Optimal. Leaf size=85 \[ \frac {i \text {Li}_2\left (-e^{2 i \sec ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac {i \sec ^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac {\sec ^{-1}\left (c e^{a+b x}\right ) \log \left (1+e^{2 i \sec ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2282, 5218, 4626, 3719, 2190, 2279, 2391} \[ \frac {i \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac {i \sec ^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac {\sec ^{-1}\left (c e^{a+b x}\right ) \log \left (1+e^{2 i \sec ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 3719
Rule 4626
Rule 5218
Rubi steps
\begin {align*} \int \sec ^{-1}\left (c e^{a+b x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sec ^{-1}(c x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,e^{-a-b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac {i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )^2}{2 b}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac {i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )^2}{2 b}-\frac {\cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )}\right )}{b}+\frac {\operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac {i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )^2}{2 b}-\frac {\cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )}\right )}{b}-\frac {i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )}\right )}{2 b}\\ &=\frac {i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )^2}{2 b}-\frac {\cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )}\right )}{b}+\frac {i \text {Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac {e^{-a-b x}}{c}\right )}\right )}{2 b}\\ \end {align*}
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Mathematica [B] time = 0.86, size = 280, normalized size = 3.29 \[ x \sec ^{-1}\left (c e^{a+b x}\right )-\frac {e^{-a-b x} \left (-4 \sqrt {1-c^2 e^{2 (a+b x)}} \text {Li}_2\left (\frac {1}{2} \left (1-\sqrt {1-c^2 e^{2 (a+b x)}}\right )\right )+\sqrt {1-c^2 e^{2 (a+b x)}} \left (\log ^2\left (c^2 e^{2 (a+b x)}\right )+2 \log ^2\left (\frac {1}{2} \left (\sqrt {1-c^2 e^{2 (a+b x)}}+1\right )\right )-4 \log \left (\frac {1}{2} \left (\sqrt {1-c^2 e^{2 (a+b x)}}+1\right )\right ) \log \left (c^2 e^{2 (a+b x)}\right )\right )+4 \sqrt {c^2 e^{2 (a+b x)}-1} \left (2 b x-\log \left (c^2 e^{2 (a+b x)}\right )\right ) \tan ^{-1}\left (\sqrt {c^2 e^{2 (a+b x)}-1}\right )\right )}{8 b c \sqrt {1-\frac {e^{-2 (a+b x)}}{c^2}}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \operatorname {arcsec}\left (c e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 116, normalized size = 1.36 \[ \frac {i \mathrm {arcsec}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2 b}-\frac {\mathrm {arcsec}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (\frac {{\mathrm e}^{-b x -a}}{c}+i \sqrt {1-\frac {{\mathrm e}^{-2 b x -2 a}}{c^{2}}}\right )^{2}\right )}{b}+\frac {i \polylog \left (2, -\left (\frac {{\mathrm e}^{-b x -a}}{c}+i \sqrt {1-\frac {{\mathrm e}^{-2 b x -2 a}}{c^{2}}}\right )^{2}\right )}{2 b} \]
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 = -0.01 \[ \int \mathrm {acos}\left (\frac {{\mathrm {e}}^{-a-b\,x}}{c}\right ) \,d x \]
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 \[ \int \operatorname {asec}{\left (c e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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