Optimal. Leaf size=76 \[ \frac {\text {Li}_2\left (e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\sinh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sinh ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
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Rubi [A] time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2282, 5659, 3716, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\sinh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sinh ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 3716
Rule 5659
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (c e^{a+b x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(c x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {\sinh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {\sinh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sinh ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {\sinh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sinh ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}\\ &=-\frac {\sinh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sinh ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{b}+\frac {\text {Li}_2\left (e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 76, normalized size = 1.00 \[ \frac {\text {Li}_2\left (e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\sinh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sinh ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
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 \[ \int \operatorname {arsinh}\left (c e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 166, normalized size = 2.18 \[ -\frac {\arcsinh \left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2 b}+\frac {\arcsinh \left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+c \,{\mathrm e}^{b x +a}+\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}+\frac {\polylog \left (2, -c \,{\mathrm e}^{b x +a}-\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}+\frac {\arcsinh \left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1-c \,{\mathrm e}^{b x +a}-\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}+\frac {\polylog \left (2, c \,{\mathrm e}^{b x +a}+\sqrt {1+c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b} \]
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 \[ -b c \int \frac {x e^{\left (b x + a\right )}}{c^{3} e^{\left (3 \, b x + 3 \, a\right )} + c e^{\left (b x + a\right )} + {\left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{\frac {3}{2}}}\,{d x} + x \log \left (c e^{\left (b x + a\right )} + \sqrt {c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1}\right ) - \frac {2 \, b x \log \left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-c^{2} e^{\left (2 \, b x + 2 \, a\right )}\right )}{4 \, b} \]
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 \[ \int \mathrm {asinh}\left (c\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\right ) \,d x \]
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 \[ \int \operatorname {asinh}{\left (c e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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