Optimal. Leaf size=84 \[ -\frac {i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {i \sin ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sin ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 84, 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, 4625, 3717, 2190, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {i \sin ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sin ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 3717
Rule 4625
Rubi steps
\begin {align*} \int \sin ^{-1}\left (c e^{a+b x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(c x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {i \sin ^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {i \sin ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sin ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {i \sin ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sin ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{b}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}\\ &=-\frac {i \sin ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\sin ^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{b}-\frac {i \text {Li}_2\left (e^{2 i \sin ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}\\ \end {align*}
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Mathematica [F] time = 1.14, size = 0, normalized size = 0.00 \[ \int \sin ^{-1}\left (c e^{a+b x}\right ) \, dx \]
Verification is Not applicable to the result.
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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 \[ \int \arcsin \left (c e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 181, normalized size = 2.15 \[ -\frac {i \arcsin \left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2 b}+\frac {\arcsin \left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+i c \,{\mathrm e}^{b x +a}+\sqrt {1-c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}+\frac {\arcsin \left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1-i c \,{\mathrm e}^{b x +a}-\sqrt {1-c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}-\frac {i \polylog \left (2, -i c \,{\mathrm e}^{b x +a}-\sqrt {1-c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b}-\frac {i \polylog \left (2, i c \,{\mathrm e}^{b x +a}+\sqrt {1-c^{2} {\mathrm e}^{2 b x +2 a}}\right )}{b} \]
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 \[ \frac {-i \, b^{2} c^{2} {\left (\frac {b x \log \left (c e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-c e^{\left (b x + a\right )}\right )}{b^{2} c^{2}} + \frac {b x \log \left (-c e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (c e^{\left (b x + a\right )}\right )}{b^{2} c^{2}}\right )} + 2 \, b^{2} c e^{a} \int \frac {\sqrt {-c e^{\left (b x + a\right )} + 1} x e^{\left (b x\right )}}{\sqrt {c e^{\left (b x + a\right )} + 1} c e^{\left (b x + a\right )} - \sqrt {c e^{\left (b x + a\right )} + 1}}\,{d x} + 2 \, b x \arctan \left (c e^{\left (b x + a\right )}, \sqrt {c e^{\left (b x + a\right )} + 1} \sqrt {-c e^{\left (b x + a\right )} + 1}\right ) + i \, b x \log \left (c e^{\left (b x + a\right )} + 1\right ) + i \, b x \log \left (-c e^{\left (b x + a\right )} + 1\right ) + i \, {\rm Li}_2\left (c e^{\left (b x + a\right )}\right ) + i \, {\rm Li}_2\left (-c e^{\left (b x + a\right )}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 70, normalized size = 0.83 \[ -\frac {{\mathrm {asin}\left (c\,{\mathrm {e}}^{a+b\,x}\right )}^2\,1{}\mathrm {i}}{2\,b}-\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (c\,{\mathrm {e}}^{a+b\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2\,b}+\frac {\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (c\,{\mathrm {e}}^{a+b\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (c\,{\mathrm {e}}^{a+b\,x}\right )}{b} \]
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 \[ \int \operatorname {asin}{\left (c e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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