Optimal. Leaf size=229 \[ -\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e}-\frac {b \text {PolyLog}\left (2,-\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \text {PolyLog}\left (2,-\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e} \]
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Rubi [A]
time = 0.62, 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 = {6422, 2598}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2598
Rule 6422
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] Result contains complex when optimal does not.
time = 0.38, size = 393, normalized size = 1.72 \begin {gather*} \frac {a \log (d+e x)}{e}+\frac {b \left (\text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )-2 \left (-4 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-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 )+2 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )-\text {sech}^{-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 )-2 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )+\text {PolyLog}\left (2,\frac {\left (-e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )+\text {PolyLog}\left (2,-\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )\right )\right )}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.60, size = 527, normalized size = 2.30
method | result | size |
derivativedivides | \(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b c \dilog \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b c \dilog \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}-\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b c \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b c \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}}{c}\) | \(527\) |
default | \(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b c \dilog \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b c \dilog \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}-\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b c \,\mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b c \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b c \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}}{c}\) | \(527\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{d + e x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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