Integrand size = 21, antiderivative size = 89 \[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=-\frac {i (a+b \arcsin (c+d x))^2}{2 b d e}+\frac {(a+b \arcsin (c+d x)) \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e} \]
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Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4889, 12, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=-\frac {i (a+b \arcsin (c+d x))^2}{2 b d e}+\frac {\log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))}{d e}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e} \]
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \arcsin (x)}{e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \arcsin (x)}{x} \, dx,x,c+d x\right )}{d e} \\ & = \frac {\text {Subst}(\int (a+b x) \cot (x) \, dx,x,\arcsin (c+d x))}{d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^2}{2 b d e}-\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\arcsin (c+d x)\right )}{d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^2}{2 b d e}+\frac {(a+b \arcsin (c+d x)) \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c+d x)\right )}{d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^2}{2 b d e}+\frac {(a+b \arcsin (c+d x)) \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (c+d x)}\right )}{2 d e} \\ & = -\frac {i (a+b \arcsin (c+d x))^2}{2 b d e}+\frac {(a+b \arcsin (c+d x)) \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=\frac {b \arcsin (c+d x) \log \left (1-e^{2 i \arcsin (c+d x)}\right )+a \log (c+d x)-\frac {1}{2} i b \left (\arcsin (c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )\right )}{d e} \]
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Time = 0.58 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.73
method | result | size |
derivativedivides | \(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(154\) |
default | \(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(154\) |
parts | \(\frac {a \ln \left (d x +c \right )}{e d}+\frac {b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}\) | \(156\) |
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\[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{d e x + c e} \,d x } \]
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\[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=\frac {\int \frac {a}{c + d x}\, dx + \int \frac {b \operatorname {asin}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
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\[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{d e x + c e} \,d x } \]
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\[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{d e x + c e} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{c\,e+d\,e\,x} \,d x \]
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