frac {\sqrt {\frac {1-c x}{1+c x}} \log \left (1+\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{x (1-c x)} \, dx}{e}\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e}-\frac {b \text {Li}_2\left (-\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \text {Li}_2\left (-\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}\\ \end {align*}
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Mathematica [C] time = 0.58, size = 393, normalized size = 1.72 \[ \frac {a \log (d+e x)}{e}+\frac {b \left (\text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )-2 \left (\text {Li}_2\left (\frac {\left (\sqrt {e^2-c^2 d^2}-e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )+\text {Li}_2\left (-\frac {\left (e+\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )-\text {sech}^{-1}(c x) \log \left (\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )-\text {sech}^{-1}(c x) \log \left (\frac {\left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )+2 i \sin ^{-1}\left (\frac {\sqrt {\frac {e}{c d}+1}}{\sqrt {2}}\right ) \log \left (\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )-2 i \sin ^{-1}\left (\frac {\sqrt {\frac {e}{c d}+1}}{\sqrt {2}}\right ) \log \left (\frac {\left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )-4 i \sin ^{-1}\left (\frac {\sqrt {\frac {e}{c d}+1}}{\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )+\text {sech}^{-1}(c x) \log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right )\right )\right )}{2 e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsech}\left (c x\right ) + a}{e x + d}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.68, size = 514, normalized size = 2.24 \[ \frac {a \ln \left (c x e +c d \right )}{e}+\frac {b \,\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \,\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \dilog \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \dilog \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}-\frac {b \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \,\mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e} \]
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 \[ b \int \frac {\log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{e x + d}\,{d x} + \frac {a \log \left (e x + d\right )}{e} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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